aa r X i v : . [ phy s i c s . h i s t - ph ] A p r On a Schwarzschild like metric
M. Anastasiei and I. Gottlieb
Dedicated to the 85th birthday of Professor Cleopatra Mociut¸chi
Abstract.
In this short Note we would like to bring into the attention of peo-ple working in General Relativity a Schwarzschild like metric found by ProfessorCleopatra Mociut¸chi in sixties. It was obtained by the A. Sommerfeld reasoningfrom his treatise ”Elektrodynamik” but using instead of the energy conservinglaw from the classical Physics, the relativistic energy conserving law.
To begin with, let’s recall the laws of Newton’s mechanics :1. A particle moves with constant velocity if no force acts on it.2. The acceleration of a particle is proportional to the force acting on it.3. The forces of action and reaction are equal and opposite.These laws hold in their simplest form only in inertial frames. The fact isstated as the Galilean principle of relativity :
Galilean PR : The laws of mechanics have the same form in all inertial frames.
It implies the classical velocity addition law that has many confirmations inclassical mechanics but does not hold for the propagation of light. The speed oflight c is the same in all inertial frame. Thus the PR was extended in the formof the next two postulates : 1 These are basic for Special Relativity (SR). The SR is due to A.Einstein whocame at the conclusion that the concepts of space and time are relative i.e. de-pendent on the reference frame of an observer. Mathematically, the postulatesP1 and P2 implies that two inertial frames are connected by a Lorentz transfor-mation of coordinates. This fact leads to a new law for the addition of velocitiesand to two real physical effects : length contraction and time dilatation. TheLorentz transformations show also that is more natural to speak about spacetime as a set of points (events) with four coordinates than about space and time sep-arately. The squared differential distance between neighboring events is given bythe Lorentz invariant quadratic form ds = c dt − dx − dy − dz . (1.1)However, the non-inertial (accelerated) frames do exist. A. Einstein noticed thatthe dynamical effects of a gravitational field and an accelerated frame can not bedistinguished and he formulated the principle of equivalence (PE). P E : Every non-inertial frame is locally equivalent to some gravitational field. This is more completely restated as
P E : The gravitational field is locally equivalent to the field of inertial forcesof a convenient accelerated frame of reference. A freely falling reference frame is called a locally inertial frame. Using thisterm the PE can be also restated as
P E : At every point in an arbitrary gravitational field we can choose a locallyinertial frame in which the laws of Physics take the same form as in SR. A. Einstein considered SR as incomplete because of the role played by theinertial frames. He generalized the relativity of inertial motions to the relativityof all motions by formulating the general principle of relativity.
General PR : The form of physical laws is the same in all reference frames.
Mathematically, the general PR can be realized with the help of the principleof general covariance.
General covariance : The form of physical laws does not depend on the choiceof coordinates.
This principle says that the physical laws have to be given in tensorial form.For details and historical motivations of the principles reviewed in the above werefer to the first chapter of the book [1]2
The Einstein equations
Let’s relabel the time and the spatial variable in a spacetime M x = ( x = t, x , x , x ) . Then (1.1) is a particular form of the following (the Einstein ’sconvention on summation is implied) ds = g ij dx i dx j , i, j, k, ... = 0 , , , . (2.1)Here the entries of the matrix g ij ( x ) are the local components of a pseudoor semi- Riemannian metric g on M . We suppose that the canonical form ofthe matrix g ij ( x ) is diag(+ , − , − , − ). One says that M is a Lorentz manifold.There exists an unique linear connection ∇ with the local coefficients given bythe usual form of the Christofell symbols which has no torsion and makes g covariant constant i.e. g ij ; k = 0, where ; means the covariant