gravitational field with definite

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On the gravitational field of a mass point according to Einstein's theory

On the gravitational field of a mass point according to Einstein's theory

α
∂ Γα µν + ∂xα
β Γα µβ Γνα = 0, αβ
(4)
and if also the “equation of the determinant” |gµν | = −1 (5)
is satisfied. The field equations together with the equation of the determinant have the fundamental property that they preserve their form under the substitution of other arbitrary variables in lieu of x1 , x2 , x3 , x4 , as long as the determinant of the substitution is equal to 1. Let x1 , x2 , x3 stand for rectangular co-ordinates, x4 for the time; furthermore, the mass at the origin shall not change with time, and the motion at infinity shall be rectilinear and uniform. Then, according to Mr. Einstein’s list, loc. cit. p. 833, the following conditions must be fulfilled too: 1. All the components are independent of the time x4 . 2. The equations gρ4 = g4ρ = 0 hold exactly for ρ = 1, 2, 3. 3. The solution is spatially symmetric with respect to the origin of the co-ordinate system in the sense that one finds again the same solution when x1 , x2 , x3 are subjected to an orthogonal transformation (rotation). 4. The gµν vanish at infinity, with the exception of the following four limits different from zero: g44 = 1, g11 = g22 = g33 = −1. The problem is to find out a line element with coefficients such that the field equations, the equation of the determinant and these four requirements are satisfied. §2. Mr. Einstein showed that this problem, in first approximation, leads to Newton’s law and that the second approximation correctly reproduces the known anomaly in the motion of the perihelion of Mercury. The following calculation yields the exact solution of the problem. It is always pleasant to avail of exact solutions of simple form. More importantly, the calculation proves also the uniqueness of the solution, about which Mr. Einstein’s treatment still left doubt, and which could have been proved only with great difficulty, in the way shown below, through such an approximation method. The following lines therefore let Mr. Einstein’s result shine with increased clearness. §3. If one calls t the time, x, y , z , the rectangular co-ordinates, the most general line element that satisfies the conditions 1-3 is clearly the following: ds2 = F dt2 − G(dx2 + dy 2 + dz 2 ) − H (xdx + ydy + zdz )2 where F , G, H are functions of r = x2 + y 2 + z 2 . The condition (4) requires: for r = ∞ : F = G = 1, H = 0. When one goes over to polar co-ordinates according to x = r sin ϑ cos φ, y = r sin ϑ sin φ, z = r cos ϑ, the same line element reads: ds2 = F dt2 − G(dr 2 + r 2 dϑ2 + r 2 sin2 ϑdφ2 ) − Hr 2 dr 2 = F dt2 − (G + Hr 2 )dr 2 − Gr 2 (dϑ2 + sin2 ϑdφthe gµν stand for functions of the variables x, and in the variation the variables x must be kept fixed at the beginning and at the end of the path of integration. In short, the point shall move along a geodesic line in the manifold characterised by the line element ds. The execution of the variation yields the equations of motion of the point: d2 xα = ds2 where Γα µν = − 1 2 gαβ

高等数学名词(中英文对照)

高等数学名词(中英文对照)

高等数学名词(中英文)第一章函数与极限Chapter1 Function and Limit 集合set 元素element子集subset空集empty set并集union交集intersection 差集difference of set基本集basic set 补集complement set直积direct product笛卡儿积Cartesian product开区间open interval 闭区间closed interval 半开区间half open interval 有限区间finite interval区间的长度length of an interval无限区间infinite interval邻域neighborhood邻域的中心centre of a neighborhood 邻域的半径radius of a neighborhood 左邻域left neighborhood右邻域right neighborhood映射mappingX 到Y 的映射mapping of X onto Y 满射surjection单射injection一一映射one-to-one mapping双射bijection算子operator变化transformation函数function逆映射inverse mapping复合映射composite mapping自变量independent variable 因变量dependent variable定义域domain函数值value of function函数关系function relation值域range自然定义域natural domain单值函数single valued function多值函数multiple valued function单值分支one-valued branch函数图形graph of a function绝对值函数absolute value符号函数sigh function整数部分integral part阶梯曲线step curve当且仅当if and only if(iff)分段函数piecewise function上界upper bound下界lower bound有界boundedness无界unbounded函数的单调性monotonicity of a function 单调增加的increasing单调减少的decreasing单调函数monotone function函数的奇偶性parity(odevity) of a function 对称symmetry偶函数even function奇函数odd function函数的周期性periodicity of a function周期period反函数inverse function直接函数direct function复合函数composite function中间变量intermediate variable函数的运算operation of function基本初等函数basic elementary function初等函数elementary function幂函数power function指数函数exponential function对数函数logarithmic function三角函数trigonometric function反三角函数inverse trigonometric function 常数函数constant function双曲函数hyperbolic function双曲正弦hyperbolic sine双曲余弦hyperbolic cosine双曲正切hyperbolic tangent反双曲正弦inverse hyperbolic sine反双曲余弦inverse hyperbolic cosine反双曲正切inverse hyperbolic tangent极限limit数列sequence of number收敛convergence收敛于converge to发散divergent极限的唯一性uniqueness of limits收敛数列的有界性boundedness of a convergent sequence子列subsequence函数的极限limits of functions函数当x 趋于x0 时的极限limit of functions as x approaches x0左极限left limit右极限right limit单侧极限one-sided limits水平渐近线horizontal asymptote无穷小infinitesimal无穷大infinity铅直渐近线vertical asymptote夹逼准则squeeze rule单调数列monotonic sequence高阶无穷小infinitesimal of higher order 低阶无穷小infinitesimal of lower order 同阶无穷小infinitesimal of the same order 等阶无穷小equivalent infinitesimal 函数的连续性continuity of a function增量increment函数在x0 连续the function is continuous at x0左连续left continuous右连续right continuous区间上的连续函数continuous function函数在该区间上连续function is continuous on an interval不连续点discontinuity point第一类间断点discontinuity point of the first kind第二类间断点discontinuity point of the second kind初等函数的连续性continuity of the elementary functions定义区间defined interval最大值global maximum value (absolute maximum)最小值global minimum value (absolute minimum)零点定理the zero-point theorem介值定理intermediate value theorem第二章导数与微分Chapter2 Derivative and Differential速度velocity匀速运动uniform motion平均速度average velocity瞬时速度instantaneous velocity圆的切线tangent line of a circle切线tangent line切线的斜率slope of the tangent line位置函数position function导数derivative可导derivable函数的变化率问题problem of the change rate of a function导函数derived function左导数left-hand derivative右导数right-hand derivative单侧导数one-sided derivatives在闭区间[a, b] 上可导is derivable on the closed interval [a,b]切线方程tangent equation角速度angular velocity成本函数cost function边际成本marginal cost链式法则chain rule隐函数implicit function显函数explicit function二阶函数second derivative三阶导数third derivative高阶导数nth derivative莱布尼茨公式Leibniz formula对数求导法log- derivative参数方程parametric equation相关变化率correlative change rata微分differential可微的differentiable函数的微分differential of function自变量的微分differential of independent variable微商differential quotient间接测量误差indirect measurement error 绝对误差absolute error相对误差relative error第三章微分中值定理与导数的应用Chapter3 Mean Value Theorem ofDifferentials and the Application ofDerivatives罗马定理Rolle’s theorem费马引理Fermat’s lemma拉格朗日中值定理Lagrange’s mean value theorem驻点stationary point稳定点stable point 临界点critical point辅助函数auxiliary function拉格朗日中值公式Lagrange’s mean value formula柯西中值定理Cauchy’s mean value theorem 洛必达法则L’Hospital’s Rule0/0 型不定式indeterminate form of type 0/0 不定式indeterminate form泰勒中值定理Taylor’s mean value theorem 泰勒公式Taylor formula余项remainder term拉格朗日余项Lagrange remainder term麦克劳林公式Maclaurin’s formula佩亚诺公式Peano remainder term凹凸性concavity凹向上的concave upward, concave up凹向下的,向上凸的concave downward’ concave down拐点inflection point函数的极值extremum of function极大值local(relative) maximum最大值global(absolute) maximum极小值local(relative) minimum最小值global(absolute) minimum目标函数objective function曲率curvature弧微分arc differential平均曲率average curvature曲率园circle of curvature曲率中心center of curvature曲率半径radius of curvature渐屈线evolute渐伸线involute根的隔离isolation of root隔离区间isolation interval切线法tangent line method第四章不定积分Chapter4 Indefinite Integrals原函数primitive function(anti-derivative)积分号sign of integration被积函数integrand积分变量integral variable积分曲线integral curve积分表table of integrals换元积分法integration by substitution分部积分法integration by parts分部积分公式formula of integration by parts 有理函数rational function真分式proper fraction假分式improper fraction第五章定积分Chapter5 Definite Integrals曲边梯形trapezoid with曲边curve edge窄矩形narrow rectangle曲边梯形的面积area of trapezoid with curved edge积分下限lower limit of integral积分上限upper limit of integral积分区间integral interval分割partition积分和integral sum可积integrable矩形法rectangle method积分中值定理mean value theorem of integrals函数在区间上的平均值average value of a function on an intervals牛顿-莱布尼茨公式Newton-Leibniz formula微积分基本公式fundamental formula of calculus换元公式formula for integration by substitution递推公式recurrence formula反常积分improper integral 反常积分发散the improper integral is divergent反常积分收敛the improper integral is convergent无穷限的反常积分improper integral on an infinite interval无界函数的反常积分improper integral of unbounded functions绝对收敛absolutely convergent第六章定积分的应用Chapter6 Applications of the Definite Integrals元素法the element method面积元素element of area平面图形的面积area of a plane figure直角坐标又称“笛卡儿坐标(Cartesian coordinates)”极坐标polar coordinates抛物线parabola椭圆ellipse旋转体的面积volume of a solid of rotation 旋转椭球体ellipsoid of revolution, ellipsoid of rotation曲线的弧长arc length of a curve可求长的rectifiable光滑smooth功work水压力water pressure引力gravitation变力variable force第七章空间解析几何与向量代数Chapter7 Space Analytic Geometry and Vector Algebra向量vector自由向量free vector单位向量unit vector零向量zero vector相等equal平行parallel向量的线性运算linear poeration of vector三角法则triangle rule 平行四边形法则parallelogram rule 交换律commutative law结合律associative law负向量negative vector差difference分配律distributive law空间直角坐标系space rectangular coordinates坐标面coordinate plane卦限octant向量的模modulus of vector向量a 与b 的夹角angle between vector a and b方向余弦direction cosine方向角direction angle向量在轴上的投影projection of a vector onto an axis数量积,外积,叉积scalar product,dot product,inner product曲面方程equation for a surface球面sphere旋转曲面surface of revolution母线generating line轴axis 圆锥面cone 顶点vertex旋转单叶双曲面revolution hyperboloids of one sheet旋转双叶双曲面revolution hyperboloids of two sheets柱面cylindrical surface ,cylinder圆柱面cylindrical surface准线directrix抛物柱面parabolic cylinder二次曲面quadric surface 椭圆锥面dlliptic cone椭球面ellipsoid单叶双曲面hyperboloid of one sheet 双叶双曲面hyperboloid of two sheets 旋转椭球面ellipsoid of revolution 椭圆抛物面elliptic paraboloid 旋转抛物面paraboloid of revolution 双曲抛物面hyperbolic paraboloid 马鞍面saddle surface椭圆柱面elliptic cylinder 双曲柱面hyperbolic cylinder 抛物柱面parabolic cylinder 空间曲线spacecurve空间曲线的一般方程general form equations of a space curve空间曲线的参数方程parametric equations of a space curve螺转线spiral 螺矩pitch投影柱面projecting cylinder投影projection平面的点法式方程pointnorm form eqyation of a plane法向量normal vector平面的一般方程general form equation of a plane两平面的夹角angle between two planes点到平面的距离distance from a point to a plane空间直线的一般方程general equation of a line in space方向向量direction vector直线的点向式方程pointdirection form equations of a line方向数direction number直线的参数方程parametric equations of a line两直线的夹角angle between two lines垂直perpendicular直线与平面的夹角angle between a line and a planes平面束pencil of planes平面束的方程equation of a pencil of planes 行列式determinant系数行列式coefficient determinant第八章多元函数微分法及其应用Chapter8 Differentiation of Functions of Several Variables and Its Application一元函数function of one variable多元函数function of several variables内点interior point外点exterior point边界点frontier point,boundary point聚点point of accumulation开集openset闭集closed set 连通集connected set 开区域open region 闭区域closed region 有界集bounded set 无界集unbounded setn 维空间n-dimentional space二重极限double limit多元函数的连续性continuity of function of seveal连续函数continuous function不连续点discontinuity point 一致连续uniformly continuous 偏导数partial derivative对自变量x 的偏导数partial derivative with respect to independent variable x高阶偏导数partial derivative of higher order 二阶偏导数second order partial derivative 混合偏导数hybrid partial derivative 全微分total differential偏增量oartial increment偏微分partial differential 全增量total increment 可微分differentiable 必要条件necessary condition 充分条件sufficient condition叠加原理superpostition principle全导数total derivative中间变量intermediate variable 隐函数存在定理theorem of the existence of implicit function曲线的切向量tangent vector of a curve法平面normal plane向量方程vector equation 向量值函数vector-valued function 切平面tangent plane法线normal line方向导数directional derivative梯度gradient 数量场scalar field 梯度场gradient field 向量场vector field 势场potential field引力场gravitational field引力势gravitational potential曲面在一点的切平面tangent plane to a surface at a point曲线在一点的法线normal line to a surface at a point无条件极值unconditional extreme values 条件极值conditional extreme values 拉格朗日乘数法Lagrange multiplier method 拉格朗日乘子Lagrange multiplier 经验公式empirical formula最小二乘法method of least squares均方误差mean square error第九章重积分Chapter9 Multiple Integrals二重积分double integral可加性additivity累次积分iterated integral体积元素volume element三重积分triple integral直角坐标系中的体积元素volume element in rectangular coordinate system柱面坐标cylindrical coordinates柱面坐标系中的体积元素volume element in cylindrical coordinate system球面坐标spherical coordinates球面坐标系中的体积元素volume element in spherical coordinate system反常二重积分improper double integral曲面的面积area of a surface质心centre of mass静矩static moment密度density形心centroid转动惯量moment of inertia参变量parametric variable第十章曲线积分与曲面积分Chapter10 Line (Curve) Integrals andSurface Integrals对弧长的曲线积分line integrals with respect to arc hength第一类曲线积分line integrals of the first type对坐标的曲线积分line integrals with respect to x,y,and z第二类曲线积分line integrals of the second type有向曲线弧directed arc单连通区域simple connected region 复连通区域complex connected region 格林公式Green formula第一类曲面积分surface integrals of the first type对面的曲面积分surface integrals with respect to area 有向曲面directed surface对坐标的曲面积分surface integrals with respect to coordinate elements第二类曲面积分surface integrals of the second type有向曲面元element of directed surface高斯公式gauss formula 拉普拉斯算子Laplace operator 格林第一公式Green’s first formula 通量flux散度divergence 斯托克斯公式Stokes formula 环流量circulation旋度rotation,curl第十一章无穷级数Chapter11 Infinite Series一般项general term 部分和partial sum 余项remainder term等比级数geometric series几何级数geometric series公比common ratio 调和级数harmonic series柯西收敛准则Cauchy convergence criteria, Cauchy criteria for convergence 正项级数series of positive terms 达朗贝尔判别法D’Alembert test柯西判别法Cauchy test 交错级数alternating series 绝对收敛absolutely convergent条件收敛conditionally convergent柯西乘积Cauchy product 函数项级数series of functions 发散点point of divergence 收敛点pointof convergence 收敛域convergence domain 和函数sumfunction幂级数power series幂级数的系数coeffcients of power series阿贝尔定理Abel Theorem 收敛半径radius of convergence 收敛区间interval of convergence 泰勒级数Taylor series 麦克劳林级数Maclaurin series 二项展开式binomial expansion 近似计算approximate calculation舍入误差round-off error,rounding error 欧拉公式Euler’s formula 魏尔斯特拉丝判别法Weierstrass test 三角级数trigonometric series振幅amplitude角频率angular frequency初相initial phase矩形波square wave 谐波分析harmonic analysis 直流分量direct component 基波fundamental wave 二次谐波second harmonic三角函数系trigonometric function system 傅立叶系数Fourier coefficient 傅立叶级数Forrier series周期延拓periodic prolongation正弦级数sine series 余弦级数cosine series 奇延拓oddprolongation 偶延拓evenprolongation傅立叶级数的复数形式complex form of Fourier series第十二章微分方程Chapter12 Differential Equation 解微分方程solve a differential equation 常微分方程ordinary differential equation 偏微分方程partial differential equation,PDE 微分方程的阶order of a differential equation 微分方程的解solution of a differential equation 微分方程的通解general solution of a differential equation初始条件initial condition微分方程的特解particular solution of a differential equation初值问题initial value problem微分方程的积分曲线integral curve of a differential equation可分离变量的微分方程variable separable differential equation隐式解implicit solution 隐式通解inplicit general solution 衰变系数decay coefficient衰变decay齐次方程homogeneous equation 一阶线性方程linear differential equation of first order 非齐次non-homogeneous齐次线性方程homogeneous linear equation 非齐次线性方程non-homogeneous linear equation常数变易法method of variation of constant 暂态电流transient state current 稳态电流steady state current 伯努利方程Bernoulli equation 全微分方程total differential equation 积分因子integrating factor高阶微分方程differential equation of higher order悬链线catenary高阶线性微分方程linear differential equation of higher order 自由振动的微分方程differential equation of free vibration强迫振动的微分方程differential equation of forced oscillation串联电路的振荡方程oscillation equation of series circuit二阶线性微分方程second order lineardifferential equation线性相关linearly dependence线性无关linearly independence 二阶常系数齐次线性微分方程second order homogeneous linear differential equation with constant coefficient二阶变系数齐次线性微分方程second order homogeneous linear differential equation with variable coefficient特征方程characteristic equation无阻尼自由振动的微分方程differential equation of free vibration with zero damping 固有频率natural frequency 简谐振动simple harmonic oscillation,simple harmonic vibration微分算子differential operator待定系数法method of undetermined coefficient共振现象resonance phenomenon欧拉方程Euler equation 幂级数解法power series solution 数值解法numerial solution 勒让德方程Legendre equation微分方程组system of differential equations 常系数线性微分方程组system of linear differential equations with constant coefficie。

