Nonlinearity of gravity in general relativity clashes with Lorentz covariance

Let us check if the steepening of the gravity potential in the Schwarzschild solution is consistent with Lorentz transformations.

The Schwarzschild potential is steeper than the newtonian potential

The potential in the Schwarzschild solution, as measured by a faraway observer, differs from the 1 / r potential of newtonian gravity and the Coulomb force. If we in general relativity put two masses close to each other, their gravity is stronger than the sum of the individual gravities. Gravity is nonlinear, in this sense.

The potential in the Schwarzschild solution at a coordinate radius r from a mass M is obtained from the slowdown of time, or the redshift. The remaining mass-energy of a test mass m is

       m  *  sqrt(1  -  rₛ / r)

       =  m  *  sqrt(1  -  2 G / c²  *  M / r),

If M / r is not very large, this is approximately

       m  *  (1  -  G / c²  *  M / r).

The remaining mass-energy in the newtonian gravity potential is the same

       m  *  (1  -  G / c²  *  M / r).

We see that the remaining mass-energy of the test mass m goes to zero in the Schwarzschild solution at a coordinate radius which is twice the corresponding radius in newtonian gravity. The horizon is the place where it goes to zero.

A nonlinear gravity potential leads to a strange result

Consider a configuration where two equal small masses M traveling at almost the speed of light to opposite directions meet, and an initially static test mass m is accelerated toward them.

                         γ M

                           ● ----->   v

            -v   <----- ●

                          γ M

                            ^  a

                            |

                            • m

The factor

       γ  =  1 / sqrt(1  -  v² / c²)

is very large. The test mass feels the gravity of 2 γ M, which is very large.

Let a' be the acceleration of m if only one of the masses M would exist. If gravity is linear, we expect

       a = 2 a'.

Suppose then that gravity is superlinear like in general relativity. The acceleration might be 1% larger:

       a = 2.02 a'.

Let us switch to the comoving frame of the mass M going to the left:

                  > γ M

                      ● -----> v'

                      ●

                     M

                 

                     ^  a''

                     |

                      • ----> v''

                     m

In this frame, there is a small static mass M. The other mass M moves to the right at almost the speed of light, and m moves a little slower.

Now, if we remove the small static mass M, the acceleration of m drops by 1%. This is counterintuitive. Why would a small and insignificant static mass M affect the acceleration so much? The acceleration is due to m feeling the gravity of the fast moving mass, > γ M. It is strange if a small mass M can have such a large impact.

Conclusions

Why do people believe that gravity is nonlinear? Because the Einstein field equations imply that gravity is nonlinear in the Schwarzschild solution.

General relativity predicts that gravity is almost linear even with neutron stars. Nonlinearity is prominent only with black holes. We are not aware of any astronomical observation which would confirm nonlinearity of gravity. LIGO graphs about black hole mergers have a large margin of uncertainty.

Thus, it probably is not known if gravity is nonlinear. Our example configuration above shows that nonlinearity leads to a counterintuitive result. It may be difficult to design a theory of gravity which is nonlinear and Lorentz covariant.

Nonlinearity is an ugly feature. We should not assume it without an empirical verification. We conclude that the Einstein field equations probably are wrong also in this respect: gravity is presumably linear.

Linearity would mean that the radius of the horizon of a black hole is newtonian:

       G M / c²,

that is, 1/2 of the Schwarzschild radius. As far as we know, there is no reliable measurement of the horizon radius for any black hole.