derivative. Thenthe Ricci tensor is R ik = R jijk where R hijk is the curvature tensor of ∇ and thescalar curvature is R = g ij R ij . A. Einstein postulates the basic equation forgravitational processes as it follows (the Einstein equations) EE : R ik − Rg ik = κT ik , (2.2)where κ is a constant and T ik is the energy -momentum tensor. The tensor T ik refers to the free formations (of the substance having non-zero rest mass, thefree electromagnetic field, etc. This should be divergence free since the Einsteintensor (given by the left hand of the Einstein equation (2.2)) is divergence free.Alternatively, the EE can be derived from Hamilton’s principle δ Z ( bR + L ) √− gdτ = 0 , (2.3)where √− gdτ is the four dimensional volume element and in the Hilbert- PalatiniLagrange function bR + L , b is a constant, whereas the function L leads to T ik .Using the Einstein equations one may try to determine g ik assuming that T ik is given or to take a special form of g ik and to determine T ik . The second task iseasier. The first is much harder. In absence of the matter (void or empty space)we have T ik = 0 and a contraction with g ik in (2.2) leads to a simpler form ofthe EE : R ik = 0. The simplest and the most important exact solution of thislast equation is the Schwarzschild metric to be discussed in the next Section. Fordetails we refer to [2]. 3 The Schwarzschild metric
The Schwarzschild metric is the first exact solution of the equation R ik = 0 . (3.1)It was found by Karl Schwarzschild in 1916 and played an important role inconfirming the predictions of the GR theory. Now there are many ways to deduceit. It is valid in the empty space surrounding a static body with mass sphericalsymmetry. Passing to spherical coordinates ( x , x , x , x
3) = ( t, r, θ, ϕ ) , thespherical symmetry and an obvious re-scaling of r reduces ds to the followingform (see [2,p.45]) ds = Adt − Bdr − r ( dθ + sin θϕ , (3.1)where because of the chosen signature we must have A >
B > r → ∞ ). Then one computesthe Christofell symbols Γ ijk which are inserted in a formula for R ik ,( 5.27 in[2]). The Einstein equations (3.1) give first that the product AB is a constant(equal to 1 at r tends to ∞ ). Hence B = 1 /A . Then ( r/B ) ′ = 1, where ’ denotesthe derivative with respect to r and upon integration one gets the Schwarzschildmetric (SM) ds = (1 − λr ) dt − (1 − λr ) − dr − r ( dθ + sin θdϕ ) , λ = 2 GMc , (3.2)where G is the universal gravitational constant and M is the mass of the centralbody (star, planet...). We stress that this metric is valid in the empty spaceoutside of the central body.In his treatise [3], A. Sommerfeld derives the SM directly from the equivalenceprinciple in form P E . Here is his reasoning.Let be a Point P lying in the empty space surrounding a central mass M which is distributed with spherical symmetry, its center being a point O . Withinspherical coordinate P will be determined by the r, θ, ϕ . Let us consider a localframe of reference in P , one of the axes of the frame being OP . A second frame ofreference, with the coordinates ˙ r, θ.ϕ slides on the OP along the first, the relativevelocity being v , that is the velocity a material point of mass m has under theaction of the gravitational field of the central body. The metric connected to themass m will be the Minkowskian one, that is ds = c d ˙ t − d ˙ r − ˙ r ( dθ + sin θdϕ ) , (3.3)4n observer having a fixed position in P will notice a momentary contraction ofthe distance as well as a momentary dilatation of the duration, that is dr = p − v /c d ˙ r ; dt = d ˙ t/ p − v /c , (3.4)and so (3.3) becomes ds = c (1 − v /c ) dt − (1 − v /c ) − dr − r ( dθ + sin θdϕ ) , (3.5)where r = ˙ r , as we are talking about the same point P . It remains to determinethe velocity v = v ( r ), where the dependence on r is due to the fact that we atalking about an accelerated material point, therefore v varies permanently duringthe motion. In order to do this, A. Sommerfeld has used the energy conservinglaw from the classical Physics:12 mv + mV ( r ) = const., (3.6)where V ( r ) = GMr is the gravitational field potential of the central mass. When r → ∞ the constant is zero and so v c = λr , λ = 2 GM/c . (3.7)Using this in (3.4) one finds the SM in the form (3.2).The main point was here the use of the energy conserving law from (3.6). Butaccordingly to the P E in its from
