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From Kepler, to Schwarzschild and beyond to Le Verrier: The Violation of Kepler's II Law

In a certain sense, the Schwarzschild solution to the "vacuum" Einstein equations should be a relativistic generalization to the Kepler problem. And if so, it should be able to account for the difference between the observed perihelion advance to the known contributions that are accounted for by Newtonian gravitation. However, all is not as clear as it has been claimed.

The gravitational potentials in the Schwarzschild metric contain

G'(rho)={1-2m/G c^2}^(1/2) (1)

in the indeterminate metric

ds^2 = G'^2(rho)dt^2- d(rho)^2-G^2(rho)d(phi)^2, (2)

where m is the gravitational parameter, GM, rho is some radial distance, and the motion has been relegated to the plane theta=pi/2. The fact that the speed of light, c, appears in the denominator of (1), makes the gravitational potential of order of the square of the relativistic speed (v/c)^2, and, consequently, the Newtonian potential m/G=v^2 is second order. Hence no orbital motion of a conic section is possible.

The most general equation for the orbit is

(dr/d(phi))^2 = (r/h)^2{n_1^2 r^2 - h^2 n_2^2}, (3)

where h is the angular momentum per unit mass, and the indices of refraction are

n_1^2=2(E+m/r), (4)


n_2^2= 1- 2m/r c^2, (5)

the square of (1). E is the total energy which can be either positive (hyperbolic) or negative (elliptic) orbits.

In the case where (5) is unity, the orbital equation for u=1/r, is obtained by differentiating (3) with respect to the azimuthal angle, phi, to obtain

u" +u =m/h^2,

which is the equation of an ellipse with a semi-latus rectum, h^2/m, where the prime means differentiation with respect to the angle, phi. Rearranging (3) gives the energy equation

(d/dt) v^2= (d/dt)[r^2] + r^2(d(phi)/dt)^2 =2(E+m/r), (6)

of a Keplerian ellipse, where

h=r^2 d(phi)/dt, (7)

is the conserved angular momentum per unit mass. The relativistic index of refraction (5) will

change the status of (7): h will be conserved, but it will not be equal to the right-hand side.

Rearranging (3) yields

(dr/n_2)^2 = n_1^2 [r^2 d(phi)/h n_2]^2 - r^2d(phi)^2. (8)

Multiplying and dividing the first term on the right by dt^2, gives

(dr/dt)^2/[1-2m/c^2 r] + r [d(phi)/dt]^2= 2(E+m/r). (9)

Introducing r=G(rho), i.e. dr/dt = G'(rho) d(rho)/dt, results in the generalized Kepler equation

[d(rho)/dt]^2 + G^2(rho)[d(rho)/dt]^2 = 2(E+m/G), (10)

provided the conserved angular momentum, h, is now given by

h= r^2 d(phi)/dt/{1-2m/r c^2}^(1/2) (11)

instead of (7).

Expression (11) is in flagrant contrast to the Schwarzschild solution which claims that

h= r^2 d(phi)/dt/{1-2m/rc^2} (12)

is the conserved quantity. If p is an affine parameter, Weinberg in Gravitation and Cosmology

gives the conserved angular momentum as [eqn (8.4.11)]

h = r^2 d(phi)/d p, (13)

where [Weinberg, eqn (8.4.10)]

dt/dp= 1/{1-2m/rc^2} (14)

so that (12) is satisfied and not (11). Moller, in The Theory of Relativity arrives at (12) incorrectly by first claiming that

Gamma= 1/{1-2m/r+v^2/c^2}^(1/2)

in his equation (10) of Chapter XII, and then asserting that

Gamma (1-2m/rc^2) = E,

a "first integral of the equations of motion" in his equation (15). His second constant of integration, C, in his equation (17) is not a constant at all but, rather,

C = h/{1-2m/rc^2}^(1/2).

Paradoxically, his equation for the square of the particle velocity, (11),

v^2 = (dr/dt)^2/(1-2m/r) + r^2[d(phi)/dt]^2 = 2(E+m/r),

is correct!

However, under no conditions will a "test" planet ever trace out a circular orbit by satisfying the Schwarzschild solution. This is because two indices of refraction, (4) and (5), are necessary, and the Schwarzschild solution only contains (5). Whereas (4) is macroscopic and non-relativistic, (5) is relativistic and can only contribute a minute amount, as in the advance of the perihelion. That is to say from the orbital equation (3) we get

u" + u =m/h^2 + 3m/c^2 u^2 (15)

upon differentiation. The first term in (15) comes from the macroscopic, non-relativistic index of refraction (4) whereas the second term is the result of the relativistic index of refraction (5).

We may also verify that the radial equation of motion for a particle in the Schwarzschild field is incorrect. Again restricting the motion to the azimuthal plane, the radial acceleration is

claimed to be

a= d^2 r/dt^2- r [d(phi)/dt]^2 {1-2m/rc^2}, (16)

cf. McGrudder "Gravitational repulsion in the Schwarzschild field" who took the result from Weinberg equation (8.4.3). The correct expression will be shown to be

a= d^2 r/dt^2- r [d(phi)/dt]^2 {1-2m/rc^2}^(1/2), (17)

again differing by a factor of the second index of refraction, as in the expression for the angular momentum per unit mass. For a metric of the form

ds^2= d(rho)^2 +G^2(rho) d(phi)^2,

the connection is

w_(12) = G' d(phi) = - w_(21).

In a moving frame with base vectors E_1 and E_2, the velocity vector is

v= d(rho)/dt E_1 +G[d(phi)/dt] E_2.

The acceleration is the covariant derivative of the velocity or

a= {d^2(rho)/dt^2 +G[d(phi)/dt] x [-G' d(phi)/dt]} E_1+{(d/dt)[d(phi/dt G']+d(rho)/dt x G'[d(phi/dt] } E_2.

This gives explicitly

a= [d^2 rho/dt^2 - G [d(phi)/dt]^2 G'] E_1 +{G d^2(phi)/dt^2+2 [d(rho)/dt][d(phi)/dt]G'}E_2. (18)

Since r=G and G' is given by (1), the radial acceleration is not given by (16), but by expression (17) thereby nullifying the Schwarzschild solution. The tangential acceleration in (18) is


which, unlike (7), does not vanish. This means that the attraction of the central body on the orbiting planet is not entirely central, but differs from a total time derivative by the quantity

G"/G/(G'/G)^2, (19)

which is roughly of the order of m/p c^2, the second-order relativistic velocity ratio, with G=p, the semi-latus rectum of the orbit. The exact expression has a factor 6 pi multiplying this expression.

This says that the advance of perihelion of a planet violates Kepler's II law so that the assumption of the centrality of the central force is not strictly obeyed. All the other contributions to the perihelion advance obey Newtonian gravitational physics and so obey Kepler's II. The discrepancy due to the difference between this value and the observed value is due to a break-down in Kepler's second law. The magnitude of the effect is given by (19).

Kepler's I law is not violated; the orbit is still an ellipse, but the action of the Sun, located at one of the foci, is not precisely central. This causes a rate of change of the line of apsides which amounts to 43" for Mercury in a century. However, it is Kepler's II which is clearly violated, and the lack of the central nature of the attracting body could not be explained on the basis of Newtonian physics.

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