重力场现象英语

重力场现象英语

重力场现象英语Dive into the enigmatic realm of gravity, a force so fundamental yet so mysterious, that it shapes the very fabric of our universe. From the majestic dance of celestial bodies to the humble apple falling to the ground, gravity is the silent orchestrator of our physical world. It's a force that has captivated the minds of scientists for centuries, from Newton's apple to Einstein's theory of general relativity, and continues to baffle and inspire us.Gravity, the invisible hand that pulls everything with mass towards each other, is a phenomenon that is as omnipresent as it is elusive. It's the reason why we stay grounded on Earth and why the Earth orbits the Sun. It's the force that holds galaxies together, yet it's so weak compared to the other fundamental forces that it can be easily overcome by other interactions at the atomic level.The gravity field, as it's known in scientific terms, is not uniform across the universe. It varies in strength depending on the mass of the objects involved and the distance between them. This variation is what gives rise to the complex dynamics of planetary motion, the formation of black holes, and the bending of light around massive objects.But gravity is not just a cosmic force; it's deeply intertwined with the structure of spacetime itself. According to Einstein, gravity is not an interaction between masses buta curvature of spacetime caused by mass. This curvature tells matter how to move, and matter tells spacetime how to curve. It's a beautiful, interwoven relationship that defies our everyday intuition.In our quest to understand gravity, we've discovered that it is the key to unlocking the mysteries of the cosmos. It dictates the life cycle of stars, the expansion of the universe, and the formation of cosmic structures. Yet, despite its significance, gravity remains the least understood of all fundamental forces, with many questionsstill unanswered, such as the nature of dark matter and dark energy, which seem to influence the gravitational field in ways we are only beginning to comprehend.The gravity field is a testament to the elegance and complexity of the universe. It's a force that, while seemingly simple in its action, is intricate in its implications. As we continue to explore the cosmos and delve deeper into the heart of gravity, we may yet uncover the secrets that will revolutionize our understanding of the universe and our place within it.。

A2-GRAVITATION

A2-GRAVITATION

Concept of gravitational field: In order to understand the gravitational interaction we consider a gravitational field. The concept of gravitational field is that, it is an example of a field of force. According to the field concept, every mass is a source of gravitational field. If another mass is placed in this field it will experience a force of attraction due to interaction of the gravitational field on the mass.
Thus equating eqs. 1 & 2, and simplifying we get
g = GM/R2…….3 Eq. 3 is the expression for the acceleration due to gravity of the earth. It should be noted here that ‘g’ also is the gravitational field strength, from definition of field strength. Field strength = Force / unit mass
Gravitational field is represented by field lines.
The field line is the imaginary path which a test mass(unit mass) would take as it is attracted by the mass in whose field it is located. As such field lines are always directed towards the centre of the mass in whose field the test mass is situated.

初中物理 Gravity and Inverse Square Relationships NIS

初中物理  Gravity and Inverse Square Relationships NIS

Work done by the mass= -mΔV
And work done by the mass is force times distance moved so mgΔr = ons and calculus for ‘g’ gives the general formula of ‘V’
• Two equations for ‘g’ • Combining • Integrating and solving for V and
Escape Velocity
Deriving Escape Velocity:
• We can calculate the energy necessary to escape earth's gravity well completely. • Gravitational Potential (Φ): • There G is the universal gravitational constant; M is the mass of the earth and r is the distance from the center of the earth. • We want to find the difference in potential of an object at infinity (i.e., it has escaped earth forever) and at the surface of the earth. Using r0 as the radius of the earth can write this difference as
Earth
Equipotential Lines Around Earth