P E it is not only more natural but it is justcompulsory the use of the relativistic law of energy conservation as a law from SR . It is true that the SR includes also the classical laws but only in limits andwith some nuances due to different groups of symmetries. This remark belongs toProfessor Cleopatra Mociut¸chi. Based on it she uses the relativistic law of energyconservation and so she arrived to and studied in the sixties, [4-8], a Schwarzschildlike metric as follows.The relativistic law of energy conservation has the form( m − m ) c − GmMr = 0 , m = m p − v /c . (3.8)It comes out that p − v /c = 1 − µ/r. (3.9)5n so (3.5) reduces to what we call the Mociut¸chi metric (MM) : ds = c (1 − µr ) dt − (1 − µr ) − dr − r ( dθ + sin θdϕ ) , µ = GMc = λ/ , (3.10)Here some properties of the Mociut¸chi metric :1. It returns all the General Relativity tests,2. It reduces to the SM if ( µ/r ) ≈ r and t donot change significance between themselves in the case of MM, contrary toSM.5. The MM results from (2.3), L is the Lagrange function of the gravitationalfield of the central mass.The last enumerated property may be connected with the remark that if onecomputes the Einstein tensor R ik − Rg ik for the MM it comes out that four ofits components are non null. In the other words the MM is not an exact solutionof the Einstein equation with the energy-momentum tensor T ij = 0. However wederived it in the hypothesis of the absence of the matter and fields (in the emptyspace outside of a star or a planet). The contradiction could be eliminated if weassume that even in this empty space a kind of manifestation of the gravitationalfield does exist. But we can do this if we slightly modify the PE as follows : P ′ E : The only dynamical effects of a gravitational field are locally equiv-alent to the field of inertial forces of a convenient accelerated frame of reference. Moreover, some results on interaction of various fields confirm , [9-11], a moregeneral principle of equivalence : P ” E : The dynamical effects of certain fields are locally equivalent tothe field of inertial forces of a convenient accelerated frame of reference. eferences [1] Blagojevi´ c M.,
Gravitation and gauge symmetries,
Series in High EnergyPhysics,Cosmology and Gravitation, IOP Publishing Ltd., 2002[2] G.t’Hooft,
Introduction to General Relativity . Lectures Notes. Caputcollege,Utrecht University, the Netherlands, 1998[3] Sommerfeld Arnold ,
Electrodynamics Lectures on Theoretical Physics Vol-ume III , translated from the German by Edward G. Ramberg, AcademicPress, 1964)[4] Tomozei (Mociut¸chi) Cleopatra,
Studiul relativist al c ˆampului gravificrezultˆand dintr-o metric˘a bazat˘a pe o nou˘a exprimare a principiuluiechivalent¸ei ( The relativistic study of the gravitational field resulting froma metric based on a new expression of the equivalence principle ), An. S¸t.Univ., Ia¸si (s.n.), Sect.I, Tom VIII/1, 1962, p.131-138.[5] Mociut¸chi Cleopatra,
Observat¸ii privind interpretarea principiuluiechivalent¸ei locale ( Observations regarding the interpretation of localequivalence principle ), An. S¸t. Univ., Ia¸si (s.n.), sect. Ib, Tom 11 (1965), p.11-19.[6] Mociut¸chi Cleopatra,
A new generalization of Schwarzschild metric , Rev.Roum. Physics, Tom 21 nr.2, 199-207, Bucarest, 1976.[7] Mociut¸chi Cleopatra,
Equation of motion derived from a generalization ofEinstein’s equation for the gravitational field , Rev. Roum. Physics, Tom 25nr.3, 251-256, Bucarest, 1980.[8] Gottlieb I., Mociut¸chi Cleopatra,
Contribut¸ii privind interpretarea ¸si gener-alizarea principiului echivalent¸ei locale al lui Einstein (Contributions on theinterpretation and generalization of Einstein’s local equivalence principle) ,An. S¸t. Univ., Ia¸si, (s.n.), sect. Ib, Tom X, (1964), fasc.1, pp.7-20.[9] Gottlieb J.,
Einige Probleme der Deutung der allgemeinen Relativitatstheo-rie , An. S¸t. Univ., Ia¸si, (s.n.), sect. Ib, Tom XII, (1966), fasc.1, pp.135-144.[10] Gottlieb J.,
Einige Probleme der Theorie des Gravitationsfeldes , Studii ¸siCerc. de Astronomie, Tom 13, nr.1, (1968), pp.11-18.711] Gottlieb I.,