On the Quantization of the Gravitational Field

On the Quantization of the Gravitational Field

a rXiv:h ep-th/99519v126M a y1999On the Quantization of the Gravitational Field D.R.Grigore 1Dept.of Theor.Phys.,Inst.Atomic Phys.Bucharest-M˘a gurele,P.O.Box MG 6,ROM ˆANIA Abstract We present a new point of view on the quantization of the gravitational field,namely we use exclusively the quantum framework of the second quantization.More explicitly,we take as one-particle Hilbert space,H graviton the unitary irreducible representation of the Poincar´e group corresponding to a massless particle of helicity 2and apply the second quantization procedure with Einstein-Bose statistics.The resulting Hilbert space F +(H graviton )is,by definition,the Hilbert space of the gravitational field.Then we prove that this Hilbert space is canonically isomorphic to a space of the type Ker (Q )/Im (Q )where Q is a supercharge defined in an extension of the Hilbert space F +(H graviton )by the inclusion of ghosts:some Fermion ghosts u µ,˜u µwhich are vector fields and a Bosonic ghost Φwhich is a scalar field.This has to be contrasted to the usual approaches where only the Fermion ghosts are considered.However,a rigorous proof that this is,indeed,possible seems to be laking from the literature.1IntroductionOne possible way to perform the quantization of the gravitationalfield is to liniarize the classicaltheory of gravitation using the so called Goldberg variables[9],[12]and then to apply some sort of canonical quantization of the resultingfield theory.Because of the gauge invariance ofthe theory(which in this case is the invariance under general coordinates transformations)one obtains a constrained system and one tries to use a Bleuler-Gupta type formalism,that’s itto start with an Hilbert space endowed with a sesquilinear non-degenerate form and select thephysical states as a subspace of the type Q AΦ=0,A=1,...,N.Among the pioneering works in this approach we mention[26],[7],[19],[13],[17],[18].Usingthe result of this analysis many computations have been done in the literature(see[3],[14],[4],[29],[28]).A related idea is to extend the Fock space to an auxiliary Hilbert space H gh includingsomefictiousfields,called ghosts,and construct a supercharge(i.e.an operator Q verifying Q2=0)such that the physical Hilbert space is H phys≡Ker(Q)/Im(Q)(see for instance[18] and references quoted there).As a result of this procedure,it is asserted that the graviton,i.e.the elementary quantum particle must be a massless spin2particle.The construction of the supercharge relies heavily on classicalfield theory arguments,because one tries to obtain for the quantum gauge transformations expressions of the same type as the general coordinates transformations appearing in general relativity.This invariance is then promoted to a BRST invariance which should be implemented by the supercommutator with the supercharge.We will present in this paper a careful analysis of this construction establishing the equiv-alence of Hilbert spaces from above in a rigorous way.In our opinion,this sort of analysis seems to be laking form the literature.In establishing this result we will use a pure quantum point of view,as advocated in[25].To be completely rigorous,we define the graviton to be the unitary irreducible representation of the Poincar´e group corresponding to zero mass and helicity2[27].The usual Wigner representation for this elementary system is not very good for practical purposes.To have some analogy with the situation from the classicalfield theoretical description of gravity one has tofind out an analogue of the construction of the photon by some factorization procedure,more precisely to implement a construction of the same type as that of [27]ch.IV.7(see also[24],[16],[10]).The explicit construction is not so elementary as in the case of the photon and deserves a detailed presentation.One will construct the Hilbert space of the graviton as a factor space H graviton≡H′/H”where H”⊂H′⊂H for some auxiliary Hilbert space H,as in the case of the photon,but the construction of these Hilbert spaces is rather complicated as it also is the verification that this factor space carries the desired representation of the Poincar´e group.This is done in the next Section.Next,we apply the standard second quantization procedures[2],[1],[16],[27]to H graviton,that’s it we postulate that the many-gravitons system is the Bosonic Fock space associated to this one-particle Hilbert space:F graviton≡F+(H graviton).Like in the case of the photon,one can prove that this Hilbert space can also be written as a factor space F graviton≡H′/H”where H”⊂H′⊂H for some auxiliary Hilbert space H.This is done in Section3and4.Next,we prove that one can extend the Hilbert space H by including some ghosts:namely,a pair of Fermion ghosts uµ,˜uµwhich are vectorfields and a Bosonic ghostsΦwhich is a scalarfield such that we construct a supercharge Q in the extended Hilbert space H gh and prove that F graviton≃Ker(Q)/Im(Q).This is done in Section5.Finally,in Section6,we give the expression for the BRST transformation and analyse the concept of observables in this framework.The“vulnerability”of this type of approach to the construction of freefields is the fact that, although the“playground”of the construction,which is the Fock space,is a canonical object, being canonically constructed from the one-particle Hilbert space,the expressions of the free fields are deeply dependent on the representation adopted for this one-particle space.Distinct factorization procedures(if they do exist)might lead to distinct theories.It is suggested in the literature[21],[22]that a road out of this problem is to use ideas from algebraicfield theory.We stress here that in the usual approaches only the Fermion ghosts are considered.How-ever,a rigorous proof that the equality F graviton≃Ker(Q)/Im(Q)is true seems to be laking from the literature.We will present at the end of Section4some arguments which raises doubts about the correctness of the standard procedures.We also mention that a rigorous construction of the Hilbert space of the many-gravitons system is indispensable for the construction of the corresponding S-matrix in the sense of perturbation theory of Bogoliubov.This construction emphasize the basic rˆo le of causality in quantumfield theory(see also[21],[22]where it is remarked that causality is again the major physical axiom in algebraicfield theory).The recursive construction of Epstein and Glaser[6], [8]as presented in[23],[11]and[25]makes sense only if this starting point is settled,because otherwise it is not clear if the elementary physical particle of the theory is characterized by zero mass and helicity2as should be the case for the graviton.We will reconsider the construction of a consistent theory of quantum gravity,in the sense of perturbation theory,using the formalism developed here in another publication.2The Graviton as an Elementary Relativistic Free ParticlesAs we have anticipated in the Introduction,one defines the graviton as a certain unitary irreducible representation of the Poincar´e group corresponding to zero mass and helicity2.We will describe this representation using the formalism of Hilbert space bundles,as presented in [27],ch.VI.7thm6.20.First,wefix some notations.The upper hyperboloid of mass m≥0is by definition X+m≡{p∈R4| p 2=m2};it is a Borel set with to the Lorentz invariant measure dα+m(p)≡d pp2+m2.We start now to define the graviton,as a representation of zero mass and helicity2of the Poincar´e group.Proposition2.1Let us define by F the space of complex traceless symmetric4×4matrices:F≡{hρσ|hρσ=hσρ,hρρ=0}.(2.1) Then,we define the following objects:(a)The Borel set:B≡{(pµ,hρσ)|p∈X+0,hρσ∈F,hρσpσ=0}.(2.2)(b)The canonical projection on thefirst entry:π:B→X+0;(c)The action of the group SL(2,C)on B:D(A)·(pµ,hρσ)≡(δ(A)µνpν,δ(A)ρλδ(A)σωhλω)(2.3) whereδ:SL(2,C)→L↑+is the standard group homomorphism;(d)The sesquilinear forms((p,h),(p,h′))p≡(iv)We still have to prove that the sesquilinear forms are positively defined.For this we use(iii)and the fact that there exists an element A∈SL(2,C)such that P≡δ(A)·p is of the form Pµ=(1,0,0,1).Let the transformed matrix beHλω≡δ(A)ρλδ(A)σωhλω.One has to write explicitly all the constraints on the matrix H,namely the symmetry,the tracelessness and the property the gives zero when applied to the quadri-vector P.After some elementary computations one discovers that the generic form of the matrix H is:H00=H33=2F0,H03=H30=−2F0,H10=H01=F1,H20=H02=F2,H13=H31=−F1,H23=H32=−F2,H11=−H22=α,H12=H21=β;(2.4) here the(complex)numbers Fµ,µ=0,...,3andα,βare arbitrary.We have now quite elementaryhµνhµν=0then we also haveProposition2.3Let us defineF0≡{(p,h)∈F|hµνh′µν,∀h∈[h],∀h′∈[h′].Then(X+0,[B],SL(2,C),[π])is a Hilbert space bundle withfibres of dimension2.The proof is elementary:one simply has to check that all objects are well defined i.e. independent of the choice of the representatives in the equivalence classes[·].Let us denote thefibre over p by[B](p).Then we apply the standard construction from[27] VI.8of associating to the Hilbert space bundle[B]a representation of the group inSL(2,C) (which is the universal covering group of the proper orthochronous Poincar´e group). Theorem2.4Let us construct the vector spaceV={s:X+0→[F]|s is a Borel function,s(p)∈[B](p)}(i.e.the space of Borel sections of the Hilbert space bundle[B])and defines 2≡ X+0dα+0(p)((p,s(p)),(p,s(p)))p.We define now the spaceV′≡{s∈V| s 2<∞};then V′is a pre-Hilbert space with respect to the scalar product(s,s′)≡ X+0dα+0(p)((p,s(p)),(p,s′(p)))p.Let˜V′be the Hilbert space which is the completion of V′with respect to the scalar product(·,·) and let us define the operators U a,A:˜V′→˜V′by(U a,A s)(p)≡e ia·p[δ(A)·h(δ(A−1)·p)],for(a,A)∈inSL(2,C)(2.9)where h(p)∈s(p).Then U is a unitary representation of inSL(2,C)and it is equivalent to the representation U+,4⊕U+,−4(see[27]thm9.4).Proof:We want to apply the theorem6.20from[27].So,we must determine the action of the stability subgroup E∗on thefibre[B](1,0,0,1).We remind that the group E∗is formed from elements of SL(2,C)of the following type:A= z z−1a0z−1 ,∀z,a∈C,|z|=1.(2.10) We also note that thefibre[B](1,0,0,1)can be identified as the quotient space of the space of all matrices of the form(2.4)factorized to the space V of the matrices constrained by the conditionα=β=0.It follows that we can identify[B](1,0,0,1)with the space of matrices Hα,βof the form(Hα,β)11=−(Hα,β)22=α,(Hα,β)12=(Hα,β)21=β,(Hα,β)ρσ=0,in rest.(2.11) So,we have the isomorphism[B](1,0,0,1)∋[Hα,β]↔(α,β)∈C2.We have to compute the action of the group E∗on such elements so we have to compute the matrix(H′α,β)ρσ=δ(A)ρλδ(A)σω(Hαβ)λω.This matrix should be of the form Hα′β′(mod(V))and onefinds out that we have α′=α{[δ(A)11]2−[δ(A)12]2}+2βδ(A)11δ(A)12=ℜ(z4)α+ℑ(z4)β,β′=α[δ(A)11δ(A)21−δ(A)12δ(A)22]+β[δ(A)11δ(A)22+δ(A)12δ(A)21]=−ℑ(z4)α+ℜ(z4)β.(2.12) In the new variablesu≡α+iβ,v=α−iβwe have the transformation rulesu′=z−4u,v′=z4vi.e.we have the representationπ4⊕π−4of the stability subgroup E∗.The assertion from the statement follows now from thm.6.20of[27].We note that the couple(˜V′,U)is a unitary representation of the group inSL(2,C)which corresponds to zero mass and helicity2.According to the usual physical interpretation,we call this system graviton.We remark that this(true)representation of the group inSL(2,C) induces a true representation of the group P↑as it this the case with all representations of integer spin(or helicity).Moreover,this representation can be extended to the whole Poincar´e group.It is possible to express this representation in an analogous way to the photon if one considers a factorisation procedure[27].Let us consider the Hilbert space H≡L2(X+0,F,dα+0) with the scalar product<φ,ψ>≡ X+0dα+0(p)<φ(p),ψ(p)>F(2.13)where<φ,ψ>F≡3µ,ν=0φρσ(I s·p).(2.14)Let us define on H the following non-degenerate sesquilinear form:(φ,ψ)≡ X+0dα+0(p)φµ(p)ψν(p)=<φ,g⊗2ψ>;(2.15)here g∈L↑is the Minkowski matrix with diagonal elements1,−1,−1,−1.Then one easily establishes that we have(U a,Λφ,U a,Λψ)=(φ,ψ),forΛ∈L↑,(U It φ,U Itψ)=(H′/H′′)(2.20) (here by the overline we understand completion).The factor representation,denoted also by U is unitary and irreducible.By restriction to the proper orthochronous Poincar´e group it is equivalent to the representation from the preceding theorem.3Second QuantizationHere we give the main concepts and formulæconnected to the method of second quantization. We follow essentially[27]ch.VII(see also[2]and[1]).The idea of the method of second quantization is to provide a canonical framework for a multi-particle system in case one has a Hilbert space describing an“elementary”particle. One usually takes the one-particle Hilbert space H to be some projective unitary irreducible representation of the Poincar´e group.Let H be a(complex)Hilbert space;the scalar product on H is denoted by<·,·>.Onefirst considers the tensor algebraT(H)≡⊕∞n=0H⊗n,(3.1) where,by definition,the term corresponding to n=0is the divisionfield C.The generic element of T(H)is of the type(c,Φ(1),···,Φ(n),···),Φ(n)∈H⊗n;the elementΦ0≡(1,0,...)is called the vacuum.Let us consider now the symmetrisation(resp.antisymmetrisation) operators S±defined byS±≡⊕∞n=0S±n(3.2) where S±0=1and S±n,n≥1are defined on decomposable elements in the usual wayS±nφ1⊗···⊗φn≡1n+1φ∨ψ(3.7)and respectivelyA(φ)ψ≡1niφψ(3.8)where iφis the unique derivation of the algebra F+(H)verifyingiφ1=0;iφψ=<φ,ψ>1.(3.9) Remark3.1We note that the general idea is to associate to every element of the one-particle spaceφ∈H a couple of operators A♯(φ)acting in the Fock space F+(H).As usual,we have the canonical commutation relations(CCR):[A(φ),A(ψ)]=0, A(φ)†,A(ψ)† =0, A(φ),A(ψ)† =<φ,ψ>1.(3.10) The operators A(ψ),A(ψ)†are unbounded and adjoint one to the other.In the Fermionic case we define these operators on elements fromψ∈H−n by√A(φ)†ψ≡√4The Quantization of the Gravitational FieldIn this Section we apply the prescription from Section3to the Hilbert space of the graviton H graviton given by(2.4).The idea is similar to the case of the photon,namely to express the (Bosonic)Fock space of the gravitonF graviton≡F+(H graviton)(4.1) as a factorization of the type(2.4).It is natural to start with the“bigger”Fock spaceH≡F+(H)≡⊕n≥0H n(4.2)where the n th-particle subspace H n is the set of Borel functionsΦ(n)µ1,ν1;...;µn,νn :(X+0)×n→Cwhich are square summable:(X+0)×nni=1dα+0(k i)3 µ1,...,µn=03 ν1,...,νn=0|Φ(n)µ1,ν1;...;µn,νn(k1,...,k n)|2<∞(4.3)verify the following properties:(a)symmetry in the triplets(k i,µi,νi),i=1,...,n;(b)symmetry in every couple(µi,νi),i=1,...,n;(c)tracelessness in every couple(µi,νi),i=1,...,n.In H the expression of the scalar product is:<Ψ,Φ>≡Ψ(n)µ1,ν1;...;µn,νn (k1,...,k n)Φ(n)µ1,ν1;...;µn,νn(k1,...,k n)(4.4)and we have a(non-unitary)representation of the Poincar´e group given by:U g≡Γ(U g),∀g∈P;(4.5) here U g is given by(2.14).Let us define the following non-degenerate sesquilinear form on H:(Ψ,Φ)≡Ψ(n)µ1,ρ1;...;µn,ρn (k1,...,k n)Φ(n)ν1,σ1;...;νn,σn(k1,...,k n).(4.6)Then the sesquilinear form(·,·)behaves naturally with respect to the action of the Poincar´e group:(U gΨ,U gΦ)=(Ψ,Φ),∀g∈P↑,(U It Ψ,U ItΦ)=Lemma4.1Let us consider the following subspace of H:H′≡F+(H′)=⊕n≥0H′n.(4.8) Then H′n,n≥1is generated by elements of the formφ1∨···∨φn,φ1,...,φn∈H′and, in the representation adopted previously for the Hilbert space H n we can takeH′n={Φ(n)∈H n|kµ11Φ(n)µ1,ν1;...;µn,νn(k1,...,k n)=0}.(4.9)Moreover,the sesquilinear form(·,·)|H′is positively defined.Next,one has the analogue of lemma2.6:Lemma4.2Let H′′⊂H′given byH′′≡{Φ∈H′| Φ 2=0}=⊕n≥0H′′n(4.10) Then,the subspace H′′n,n≥1is generated by elements of the typeφ1∨···∨φn where at least one of the vectorsφ1,...,φn∈H′belongs to H′′.Moreover,in the representation adopted previously for the Hilbert space H n the elements of H′′n are linearly generated by functions of the type:Φ(n)µ1,ν1;...;µn,νn (k1,...,k n)=1H′/H′′.(4.12)Proof:Ifψ∈H′then we denote its class with respect to H′′by[ψ];similarly,ifΦ∈H′we denote its class with respect to H′′by[Φ].Then the application i:We define for every p∈X+0the annihilation and creation operators according to:(hρσ(p)Φ)(n)µ1,ν1;...;µn,νn (k1,...,k n)≡√√2gρσgµiνiΦ(n−1)µ1,ν1;...;ˆµi,ˆνi;...;µn,νn(k1,...,ˆk i,...,k n),∀n∈N.(4.15)We use everywhere consistently the Bourbaki convention: ∅≡0.Then one has the canonical commutation relations(CCR)[hρσ(p),hλω(p′)]=0, h†ρσ(p),h†λω(p′) =0,hρσ(p),h†λω(p′) =−ω(p) gρλgσω+gρωgσλ−1(2π)3/2 X+0dα+0(p)e ip·x hρσ(p),h(+)ρσ(x)≡1n+1andh(+)ρσ(x)Φ (n)µ1,ν1;...;µn,νn(k1,...,k n)=−1n n i=1e ik i·x×gρµi gσνi+gρνi gσµi−12 gρλgσω+gρωgσλ−12 gρλgσω+gρωgσλ−1(2π)3/2 X+0dα+0(p)e∓ip·x.(4.29) One can describe in a convenient way the subspaces H′and H′′using the following operators L(f)≡ R4dxfρ(x)∂σh(−)ρσ(x),L†(f)≡ R4dxfρ(x)∂σh(+)ρσ(x)(4.30) where f:R4→R4verify the transversality condition:∂fρProposition 4.5The following relations are true:H ′={Φ∈H|L (f )Φ=0,∀f }=∩f Ker (L (f ))(4.31)andH ′′={L (f )†Φ|∀Φ∈H ,∀f }=∪f Im (L (f )†).(4.32)It follows that we haveF graviton =n !R 4ndx 1···dx n T n (x 1,···,x n )g (x 1)···g (x n ),(4.36)where g (x )is a tempered test function that switches the interaction and T nare operator-valued distributions acting in the Fock space of some collection of free fields;in [6](see also [8])one considers in detail the case of a real free scalar field.These operator-valued distributions,which are called chronological products should verify some properties which can be argued starting from Bogoliubov axioms .•First,it is clear that we can consider them completely symmetrical in all variables without loosing generality:T n (x P (1),···x P (n ))=T n (x 1,···x n ),∀P ∈P n .(4.37)•Next,we must have Poincar´e invariance :U a,ΛT n (x 1,···,x n )U −1a,Λ=T n (Λ·x 1+a,···,Λ·x n +a ),∀Λ∈L ↑.(4.38)14In particular,translation invariance is essential for implementing Epstein-Glaser scheme of renormalisation.•The central axiom seems to be the requirement of causality which can be written compactly as follows.Let usfirstly introduce some standard notations.Denote by V+≡{x∈R4|x2> 0,x0>0}and V−≡{x∈R4|x2>0,x0<0}the upper(lower)lightcones and byV−,∀i=1,...,m,j=1,...,n we use the notation X≥Y.We use the compact notation T n(X)≡T n(x1,···,x n)and by X∪Y we mean the juxtaposition of the elements of X and Y.In particular,the expression T n+m(X∪Y)makes sense because of the symmetry property(4.37).Then the causality axiom writes as follows:T n+m(X∪Y)=T m(X)T n(Y),∀X≥Y.(4.39)•The unitarity of the S-matrix can be most easily expressed(see[6])if one introduces,the following formal series:¯S(g)=1+∞ n=1(−i)nexists,in the weak sense,and are independent of the test function g.One also calls the limit performed above,the infrared limit.•Finally,one demands the stability of the vacuum i.e.lim ǫց0<Φ0,S(gǫ)Φ0>=1⇔limǫց0<Φ0,T n(g⊗nǫ)Φ0>=0,∀n∈N∗.(4.47)A renormalisation theory is the possibility to construct such a S-matrix starting from the first order term:T1(x)≡L(x)(4.48) where L is a Wick polynomial called interaction Lagrangian which should verify the following axioms:U a,ΛL(x)U−1a,Λ=L(Λ·x+a),∀Λ∈L↑,(4.49) [L(x),L(y)]=0,∀x,y∈R4s.t.(x−y)2<0,(4.50)L(x)†=L(x)(4.51) andL≡limǫց0L(gǫ)(4.52) should exists,in the weak sense,and should be independent of the test function g.Moreover, we should have<Φ0,LΦ0>=0.(4.53) To construct such an operator is not exactly an easy matter.In fact,the set of relations (4.49),(4.50)and(4.51)is a problem of constructivefield theory in the particular case when the Hilbert space is of the Fock type.In the analysis of Epstein and Glaser[6],[8]it is argued that the most natural candidates for interaction Lagrangian L(x)are the Wick polynomials.In the familiar physicists language,the problem is to prove that no ultraviolet divergences and no infrared divergences appear,i.e.the operators T n arefinite and well defined and the adiabatic limit exists.The only freedom left in this case for a renormalisation theory is the non-uniqueness of the T n’s due tofinite normalization terms(also called counterterms)which are distributions with the support{x1=···=x n=0}.The whole construction can be based on the operation of distribution splitting;there are very effective ways to perform this operation explicitly[20].In the case of gravitation,one should modify the preceding axioms of perturbation theory as follows.One constructs the whole theory in the auxiliary Hilbert space H and imposes the axioms as presented above;then one requires that the resulting S-matrix factorizes,in the adiabatic limit,to the“physical”Hilbert space F graviton.We will give the explicit form of this factorization axiom at the end of the next Section for the case of quantization with ghosts.If the adiabatic limit do not exists(as it is presumably the case for zero mass systems as the graviton)one has to consider the factorization axiom as an heuristic relation and should replace it by another postulate(see[20],[5]).165Quantisation with Ghost FieldsIn this subsection we give the complete analysis of another realisation of F graviton which is essential for the construction of the perturbation theory in the sense of the preceding Section. We we exhibit a construction which seems to be new in the literature and give complete proofs, following the lines of[10].First,we give consider the Hilbert spaceH gh=∞n,m,l,s=0H nmls(5.1)where H nmls consists of Borel functionsΦ(nmls)µ1ν1,...,µn,νn;ρ1,...,ρm;σ1,...,σl:(X+0)n+m+l+s→C such that:(1)they are square modulus summable with respect to the Lorentz invariant measure:∞n,m,l,s=0 (X+0)n+m+l dα+0(K)dα+0(P)dα+0(Q)dα+0(R)3µi,νi,ρ1,σi=0|Φ(nmls)µ1ν1,...,µn,νn;ρ1,...,ρm;σ1,...,σl(K;P;Q;R)|2≤∞(5.2)where we are using the condensed notations:K≡(k1,...,k n),P≡(p1,...,p m),Q≡(q1,...,q l)and R≡(r1,...,r s).(2)they verify the following(anti)symmetry properties:(a)symmetry at the permutation of the triplets(k i,µi,νi)↔(k j,µj,νj),i,j=1,...,n;(b)symmetry at the changeµi↔νi,i=1,...,n;(c)antisymmetry at the permutation of the couples(p i,ρi)↔(p j,ρj),i,j=1,...,m;(d)antisymmetry at the permutation of the couples(q i,σi)↔(q j,σj),i,j=1,...,l;(e)antisymmetry in the variables r i,i=1,...,s.(3)tracelessness in every couple(µi,νi),i=1,...,n.In this representation we define the annihilation operators according to the expressions:(hαβ(t)Φ)(nmls)µ1ν1,...,µn,νn;ρ1,...,ρm;σ1,...,σl (K;P;Q;R)≡√m+1Φ(n,m+1,l,s)µ1ν1,...,µn,νn;α,ρ1,...,ρm;σ1,...,σl(K;t,P;Q;R)(5.4)(cα(t)Φ)(nmls)µ1ν1,...,µn,νn;ρ1,...,ρm;σ1,...,σl(K;P;Q;R)≡(−1)m√and(d(t)Φ)(nmls)µ1ν1,...,µnνn;ρ1,...,ρm;σ1,...,σl (K;P;Q;R)≡√2gαβgµiνi ×Φ(n−1,mls)µ1ν1,...,ˆµiˆνi,...,µn,νn;ρ1,...,ρm;σ1,...,σl(k1,...,ˆk i,...,k n;P;Q;R)(5.7)(b∗α(t)Φ)(nmls)µ1ν1,...,µnνn;ρ1,...,ρm;σ1,...,σl(K;p1,...,p m;Q;R)≡ω(t)×mi=1(−1)i−1δ(t−p i)gαρiΦ(n,m−1,l,s)µ1ν1,...,µn,νn;ρ1,...,ˆρi,...,ρm;σ1,...,σl(K;p1,...,ˆp i,...,p m;Q;R)(5.8)(c∗α(t)Φ)(nmls)µ1ν1,...,µn,νn;ρ1,...,ρm;σ1,...,σl(K;P;q1,...,q l;R)≡(−1)mω(t)×li=1(−1)i−1δ(t−q i)gασiΦ(nm,l−1,s)µ1ν1,...,µn,νn;ρ1,...,ρm;σ1,...,ˆσi,...,σl(K;P;q1,...,ˆq i,...,q l;R)(5.9) and(d∗(t)Φ)(nmls)µ1ν1,...,µnνn;ρ1,...,ρm;σ1,...,σl(K;P;Q;r1,...,r s)≡ω(t)×si=1δ(t−r i)Φ(n,m,l,s−1)µ1ν1,...,µn,νn;ρ1,...,ρm;σ1,...,σl(K;P;Q;r1,...,ˆr i,...,r s).(5.10) We note that the Hilbert space H given by(4.2)can be naturally embedded into H gh asfollows:H∼∞n=0H n000.(5.11)Moreover,this embedding preserves the expressions of the operators h#αβ.For this reason we call the Fock space H gh the ghost extension of H.Remark the choice of the various statistics which seems to be essential for the whole analysis.They verify the canonical(anti)commutation relations(4.16)and{bα(k),bβ(q)}=0,{b∗α(k),b∗β(q)}=0,{bα(k),b∗β(q)}=2ω(q)gαβδ(k−q)1(5.12) {cα(k),cβ(q)}=0,{c∗α(k),c∗β(q)}=0,{cα(k),c∗β(q)}=2ω(q)gαβδ(k−q)1(5.13)18[d(k),d(q)]=0,[d∗(k),d∗(q)]=0,[d(k),d∗(q)]=2ω(q)δ(k−q)1(5.14) {b#α(k),c#β(q)}=0,[b#α(k),d#(q)]=0,[c#α(k),d#(q)]=0(5.15)[h#αβ(k),b#γ(q)]=0,[h#αβ(k),c#γ(q)]=0,[h#αβ(k),d#(q)]=0.(5.16)In this Hilbert space a(non-unitary)representation of the Poincar´e group acts in an obvious way and the creation and annihilation operators defined above behave naturally with respect to these Poincar´e transformations.Then we can define,beside the gravitationalfield(see the preceding Section)the following Fermionicfieldsu(x)≡1(2π)3/2 X+0dα+0(q) −e−iq·x c(q)+e iq·x b∗(q) (5.18) and the BosonicfieldΦ(x)≡12(d(k)c∗ν(k)+d∗(k)bν(k)) (5.21)called supercharge.Its properties are summarised in the following proposition which can be proved by elementary computations:Proposition5.1The following relations are valid:QΦ0=0;(5.22) Q,h†µν(k) =−12gρσgµν kρc∗σ(k),(5.23)19Q,b ∗µ(k ) =k νh †µν(k )+12k µc ∗µ(k );(5.24)[Q,h µν(k )]=12g ρσg µνk ρb σ(k ),(5.25){Q,b µ(k )}=0,{Q,c µ(k )}=k νh µν(k )+12k µb µ(k );(5.26)Q 2=0;(5.27)Im (Q )⊂Ker (Q )(5.28)andU g Q =Q U g ,∀g ∈P .(5.29)Moreover,one can express the supercharge in terms of the ghosts fields as follows:Q = R 3d 3x ∂µh µν(x )↔∂0u ν(x )+1n +12nn i =1[(k i )µi δανi +(k i )νi δαµi−14ll i =1(−1)i −1(q i )σi ×Φ(n,m,l −1,s +1)µ1,ν1,...,µn νn ;ρ1,...,ρm ;σ1,...,ˆσi ,...,σl (K ;P ;q 1,...,ˆqi ,...,q l ;q i ,r 1,...,r l )+1m +1Now we introduce on H gh the Krein operator which in compact tensorial notations looks: (JΦ)(nmls)(K;P;Q;R)≡(−1)ml+n g⊗2n+m+lΦ(nlms)(K;Q;P;R).(5.32) The properties of this operator are given below and can be proved by elementary computa-tions:Proposition5.3The following relations are verified:J∗=J−1=J(5.33) Jbµ(p)J=cµ(p),Jd(p)J=d(p),Jh∗µν(p)J=h†µν(p)(5.34)JQJ=Q∗(5.35) andU g J=J U g,∀g∈P.(5.36) Here O∗is the adjoint of the operator O with respect to the scalar product<·,·>on H gh.We can define now the sesquilinear form on H gh according to(Ψ,Φ)≡<Ψ,JΦ>;(5.37) then this form is non-degenerated.It is convenient to denote the conjugate of the arbitrary operator O with respect to the sesquilinear form(·,·)by O†i.e.(O†Ψ,Φ)=(Ψ,OΦ).(5.38) Then the following formula is available:O†=JO∗J.(5.39) As a consequence,we havehµν(x)†=hµ(x),u(x)†=u(x),˜u(x)†=−˜u(x),Φ†(x)=Φ(x).(5.40) From(5.36)it follows that we have:(U gΨ,U gΦ)=(Ψ,Φ),∀g∈P↑,(U It Ψ,U ItΦ)=。

新发展大学英语阅读与写作3课文翻译What Is a Scientific Theory 什么是科学理论

新发展大学英语阅读与写作3课文翻译What Is a Scientific Theory 什么是科学理论

What Is a Scientific Theory?In order to talk the mature of the universe and to discuss questions such as whether it has a beginning or an end, you have to be clear about what a scientific theory is. I shall take the simple-minded view that a a theory is just a model of the universe, of a restricted part of it, and a set of rules that relate quantities in the model to observations that we make, It exists only in our minds an does not have any other reality (whatever that night mean). A theory is a good theory if it satisfies two requirements: It must accurately describe a large class of observations on the basis of a model that contains only a few arbitrary elements, and it must make definite predictions about the results of future observations. For exampl e, Aristotle’s theory that everything was made out of four elements, earth, air, fire, and water, was simple enough to qualify, but it did not make any definite predictions. On the other hand, Newton’s theory was proportional to a quantity called their mass and inversely proportional to the square of the distance between them. Yet it predicts the motions of the sun, the moon, and the planets to a high degree of accuracy.Any physical theory is always provisional, in the sense that it is only a hypothesis: you can never prove it. No matter how many times the results of experiments, agree with some theory, you can never be sure that the next time the result will not contradict the theory. On the other hand, you can disprove a theory by finding even a single observation that disagrees with the predictions of the theory. As philosopher of science Karl Popper has emphasized, a good theory is characterized by the fact that it makes a number of predictions that could in principle be disproved or falsified by observation. Each time new experiments are observed to agree with the predictions the theory survives, and our confidence in it is increased; but if ever a new observation is found to disagree, we have to abandon or modify the theory. At least that is what is supposed to happen, but you can always question the competence of the person who carried out the observation.In practice, what often happens is that a new theory is devised that is really an extension of the previous theory. For example, very accurate observations of the planet Mercury revealed a small difference between its motion and the predictions of Newton’s theory of gravity. Einstein’s general theory of relativity predicted a slightly different motion from Newton’s theory. The fact that Einstein’s predictions matched what was seen, while Newton’s did not, was one of the crucial confirmations of the new theory. However, we still use Newton’s theory for all practical purposes because the difference between its predictions and those of general relativity is very small in the situations that we normally deal with. (Newton’s theory also has the great advantage that it is much simpler to work with than Einstein’s!)The eventual goal of science is to provide a single theory that describes the whole universe. However, the approach most scientists actually follow is to separate the problem into two parts. First, there are the laws that tell us how the universe changes with time. (If we know what the universe is like at any one time, these physical laws tell us how it will look at any later time.) Second, there is the question of the initial state of the universe. Some people feel that science should be concerned with only the first part; they regard the question of the initial situation as a matter for meta-physics of religion. They would say that God, being omnipotent, could have made it develop in a company way he wanted. That may be so, but in that case he also could have made it develop in a completely arbitrary way. Yet it appears that he chose to make it evolve in a very regular way according to certain laws. It therefore seems equally reasonable to suppose that there are also laws governing the initial state.It turns out to be very difficult to device a theory to describe the universe all in one go. Instead, we break the problem up into bits and invent a number of partial theories. Each of these partial theories describes and predicts a certain limited class of observations, neglecting the effects of other quantities, or representing them by simple sets of numbers. It may be that this approach is completely wrong. If everything in the universe depends on everything else in a fundamental way, it might be impossible to get close to a full solution by investigating parts of the past. The classic example again is the Newtonian theory of gravity, which tells us that the gravitational force between two bodies depends only on one number associated with each body, its mass, but is otherwise independent of what the bodies are made of. Thus one does not need to have a theory of the structure and constitution of the sun and the planets in order to calculate their orbits.Today scientists describe the universe in terms of two basic partial theories –the general theory of relativity and quantum mechanics. They are the great intellectual achievements of the first half of this century. The general theory of relativity describes the force of gravity and the large-scale structure of the universe, that is , the structure on scales from only a few miles to as large as a million million million (1 with zeros after it) miles, the size of the observable universe. Quantum mechanics, on the other hand, deals with phenomena ion extremely small scales, such as a millionth of a millionth of an inch. Unfortunately, however, these two theories are known to be inconsistent with each other – they cannot both be correct. One of the major endeavors in physics today, is the search for a new theory, and we may still be a long way from having one, but we do already know many of the properties that it must have.o什么是科学理论?为了谈宇宙的成熟和讨论这样的问题是否有一个开始或结束,你必须清楚科学理论是什么。

The gravitational field energy density for symmetrical and asymmetrical systems

The gravitational field energy density for symmetrical and asymmetrical systems

The gravitational field energy density for symmetrical and asymmetrical systemsRoald SosnovskiyTechnical University, 194021, St. Petersburg, RussiaE-mail:rosov2@yandexAbstract. The relativistic theory of gravitation has the considerable difficulties by description of the gravitational field energy. Pseudotensor t 00 in the some cases cannot be interpreted as energy density of the gravitational field. In [1] the approach was proposed, which allow to express the energy density of such a field through the components of a metric tensor. This approach based on the consideration of the isothermal compression of the layer consisted of the incoherent matter. It was employ to the cylindrically and spherically symmetrical static gravitational field. In presented paper the approach is developed.1. Introduction. The problem of the gravitational field energy discussed a long time [2], [3]. However, pseudotensor differs from author to author reflecting the ambiguity in defining gravitational field energy density [3]. In [1] the approach has proposed allows one to express the energy density of such a field through the components of a metric tensor. This approach based on the consideration of the isothermal compression of a layer consisted of the incoherent matter in the field of the infinitesimal thin material shell by fulfillment of the requirements [4]: (a) the local energy conservation law should be fulfilled and (b) the correspondence principle should be fulfilled including the energy part.µνt In the presented paper proved, that this approach can be used for asymmetrical systems. Here proved, that the requirement of the invariance of the gravitational field energy density [4] fulfilled. For the cylindrically and spherically symmetrical systems is obtained field energy density formulas, contained only the metric tensor component and his derivative.2. The differential of the gravitational field energyIn [1] has obtained the formulas of the gravitational field energy for the special coordinates, connected with type of symmetry. Here it considered the formula of the field energy for the arbitrary static coordinates systems. The solution is analogous to one in[1].2.1. The isothermal compression. Here it considered the movement of the particles layers when acquired energy of particles has eradiated or dissipated. The movement considered as consisted of discrete infinitesimal steps, when the particles fall free, and in end of step energy of particles has dissipated. Concrete ways of dissipation no discussed. Sufficiently to suppose that such way can be on principle approximately realized. For example, free fall of particles in thin lay on the solid surface with following cooling of the solid.The particles considered as test-particles. However, the change of field, caused with accumulation of the matter on solid surface, calculated after every step. Assume αx is the initial coordinate system, admissible for the system configuration, with metric .0,0=µµνg g Let us consider the displacement of the particles layer from position 111x x = to position 1111x d x x +=. The free particles fall equations are[5] 0=∂∂−⎟⎠⎞⎜⎝⎛∂∂µµτxL x L d d & (1) where τ is the intrinsic time, ·τµµd x d x =& and()νµµνσσx x g x x L &&&21,=, ()321,,x x x g g µνµν= (2) For static system (1),(2) lead to 0000g C x =& (3) where C 0 is constant on all step. From (3) and from equationk i ik x x g xg c &&&+=02002 (4) we get, so far as i x&is small, 000g c C = (5) From formulas (1), (2) for i, k =1,2,3 result02,0021x g xg i k ik &&&= (6) and from (5),(6)002,0021g c g g x i ik kδτ=& (7) where τ is the intrinsic time of particles movement. Components k u of the maximum velocity of free particles fall near by point ),,(321x x x x r may written00,00g g g u iik k β= (8)where ß is infinitesimal coefficient.2.2. The static gravitational field energy. General formulas. The energy of the particles by free fall can obtained from the relation [6]0002g g u mc E ννδ=, τννcd x d u = (9) where δm is rest mass of the particles group. From (3) and so far as 0000g g ννδ= the change of particles energy on way i x d isi i x d g g mc dE ,000022δ−= (10) This energy has dissipated on way i x d . If the local energy conservation law fulfilled then the energy change of particles must result from change of field energy dE f on way i x d . Thereforei i f x d g mc dE ,000022δ= (11)From (8) follow, that the components of the of the particle coordinates mean change is00,00g g g x d kik i λ= (12)and scalar displacement is equalk i ik g g g g dl ,00,0000λ= (13) wherek i ik g g g dlg ,00,0000=λ (14)If substitute λ in (12) and then i x d in (11) then we get dl g g g g mc E d k i ik f ,00,000022δδ−= (15)Here δm and δl are scalars, 00g by 00=i g no depend on the space coordinates transformation. The quantity k i ik g g g ,00,00 is invariant by the space coordinates transformation k i k i x A x = (16)Therefore, d δE f is also invariant.3. Energy of the asymmetrical gravitational field.3.1. The object. Considered the static field of the asymmetric convex smooth infinitesimally thin material shell with surface mass density σc .The quantity σc is a single-valued function of the coordinates of shell point ()3211,x x x x c c =, ()32,x x c c σσ=. It considered the space between a shell and some convex smooth external surface ()3211,x x x x e e = with surface mass density ()32,x x e e σσ=. Assumed, that it known how the metric ik g of the space region between c x and e x to find. This is possible at least by miens of the computer methods [7].3.2. The calculations order. Is considered the motion of N j discrete test particles layers from the external surface to the shell. The motion is discrete; the number of steps is N q . For every layer position, the calculations made for N k · N n points. The every point position determined in coordinate system i x . For every point P(k,n,q) the volume element is built at the vectors ),,(321ii i i i x d x d x d z d z d r r =, i =1,2,3. Let this vectors create the coordinate system ()0000,,,,g g x d j q n k B dz k i k i == (17)One side (dz 2,dz 3) of this element is disposed at the layer position q and the opposite side at the layer position q+1. The vector l d r describe the fall of particle from point P(k,n,q) up to point F(k',n',q+1) at the layer position q+1. For every point P(k,n,q) and every layer j are calculated from (15) the field energy differential d δE f and masse density σ(k',n',q+1,j) for point P(k',n',q+1). Afterwards the layer j arrive the position q = N q metric components g ik calculated for all points P(k,n,q) . The method of such calculation no considered because that does not matter for the purpose of this paper.3.3. The gravitational field integral energy and energy density invariance. Let the volume element is built at the vectors dz i .The mass of the particles group, passed through this element, is equal()()()j q n k dS j q n k j q n k m ,,,,,,,,,⋅=δσδ (18)where δσ(k,n,q,j) is the matter density in the particles layer j and dS(k,n,q,j) – the area of the element (dz 2,dz 3). The mass δm of this element, which is considered as in the one point concentrated, fall from point P(k,n,q) in point F(k,n,q+1) with coordinates z i + dl i . Components dl i can be calculated in coordinates dz i from (12), (17). Component dl 1 = dz 1 andq q pkp k g g g g dl dl ,001,001=,k=2,3 (19)By means interpolation can be calculate the mass δm(n,q+1,j) and mass density δσ(n,q+1,j) for points P(k,n,q+1). Consider the successive pass of the layers through the element of area ()32,dz dz with point P(k,n,q). From (15),(17) after step j = N j the field energy change in volume element is equal ()j q n k s i isN j f g g g g dl dS c q n k dE j ,,,,00,0000222,,⎥⎦⎤⎢⎣⎡⋅⋅=∑δσ (20) where [] depend on (k,n,q,j). The quantities under the symbol Σ are the invariants, therefore dE f (k,n,q) is invariant. The sum of energy in all points of the field is also invariant.The energy density in point P(n,q) is given by()()),(,,,q n dV q n dE q n k w f =, 321dz dz dz g dV я= (21)where dV(n,q) is the volume of the volume element, built at the vectors ()321,,z d z d z d r r r for step j=N j ; z g is determinant of the metric components. The quantity dV is scalar, therefore w(n,q) is invariant.Thus, the approach based on the consideration of the isothermal compression of the layer consisted of the incoherent matter, can be used for asymmetrical systems.4. The transformation of the formulas for field energy density of the symmetrical systems.The formulas for these quantities in the paper [1] maintain, besides the metric tensor components, the field source mass M and the distance to symmetry centre R. As the metric tensor components are the functions of M and R, it is possible to except M and R from these formulas.4.1. The cylindrical symmetry. In [1] there are the formulas000a R R g ⎟⎟⎠⎞⎜⎜⎝⎛=, 204c GM a z = (22) where R – radius, R 0 – radius of the field source, M z – the linear mass density. From (23) followRc GM g g z 2001,004= (23) and energy density 002001,0040022322g g g G c g R GM w z ⎟⎟⎠⎞⎜⎜⎝⎛−=−=ππ (24)4.2. The spherical symmetry. From [1] in this caseRc GM g 20021−=; dx 1=dR; dx 2=Rd θ; dx 3=RSin θd φ (25) and energy density⎥⎦⎤⎢⎣⎡+⎟⎠⎞⎜⎝⎛−=R c GM R c GM GR g сw 2222221ln 1600π (26)or from (25) []G g c g g g GR g g сw ππ321ln 11621,004000020020021,004−≅−+−= (27)4.References1.R.Sosnovskiy.gr-qc 05070162.K.S.Virbhadra.A comment on the energy-momentum pseudotensor of Landau and Lifshitz. Phys. Lett.A 157(1991)1953.J.Katz. gr-qc 05100924. N.V.Mitzkevitsch. Physical fields in general relativity. Nauka, Moskow, 19695.J.L.Martin. General Relativity . N.Y.,19886.A.Logunov. Lectures in relativity and gravitation. A modern Look. Nauka, Moskow,19907. L.Lehner. gr-qc 0106072。

数学微积分英汉词典

数学微积分英汉词典

MATHEMATICAL TERMS(Part 1) calculus 微积分definition 定义theorem 定理lemma 引理corollary推论prove 证明proof 证明show 证明solution 解formula 公式if and only if ( iff ) 当且仅当∀x∈X for all x∈X∃x∈X there exists an x∈Xsuch that 使得given 已知set集合finite set有限集infinite set 无限集interval区间open interval开区间closed interval 闭区间neighborhood 邻域number 数natural number 自然数integer 整数odd number 奇数even number 偶数real number 实数rational number 有理数irrational number 无理数positive number 正数negative number 负数mapping 映射function 函数monotone function 单调函数increasing function 增函数decreasing function 减函数bounded function 有界函数odd function 奇函数even function 偶函数periodic function 周期函数composite function 复合函数inverse function 反函数domain 定义域range 值域variable 变量independent variable自变量dependent variable因变量sequence 数列convergent sequence收敛数列divergent sequence 发散数列bounded sequence 有界数列decreasing sequence 递减数列increasing sequence 递增数列limit极限one-sided limit 单侧极限left-hand limit 左极限right-hand limit 右极限The Squeeze Theorem 夹挤定理infinity 无穷大infinitesimal 无穷小equivalent infinitesimal 等价无穷小infinitesimal of higher order 高阶无穷小order of infinitesimal 无穷小的阶infinitesimals of the same order 同阶无穷小increment 增量continuous function 连续函数continuity 连续性f(x) is continuous at x 在x连续f(x) is discontinuous at x 在x间断discontinuity 间断点discontinuity of the first (second) kind 第一(二)类间断点removable discontinuity 可去间断点jump discontinuity 跳跃间断点infinite discontinuity 无穷间断点intermediate value 介值The Intermediate Value Theorem 介值定理zero point 零点The Zero Point Theorem 零点定理root 根equation 方程uniform continuity 一致连续derivative 导数rate of change 变化率velocity 速度instantaneous velocity 瞬时速度tangent (line) 切线normal (line) 法线slope 斜率left-hand derivative 左导数right-hand derivative 右导数f(x) is differentiable at x f(x) 在x处可导(可微) differentiation 求导The Chain Rule 链式法则differentiation formulas 求导公式implicit function 隐函数explicit function 显函数implicit differentiation 隐函数求导logarithm 对数Logarithmic differentiation 对数求导法parameter 参数parametric equation 参数方程parametric curve 参数曲线hyperbolic function 双曲函数hyperbolic sine 双曲正弦hyperbolic cosine 双曲余弦hyperbolic tangent 双曲正切hyperbolic cotangent 双曲余切differential 微分differential quotient 微商approximate value 近似值error 误差relative error 相对误差absolute error 绝对误差invariance of differential form 微分形式不变性higher derivative 高阶导数first derivative 一阶导数second derivative 二阶导数third derivative 三阶导数nth derivative n阶导数twice differentiable 二阶可导acceleration 加速度mean value 中值The Mean Value Theorem 中值定理Rolle’s Theorem 罗尔定理Lagrange’s Mean Value Theorem 拉格朗日中值定理Cauchy’s Mean Value Theorem 柯西中值定理equality 等式inequality 不等式indeterminate form 不定型(未定式) indeterminate form of type 00(∞∞) 00(∞∞)型未定式L’Hospital’s Rule 洛必达法则Taylor’s formula 泰勒公式polynomial 多项式nth-degree polynomial n次多项式remaind 余项Lagrange’s form of remainder 拉格朗日型余项P eano’s form of remainder 皮亚诺型余项Maclaurin formula 麦克劳林公式Taylor polynomial 泰勒多项式Mauclaurin polynomial 麦克劳林多项式polynomial approximation 多项式逼近accuracy 精确度margin 边际marginal cost 边际成本marginal revenue 边际收益elasticity 弹性density 密度mass 质量extreme value 极值local maximum value 极大值local minimum value 极小值(absolute) maximum 最大值(absolute) minimum 最小值stationary point 驻点(稳定点)critical point 临界点The First (Second) Derivative Test (极值的)一(二)阶判别法convex 凸的convex curve 凸曲线concave 凹的concave curve 凹曲线convex function 凸函数point of inflection 拐点asymptote 渐近线horizontal asymptote 水平渐近线vertical asymptote 垂直渐近线slant asymptote 斜渐近线curve sketching 作图sketch a curve 作图curvature 曲率The bisection method 二分法The secant method 弦位法Newton’s method 牛顿(切线)法The tangent method 切线法differential calculus 微分学integral 积分integral calculus 积分学definite integral 定积分indefinite integral 不定积分partition 分割Riemann sum 黎曼和integral sign 积分符号integrand 被积函数upper (lower) limit of integration 积分上(下)限integration 积分(求积)integrable 可积的f(x) is integrable on [a, b]integrable function 可积函数integrability 可积性sufficient condition 充分条件necessary condition 必要条件piecewise continuous 分段连续property 性质The mean value theorem of integral 积分中值定理The fundamental theorem of calculus 微积分基本定理Newton-Leibniz formulaprimitive function (anti-derivative) 原函数(反导数)The substitution rule for integration 换元积分法The inverse of the chain rule 反链式法(凑微分法)integration by parts 分部积分法rational function 有理函数fraction 分式irreducible fraction 最简分式partial fraction 部分分式partial fraction decomposition 部分分式分解vector 矢量free vector 自由矢量zero vector 零矢量magnitude of a vector 矢量的模unit vector 单位矢量scalar product 数量积dot product 点积vector product 矢量积cross product 叉积a is perpendicular (orthogonal) tob a与b垂直coordinate 坐标coordinate system 坐标系coordinate axis 坐标轴x-axis x轴coordinate plane 坐标面direction angle 方向角direction cosine 方向余弦rectangular coordinate system 直角坐标系octant 卦限the first octant 第一卦限variable 变量function of two (three) variables 二(三)元函数function of several variables 多元函数independent variable 自变量dependent variable 因变量domain 定义域range 值域set of points 点集neighborhood 邻域interior point 内点boundary point 边界点bound 边界open set 开集closed set 闭集connected set 连通集region 区域open region 开区域closed region 闭区域bounded region 有界区域unbounded region 无界区域cluster point 聚点double limit 二重极限iterated limit 累次极限continuity 连续性increment 增量total increment 全增量partial increment 偏增量partial derivative 偏导数partial derivative of f(x,y) with respect to x ( y ) f(x,y)关于x(y)的偏导数higher partial derivative 高阶偏导数mixed partial derivative 混合偏导数Laplace equation 拉普拉斯方程total differential 全微分differentiable 可微chain rule 链式法则implicit function 隐函数implicit differentiation 隐函数微分法Jacobian determinant 雅可比行列式curve 曲线space curve 空间曲线tangent vector 切矢tangent line 切线normal plane 法平面surface 曲面normal vector 法矢normal line 法线tangent plane 切平面sphere 球面cylinder 柱面cone 锥面directional derivative 方向导数gradient 梯度gradient vector 梯度矢量f del flevel curve 等值线level surface 等值面local extremum 极值local maximum 极大值local minimum 极小值extreme value 最值absolute maximum (minimum) 最大(最小)值stationary point (critical point) 驻点(临界点)conditional extremum 条件极值Lagrange multiplier 拉格朗日乘数method of Lagrange multiplier 拉格朗日乘数法objective function 目标函数constraint 约束条件method of least square 最小二乘法field 场scalar field 数量场vector field 矢量场gradient field 梯度场potential field 势场potential function 势函数conservative field 保守场gravitational field 引力场force field 力场velocity field 速度场multiple integral 重积分double integral 二重积分iterated integral 累次积分region 区域region of integration 积分区域type X (Y) region X(Y)型区域order of integration 积分秩序reverse the order of integration 交换积分秩序polar coordinates 极坐标double integrals in polar coordinates 极坐标下的二重积分volume 体积lamina 平面薄片mass 质量density 密度moment about x-axis 关于x轴的(静)力矩center of mass 重心moment of inertia 转动惯量surface 曲面area of a surface 曲面的面积triple integral 三重积分rectangle 矩形rectangular coordinates 直角坐标系cylinder 柱面cylindrical coordinates 柱面坐标系sphere 球面spherical coordinates 球面坐标系change of variables in multiple integrals 重积分的变量替换Jacobian determinant 雅可比行列式line integral 曲线积分line integral with respect to arc length 对弧长的曲线积分(第一型)line integral with respect to x ( y ) 对坐标x(y)曲线积分(第二型)line integral of a vector field 向量场的曲线积分smooth curve 光滑曲线piecewise smooth curve 逐段光滑曲线oriented curve 有向曲线orientation of a curve 曲线的方向work 功the line integral is independent of path 曲线积分与路径无关connected region 连通区域simply-connected region 单连通区域closed curve 闭曲线Green’s theorem 格林定理(公式)positive orientation of a curve 曲线的正向Fundamental theorem for line integrals 曲线积分的基本定理surface integral 曲面积分surface integral of a scalar field 数量场的曲面积分(第一型)surface integral of a vector field 向量场的曲面积分(第二型)orientable surface 可定向曲面oriented surface 有向曲面Möbius strip 莫比乌斯带Klein bottle 克莱因瓶one-sided surface 单侧曲面two-sided surface 双侧曲面closed surface 闭曲面flux 流量、通量electric flux 电通量divergence 散度rotation (curl) 旋度Gaus s’ theorem 高斯定理(公式)The divergence theorem 散度定理(公式)Stokes’ theorem 斯托克斯定理(公式)curl theorem 旋度定理(公式)circulation of v around L v沿L的环流量Hamilton operator 哈密顿算子harmonic field 调和场11。

gravitational field

gravitational field

补充: Ek=GMm/2r
W=m×g
365
687 4333
3.36*1018
3.36*1018 3.36*1018
Saturn Uranus
Neptune Moon geostationary satellite
1426 2870
4498 0.3844 0.0424
10759 30660
60148 27.3 1
3.36*1018 3.37*1018
gravitational field
gravitational potential (φ) : work done in bringing unit mass from infinity to that point.
Try to explain the “-” sign in the gravitational potential. The gravitational force is attractive. When 2 masses come closer, positive work is done by the gravitational force.
potential well
gravitational field
3. orbiting under gravity
(1). orbital speed (first cosmic speed): , 7.9km/s r v
(2). general case: Fc=Fg
For a certain orbit, all masses travel at the same speed.
Exercise 1: The earth has radius 6400 km. The gravitational field strength on the surface is 9.81N/kg. Use this information to determine the mass of the earth and its mean density.

牛顿的万有引力定律英语

牛顿的万有引力定律英语

牛顿的万有引力定律英语Newton's Law of Universal Gravitation is a fundamental principle in physics that describes the gravitational force between any two objects in the universe. This law was formulated by the renowned English mathematician and physicist Sir Isaac Newton in the late 17th century and has since become a cornerstone of classical mechanics.The origins of Newton's work on gravity can be traced back to his early life and education. Born in 1642 in Woolsthorpe Manor, Lincolnshire, England, Newton was a precocious child with a keen interest in the natural world. As a student at the University of Cambridge, he began to develop his theories on the motion of celestial bodies, building upon the work of earlier scientists such as Galileo Galilei and Johannes Kepler.One of the key events that led to the formulation of the Law of Universal Gravitation was the observation of the motion of the planets around the Sun. Kepler had already established three laws of planetary motion, but the underlying cause of these patterns remained a mystery. Newton, through his mathematical and scientificprowess, was able to unify these observations into a single, comprehensive theory.The essence of Newton's Law of Universal Gravitation can be summarized as follows: every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This can be expressed mathematically as the equation:F =G * (m1 * m2) / r^2Where F is the force of gravity between the two objects, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.The implications of this law are far-reaching and have had a profound impact on our understanding of the universe. It explains the motion of the planets around the Sun, the behavior of tides, the acceleration due to gravity on Earth, and even the motion of galaxies and the large-scale structure of the cosmos.One of the most remarkable aspects of Newton's Law of Universal Gravitation is its universality. It applies not only to the motion of celestial bodies but also to everyday objects on Earth. The sameforce that keeps the Moon in orbit around the Earth also governs the fall of an apple from a tree. This unification of the terrestrial and celestial realms was a groundbreaking achievement that revolutionized our understanding of the physical world.Newton's work on gravity also had a significant impact on the development of other areas of physics. His laws of motion, which describe the relationship between an object's mass, acceleration, and the forces acting upon it, are fundamental to the study of classical mechanics. These laws, combined with the Law of Universal Gravitation, form the foundation of Newtonian mechanics, which dominated the field of physics for over two centuries.Despite the enduring success of Newton's theory, it is important to note that it is not a complete or perfect description of gravity. In the early 20th century, Albert Einstein's theory of general relativity provided a more comprehensive and accurate understanding of gravitational phenomena, particularly in the realm of high-energy physics and the behavior of massive objects in the universe.General relativity, which describes gravity as a distortion of space-time rather than a force, has been extensively tested and verified through numerous experiments and observations. However, Newton's Law of Universal Gravitation remains a highly useful and accurate approximation for the vast majority of everyday situationsand is still widely used in various fields, such as astronomy, engineering, and space exploration.In conclusion, Newton's Law of Universal Gravitation is a landmark achievement in the history of science that has profoundly shaped our understanding of the physical world. Its simplicity, elegance, and universal applicability have made it a cornerstone of classical physics, and its influence continues to be felt in the ongoing pursuit of scientific knowledge and the exploration of the universe.。

引力场中真空折射率

引力场中真空折射率

引力场中真空折射率英文回答:The refractive index of vacuum in a gravitational field is a fascinating topic to explore. When discussing the refractive index, we are essentially referring to how light propagates through a medium. In a vacuum, where there is no material medium, the refractive index is typically considered to be 1. However, in the presence of a gravitational field, such as near a massive object like a black hole, the refractive index can be affected.According to general relativity, gravity can cause the curvature of spacetime. This curvature can influence the path of light, leading to a phenomenon known as gravitational lensing. Gravitational lensing occurs when the gravitational field of a massive object bends the path of light passing nearby. This bending can result in the light being deflected and distorted, similar to how a regular lens can bend and focus light.In the context of refractive index, we can think of the gravitational field as a medium that affects the propagation of light. The presence of a gravitational field can alter the speed of light and change its direction. This change in speed and direction can be quantified by a modified refractive index.For example, imagine you are near a massive star, and you are observing a distant object through a telescope. The light from the distant object would pass through the gravitational field of the star before reaching your telescope. The gravitational field would cause the light to deviate from its original path, resulting in a change in direction. This change in direction can be thought of as a change in the refractive index of the vacuum in the gravitational field.Another example is the phenomenon of gravitational time dilation. According to general relativity, time can be dilated in the presence of a gravitational field. This means that time can appear to pass slower in a strongergravitational field compared to a weaker one. This time dilation can also affect the refractive index of the vacuum. The change in the passage of time can lead to a change in the speed of light, which in turn affects the refractive index.In summary, the refractive index of vacuum in a gravitational field can be different from the standardvalue of 1. The presence of a gravitational field can alter the speed and direction of light, leading to a modified refractive index. This phenomenon is observed ingravitational lensing and gravitational time dilation. Understanding the effects of gravity on the refractiveindex can provide insights into the behavior of light in extreme gravitational environments.中文回答:真空在引力场中的折射率是一个非常有趣的研究课题。

地球物理场英语

地球物理场英语

地球物理场英语The Earth's Geophysical FieldsThe Earth is a complex and dynamic planet, with a wide range of geophysical fields that play a crucial role in shaping its environment and influencing various natural phenomena. These fields are the result of complex interactions between the Earth's internal structure, its rotation, and the external forces acting upon it. Understanding and studying these geophysical fields is essential for a deeper understanding of the Earth's systems and processes.One of the most fundamental geophysical fields is the Earth's gravitational field. This field is generated by the mass of the Earth and is responsible for the attraction of objects towards the planet's surface. The strength of the gravitational field varies depending on the location, with higher values near the Earth's surface and lower values at higher altitudes. This field is not only important for the motion of objects on the Earth's surface but also plays a crucial role in the orbits of satellites and the dynamics of the solar system.Another important geophysical field is the Earth's magnetic field. This field is generated by the Earth's core, which is primarilycomposed of molten iron and nickel. The rotation of the Earth's core, combined with the convection of the molten material, creates a dynamo effect that generates the magnetic field. The magnetic field is responsible for protecting the Earth from harmful solar radiation and cosmic rays, and it also plays a role in the navigation of many species, including migratory birds and marine animals.The Earth's electromagnetic field is also a significant geophysical field. This field is generated by the movement of charged particles in the Earth's atmosphere and ionosphere, as well as by the interaction between the Earth's magnetic field and the solar wind. The electromagnetic field is important for various communication and navigation systems, as well as for the study of space weather and the effects of solar activity on the Earth's environment.In addition to these fundamental geophysical fields, the Earth also has a range of other fields that are equally important. For example, the Earth's seismic field, which is generated by the propagation of seismic waves through the Earth's interior, is crucial for the study of the planet's internal structure and the detection of earthquakes and other geological events. The Earth's thermal field, which is generated by the heat flow from the planet's interior, is also an important geophysical field that influences the distribution of temperature and the movement of tectonic plates.The study of these geophysical fields is not only important for our understanding of the Earth's systems but also has practical applications in a wide range of fields, including geology, climatology, environmental science, and engineering. For example, the study of the Earth's gravitational field is used in the development of satellite navigation systems, while the study of the Earth's magnetic field is used in the exploration for natural resources and the development of new technologies for renewable energy.In conclusion, the Earth's geophysical fields are a complex and fascinating aspect of our planet's systems. By studying these fields, scientists and researchers can gain valuable insights into the Earth's history, its present-day processes, and its future evolution. As we continue to explore and understand these fields, we can better appreciate the incredible complexity and beauty of our home planet.。

gravitational field 课件

gravitational field 课件
Newton and rce
How to calculate this attractive force (gravitational force)?
Gravitational Force
Gravitational force Masses Product of masses Distances Square of distances
Gravitational Potential
V
Gravitational Field Lines
Draw the gravitational field lines for a spherical object of radius r on a plane.
Equipotential & Potential Wall
equipotential surfaces
Equipotential lines Field lines
They are perpendicular to each other. • Addition of potentials: vectors
What is the potential at A?
An object with a mass of 48Kg measured on Earth is taken to the Moon. What is the weight of the object on the Moon‟s surface of the acceleration due to gravity on Moon is one-sixth of that on Earth? A. 8N B. 48N C. 288N D. 80N
moon
20 00 Kg

Gravitational Faraday rotation in a weak gravitational field

Gravitational Faraday rotation in a weak gravitational field

a rXiv:as tr o-ph/41295v115Jan24Gravitational Faraday rotation in a weak gravitational field Mauro Sereno ∗Dipartimento di Scienze Fisiche,Universit`a degli Studi di Napoli “Federico II”,and Istituto Nazionale di Fisica Nucleare,Sez.Napoli,via Cinthia,Compl.Univ.Monte S.Angelo,80126Napoli,Italia (Dated:December 10,2003)Abstract We examine the rotation of the plane of polarization for linearly polarized light rays by the weak gravitational field of an isolated physical system.Based on the rotation of inertial frames,we review the general integral expression for the net rotation.We apply this formula,analogue to the usual electromagnetic Faraday effect,to some interesting astrophysical systems:uniformly shifting mass monopoles and a spinning external shell.PACS numbers:04.20.Cv,04.70.Bw,04.25.Nx,95.30.Sf Keywords:Gravitational lensingI.INTRODUCTIONElectromagnetic theory in a curved space-time,in the approximation of geometric optics, provides some of the most well-known and stringent tests of Einstein’s general theory of gravitation.Under geometric optics,a ray follows a null geodesic regardless of its polariza-tion state and the polarization vector is parallel transported along the ray[1].In the last decades,observations of both bending of light and gravitational time delay have revealed themselves as a powerful tool in observational astrophysics and cosmology.These phenom-ena are fully accounted for in the gravitational lensing theory[2,3].On the other hand, effects of polarization along the light path have not yet been measured.The polarization vector of a linearly polarized electromagnetic wave rotates due to the properties of the space-time.The gravitational rotation of the plane of polarization in stationary space-times is a gravitational analogue of the electro-magnetic Faraday effect, i.e.,the rotation that a light ray undergoes when passing through plasma in the presence of a magneticfield.The analogy wasfirst noted in[4],where the problem of nonlinear interaction on gravitational radiation was considered.Thefirst discussion of this relativistic effect goes back to1957,when Skrotskii[5]applied a method previously developed by Rytov[6]to consider geometric optics in a curved space-time.For this historical reason,the gravitational effect on the polarization of light rays is also known as Skrotskii or Rytov effect.In1958,Balazs[7]further stressed how the gravitationalfield of a rotating body may behave as an optically active medium.In1960, Plebanski[8]solved the Maxwell’s equations in the gravitationalfield of an isolated physical system.He showed how the polarization vector changes its direction due to the deflection of the light ray,and,in addition to this change,how a rotation of the plane of polarization around the propagation vector may occur.Ten years later,Godfrey[9]took a very different approach.Following Mach’s principle,he considered dragging of inertial frames along with a rotating body and obtained an approximate expression for the rotation of the polarization vector of a light ray propagating along the rotation axis of a Kerr black hole.Trajectories initially propagating parallel to the symmetry axis of a central spinning body were studied in[10],where the problem was formulated in a cylindrical-like Kerr solution.A different situation was considered in[11],where it was discussed how the polarization features of X-ray radiation emitted from an accretion disk surrounding a rotating black hole are alsostrongly affected by general-relativistic effects.The relativistic rotation of the plane of polarization was further studied in[12].Solving the equations of motion of a light ray in the first post-Minkowskian approximation,a formula describing the Skrotskii effect for arbitrary translational and rotational motion of gravitating bodies was derived.Finally,the Skrotskii effect on light rays propagating in the vacuum region outside the event horizon of a Kerr black hole has been discussed in[13,14].In particular,the formu-lation in[14]stressed in an illuminating way the analogy with the usual Faraday effect.In this paper,we explore the gravitational Faraday rotation by the gravitationalfield of an isolated system(lens)when the source of radiation and the observer are remote from the gravitational lens.We restrict to the weak gravitationalfield far from the lens,and analyze it using linearized theory.This approximation holds for almost all gravitational lensing phenomena.We consider gravitational Faraday rotation by usual astrophysical systems, such as a system of shifting stars acting as lenses or a galaxy deflecting light rays emitted from background sources.The paper is as follows.In Section II,we extend the argument of Godfrey[9]on dragging of inertial frames to reobtain the well known general formula for the angle of rotation of the plane of polarization of a linearly polarized electromagnetic wave in a stationary space-time. This heuristic approach allows us to face the problem without integrating the equation of motion.In Sec.III,the weak-field,slow motion approximation is introduced and the weak field limit of the gravitational Faraday rotation is performed.In Section IV,we evaluate the Faraday rotation for some systems of astrophysical interest.We examine a system of uniformly moving lenses and a rotating external shell.Section V is devoted to somefinal considerations.II.DERIV ATION OF THE GRA VITATIONAL F ARADAY ROTATIONLet us consider a stationary space-time embedded with a metric gαβ[21].Such a metric can be written as[15]ds2=h dx0−A i dx i 2−dl2P(1) where we have introduced the notationh≡g00,A i≡−g0ianddl P2≡ −g ij+g0i g0j√22 obs sou√determined by off-diagonal components of the metric.To our aim,it is enough to writeds 2≃ 1+2φc 3− 1−2φ| x − x ′|d 3x ′;(6)φ/c 2is of order ∼O (ε2).Vis a vector potential taking into account the gravito-magnetic field produced by mass currents.To O (ε3),V (t, x )≃−G ℜ3(ρ v )(t, x ′)c 2+O (ε4),(8)A i ≃4c 2+O (ε3) dl E ,(10)where dl E ≡c 3 p ∇× V ·ˆk dl E +O (ε5),(11)where p is the spatial projection of the null geodesics and ˆkis the unit tangent vector.It is useful to employ the spatial orthogonal coordinates (l,ξ1,ξ2)≡(l,ξ),centred on the lens and such that the l -axis is along the incoming,unperturbed light ray direction e in .The lens plane,(ξ1,ξ2),corresponds to l =0.The three-dimensional position vector to the light ray x can be written as x =ξ+l e in .To calculate the Skrotskii effect to order O(ε3)we can adopt the Born approximation, which assumes that rays of electromagnetic radiation propagate along straight lines,i.e,the bending of the ray may be neglected.The integration along the line of sight(l.o.s.)is accurate enough to evaluate the main contribution to the net rotation[17].To this order, we can employ the unperturbed Minkowski metricηαβ=(1,−1,−1,−1)and a constant unit propagation vector of the signal,ˆk(0)=(1,0,0).The Faraday rotation to order O(ε3)readsΩSk≃−2| x− x i|.(13) The rotation angle from a system of N shifting lenses isΩSk≃−2c3Ni M i∆ξ(i)1v(i)2−∆ξ(i)2v(i)1B.Rotating shellThe gravito-magnetic potential takes a very simple form in the case of a spherically symmetric distribution of matter in rigid rotation.We limit to a slow rotation so that the deformation caused by rotation is negligible and the body has a nearly spherical symmetry.Taking the centre of the source as the spatial origin of a background inertial frame,we get V ≃−4πx 3 x 0ρ(r )r 4dr + +∞xρ(r )rdr ω× x =−G x 3−4π3 x 0ρ(r )r 4dr ωis the angular momentum contributed from the matter within a radius x ≡| x |.Einstein’s gravitational theory predicts peculiar phenomena inside a rotating shell.It is interesting to calculate the gravito-magnetic potential for such a system.The gravito-magnetic potential in Equation (15),inside a uniform spherical shell of mass M ,radius R and rotating with constant frequency,reduces to (see also [18])V In ( x )≃−GM 3ω× x2 J × x 3MR 2 ω.It is ∇× V In ( x )≃−2GM2J −3( J ·ˆx )ˆx R 2−ξ2and l Out =+ c 3l In−∞∇× V Out l .o .s .dl E + l Out l In ∇× V In l .o .s .dl E + +∞l Out ∇× V Out l .o .s .dl E +O (ε5)=4GM 1− ξThe result vanishes if the angular velocity lies in the lens plane.Since the gravitational Faraday rotation outside a rotating body,when the light path does not enter the lens,is ∼G2M TOTξ3,i.e,of order O(ε5)[13,14,19],the effect on the light ray can be neglectedat this order of approximation.The case of a rotating external sphere offinite thickness can be easily solved just integrat-ing the result in Eq.(20).Every infinitesimal shell of radius r′with mass dM=4πρ(r′)r′2dr′and angular velocityω(r′)contributes an angledΩSk≃16πG r′2−ξ2r′dr′+O(ε5).(21) Integrating from the impact parameter,ξ,to the external shell radius R,we get the total gravitational Faraday rotation which a light ray undergoes because of the spin of the external shell.We getΩSk= dΩSk≃16πG r′2−ξ2r′dr′+O(ε5).(22) Let us consider a homogeneous sphere of constant density in rigid rotation.The plane of polarization of a light ray,that penetrates through this rotating body,is rotated ofΩSk≃16πGc3J l.o.s.(R2−ξ2)3/2microlensing events on the Galactic scale,a star,acting as deflector,moves relatively to a background source.This is the case of a shifting lens.A distant quasar lensed by a foreground galaxy may form images inside the galaxy radius. In such an astrophysical configuration,photons propagate inside a rotating shell.Since the Faraday rotation due to external rotating shell is of order O(ε3),it could induce a detectable effect.High quality data in totalflux density,percentage polarization and polarization position angle at radio frequencies already exist for multiple images of some gravitational lensing systems,like B0218+357[20].The prospects to detect the gravitational Faraday rotation will be the argument of a forthcoming paper.AcknowledgmentsThe author wishes to thank the Dipartimento di Fisica“E.Caianiello”,Universit`a di Salerno,Italia,for the hospitality when hefirst worked on the idea illustrated in the paper.[1]Misner,C.,Thorne,K.S.,Wheeler,J.A.,1973,Gravitation,Freeman,San Francisco[2]Petters A.O.,Levine H.,Wambsganss J.,2001,Singularity Theory and Gravitational Lensing,Birkh¨a user,Boston[3]Schneider P.,J.Ehlers J.,Falco E.E.,1992,Gravitational Lenses,(Springer,Berlin)[4]Piran,T.,Safier,P.N.,1985,Nat,318,271[5]Skrotskii,G.B.,1957,Dokl.Akad.Nauk USS,114,73[6]Rytov,S.M.,C.R.(Dokl.)Acad.Sci.URSS,18,263[7]Balazs,N.L.,1958,Phys.Rev.110,236[8]Plebanski,J.,1960,Phys.Rev,118,1396[9]Godfrey,B.B.,1970,Phys.Rev.D,1,2721[10]Su,F.S.O.,Mallett,R.L.,1980,ApJ,238,1111[11]Connors,P.A.,Piran,T.,Stark,R.F.,1980,ApJ,235,224[12]Kopeikin,S.,Mashoon,B.,2002,Phys.Rev.D,65,064025[13]Ishihara,H.,Takahashi,M.,Tomimatsu,A.,1988,Phys.Rev.D,38,472[14]Nouri-Zonoz,M.,1999,Phys.Rev.D,60,024013[15]Landau L.D.,Lifshits E.M.,1985,Teoria dei Campi,Editori Riuniti,Roma[16]Mashoon,B.,1975,Phys.Rev.D,11,2679[17]Sereno,M.,2003,Phys.Rev.D,67,064007;[astro-ph/0301290].[18]Ciufolini,I.,Kopeikin,S.,Mashoon,B.,Ricci,F.,2003,Phys.Lett.A,308,101[19]Sereno,M.,2003,in preparation.[20]Biggs,A.D.,Browne,I.W.A.,Helbig,P.,Koopmans,L.V.E.,Wilkinson,P.N.,Perley,R.A.,1999,MNRAS,304,349[21]Latin indices run from1to3,whereas Greek indices run from0to3.[22]The controvariant components of spatial three-vectors are equal to the spatial componentsof the corresponding four-vectors.Operations on such three-vectors are defined in the three-dimensional space with metricγαβ.。

引力波中的相关英语高考考点

引力波中的相关英语高考考点

引力波中的相关英语高考考点英语可能会在阅读理解中出关于引力波的题目。

相关词汇一定要搞清楚。

引力波 gravitational wave1.由“广义相对论”所预言的“引力子”和“引力波”不存在。

According to the “ general relativity ” predict “ graviton ” and“ gravitational waves ” does not exist.2.因此,高斯束谐振系统对高频遗迹引力波的频率和传播方向具有良好的选择效应。

Therefore, GBRS have a useful selective effect with respect to the frequency and propagation direction of relic HFGWs.3.引力规范理论中的一类引力波方程A Class of Gravitational Waves Equation in Gravitational Gauge Theory4.对物质体系在发射和接收引力波时的能量转换作了新解释.A new interpretation for the energy exchanges of the matter system is given when there exists the gravitational wave.5.谐和条件下的对角度规引力波方程Gravitational Wave Equations under Diagonal Metric and Harmonic Coordinate Conditions6.杨振宁场引力波的极化Polarization of the gravitational waves of yang's gravitational field7.宇宙常数Λ≠0的平面引力波The Plane Gravitational Waves with the Cosmological Constant Λ≠ 08.一种标&张量引力理论的引力波辐射Radiation of gravitational waves in a scalar-tensor theory of gravitation9.De Sitter弯曲时空中遗迹引力波及其能量动量赝张量的表述和正定性问题Relic Gravitational Wave and Positive Definite and Expression of Their Energy-Momentum Pseudo-Tensor in De sitter Background Spacetime of the Curve10.在室内模型激光干涉引力波探测器的基础上,几个野外大型激光干涉引力波探测器正在紧张地建设中。

Gravitational Radiation from Oscillating Gravitational Dipole

Gravitational Radiation from Oscillating Gravitational Dipole
Gravitational Radiation from Oscillating Gravitational Dipole
Fran De Aquino
Maranhao State University, Physics Department, S.Luis/MA, Brazil. deaquino@uema.br
1. INTRODUCTION
When the gravitational field of an object changes, the changes ripple outwards through space and take a finite time to reach other objects. These ripples are called gravitational radiation or gravitational waves . The existence of gravitational waves follows from the General Theory of Relativity. In Einstein's theory of gravity the gravitational waves propagate at the speed of light. Just as electromagnetic waves (EM), gravitational waves (GW) too carry energy and momentum from their sources. Unlike EM waves, however, there is no dipole radiation in Einstein's theory of gravity. The dominant channel of emission is quadrupolar. But the recent experimental discovery of negative gravitational mass1,2 suggest the possibility of dipole radiation. This fact is highly relevant because a gravitational wave transmitter can be designed to generate detectable levels of gravitational radiation in the laboratory. Here, we will study the theory and design of the oscillating gravitational dipole. property of mass has two distinct aspects, gravitational mass mg and inertial mass mi . The inertial mass is the mass factor in Newton's 2nd Law of Motion (F = mi a ), while the gravitational mass produces and responds to gravitational fields. It supplies the mass factors in Newton's famous inverse-square law of gravity 2 ( F12 = Gmg1mg 2 r12 ; G is the Newton's gravitational constant). According to the weak form of Einstein’s General Relativity equivalence principle, the gravitational and inertial masses are equivalent. However recent calculations 3 have revealed that they are correlated by an adimensional factor, which depends on the incident radiation upon the particle. It was shown that there is a direct correlation between the radiation absorbed by the particle and its gravitational mass, independently of the inertial mass. It was also shown that only in the absence of electromagnetic radiation this factor becomes equal to one and that, in specific electromagnetic conditions, it can be reduced, nullified or made negative. This means that we can reduce, nullify or make negative the gravitational mass of a body. This unexpected theoretical result has been confirmed by two experiments using Extremely Low Frequency (ELF) radiation upon ferromagnetic material1,2.

星体的重力势能和所受引力

星体的重力势能和所受引力

星体的重力势能和所受引力The gravitational potential energy of a celestial bodyis closely related to the gravitational force it experiences. Let's explore this concept from various perspectives.Firstly, let's understand what gravitational potential energy is. Gravitational potential energy is the energy possessed by an object due to its position in agravitational field. In the context of celestial bodies, such as stars or planets, this potential energy arises from the gravitational force between them. The greater the massof the celestial body, the stronger its gravitational force, and thus the higher its gravitational potential energy.From a mathematical perspective, the gravitational potential energy (U) of an object can be calculated using the formula U = -GMm/r, where G is the gravitational constant, M and m are the masses of the two objects, and ris the distance between their centers of mass. Thisequation indicates that the potential energy is inversely proportional to the distance between the objects. As the distance increases, the potential energy decreases, and vice versa.Now, let's consider the gravitational force experienced by a celestial body. The gravitational force is the attractive force between two objects with mass. In the case of celestial bodies, it is the force that keeps them in their orbits or causes them to attract other objects towards them. The force of gravity is directly proportional to the product of the masses of the two objects and inversely proportional to the square of the distance between them, as described by Newton's law of universal gravitation.The gravitational force exerted by a celestial body can be calculated using the equation F = (GMm)/r^2, where F is the gravitational force, G is the gravitational constant, M and m are the masses of the two objects, and r is the distance between their centers of mass. This equation shows that the force of gravity decreases as the distance betweenthe objects increases.The relationship between gravitational potential energy and gravitational force can be understood by considering the work done by gravity. When an object moves in a gravitational field, the force of gravity does work on it, which is equal to the change in gravitational potential energy. If an object is moved closer to a celestial body, the work done by gravity is negative, indicating a decrease in potential energy. Conversely, if the object is moved away, the work done by gravity is positive, indicating an increase in potential energy.In summary, the gravitational potential energy of a celestial body is determined by its mass and the distance between it and other objects. The greater the mass, the higher the potential energy, and the closer the distance, the higher the potential energy. The gravitational force experienced by a celestial body is directly related to its mass and the distance between it and other objects. The force of gravity decreases with increasing distance. Understanding these concepts helps us comprehend thedynamics of celestial bodies and their interactions in the vast expanse of the universe.。

Gravitational field, potential and energy - SJHS-I

Gravitational field, potential and energy - SJHS-I

The gravitational potential energy of a mass at any point is defined as the work done in moving the mass from infinity to that point.
.
Q. What do the indicated properties of these two graphs represent?
Ep = - GMm r
E.g. Calculate the potential energy of a 5kg mass at a point 200km above the surface of Earth.
( G = 6.67 10-11 N m2 kg-2 , mE=. 6.0 1024 kg, rE= 6.4 106 m )
Thus for a field due to a (point or spherical) mass M:
true zero of GPE is arbitrarily taken not as Earth’s
surface but at ‘infinity’. If work must be done to
Ep = 0
‘Iass from near Earth to zero at infinity then at all points
a
b
.
.
Gravitational Potential Whereas gravitational force on an object on Earth depends upon the mass of the object itself, gravitational field strength is a measure of the force per unit mass of an object at a point in Earth’s field. Similarly, whereas the GPE of say a satellite, depends upon both the mass of Earth and the satellite itself, gravitational potential is a measure of the energy per unit mass at a point in Earth’s field.
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1
Introduction
During several Earth flybys carried out since 1990, some spacecrafts have experienced an unexpected and until now unexplained anomalous velocity increase by a few mm/s. This phenomenon is called the flyby anomaly and looks like the effect of an instantaneous acceleration of the spacecraft at the time of closest approach to Earth. The measured velocity differences could be approximately reproduced by estimation of a sudden change of the velocity vector at perigee [2]. Another approximate simulation was possible by means of estimation of the gravitational harmonic coefficients J2, C21, S21, C22 and S22 of the Earth’s gravity field. However, the estimated values for these coefficients were by far unreasonable, and moreover different for the Galileo1 and the NEAR flyby [2]. Many efforts have been made to find a reason for this acceleration, but none of them were able to give an explanation based on known physics. Likewise without success were attempts to modify the law of gravitation in order to get better agreement with the observations. A good summary of these efforts is given by [2, 4, 5]. Additionally, a comprehensive overview of gravity tests is provided by [6]. Further attempts were made to find at least phenomenological patterns in the flyby data, i.e., relationships between eccentricity, perigee altitude and perturbations by the Sun, Moon or planets [3, 4]. Due to the poor database of only a few flybys, these attempts were without success, as well. Hence it is yet unclear, whether an unconsidered interaction based on known physics is responsible for this mysterious effect, or if the known physical laws, especially General Relativity, have to be modified. After more than 15 years of careful analyses it seems unlikely to find an explanation based on the standard physical laws. Therefore, in this investigation another attempt was made to find an addendum to Newton’s law of gravitation, which could simulate all observed anomalies, at first without any regards to a consistent theory. To this end three basic assumptions for the structure of the additional gravitational field term were made:
Simulation of the flyby anomaly by means of an empirical asymmetric gravitational field with definite spatial orientation.
H. J. BusaБайду номын сангаасk
Wulfsdorfer Weg 89, 23560 Lübeck, Germany 27. November 2007
1
1. 2. 3.
No spherical symmetry to the center of the Earth No fixed orientation of the asymmetry to the surface coordinate system of the Earth Short interaction distance
The reason for item 1 seems to be obvious, since in the case of spherical symmetry and short interaction distance of the additional field one would expect no difference to the velocity magnitude of a Newtonian trajectory for long and equal geocentric distances because the trajectory would be symmetric, too. This has been proved by numerical integration of trajectories with Newtonian and symmetric non-Newtonian gravity fields. Item 2 is necessary, because otherwise the asymmetry would easily be found by analysis of the orbits of Earth satellites used for determination of the Earth’s gravity field. This point will be discussed later in more detail. Item 3 is justified by the lack of an unexplained acceleration in the motions of geostationary satellites, and by the nearly instantaneous onset of the acceleration near the point of closest approach (CA) of a flyby trajectory. Based on these assumptions, a more specific mathematical formulation for a numerical simulation was found. This formula was implemented in a computer program for simulating the above-mentioned flybys. The free parameters of the equation were varied in order to achieve compliance with the measured anomaly values of all flybys. Clearly such simulation is far away from a consistent theory, but, if successful, can potentially show the direction of a next step.
Abstract
All anomalous velocity increases until now observed during the Earth flybys of the spacecrafts Galileo, NEAR, Rosetta, Cassini and Messenger have been correctly calculated by computer simulation using an asymmetric field term in addition to the Newtonian gravitational field. The specific characteristic of this term is the lack of coupling to the rotation of the Earth or to the direction of other gravitational sources such as the Sun or Moon. Instead, the asymmetry is oriented in the direction of the Earth’s motion within an assumed unique reference frame. With this assumption, the simulation results of the Earth flybys Galileo1, NEAR, Rosetta1 and Cassini hit the observed nominal values, while for the flybys Galileo2 and Messenger, which for different reasons are measured with uncertain anomaly values, the simulated anomalies are within plausible ranges. Furthermore, the shape of the simulated anomaly curve is in qualitative agreement with the measured Doppler residuals immediately following the perigee of the first Earth flyby of Galileo. Based on the simulation, an estimation is made for possible anomalies of the recently carried out flybys of Rosetta at Mars on 25.02.07 and at the Earth on 13.11.07, and for the forthcoming Earth flyby on 13.11.09. It is discussed, why a so modelled gravitational field has not been discovered until now by analysis of the orbits of Earth satellites, and what consequences are to be considered with respect to General Relativity.
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