# Doubts about Einstein Field Equations

1. Aug 16, 2010

### petergreat

Why should we trust the Reissner–Nordström solution of charged black holes? It relies on coupling between Einstein tensor and EM stress-energy tensor, which has NO experimental support whatsoever.

Is there any chance we can test this?

2. Aug 16, 2010

### bcrowell

Staff Emeritus
Why do you say that? I'm not an expert on experimental tests of GR, but it seems to me that there is quite a bit of experimental evidence on this.

Of course there are classic tests like the deflection of light by the sun and the Pound-Rebka experiment, both of which showed that light interacts gravitationally pretty much as predicted by standard GR.

As an example of an alternative rule for coupling, I could imagine $G_{ab}=c_1T_{ab}+c_2g_{ab}T^c_c$ for the Einstein field equations, but then the equivalence principle would be violated by the second term, which vanishes for EM waves. The equivalence principle has a lot of support from extremely precise experiments.

In general, I think it's quite difficult to come up with alternative forms of the Einstein field equations that are consistent with observational support for mass-energy conservation, momentum conservation, and the equivalence principle.

The early universe was radiation-dominated, so cosmological observations that probe that era should be very direct tests of whether light produces gravitational fields in the way described by GR. Don't modern high-precision CMB measurements probe this in some detail?

Do you know of any currently viable alternative theory of gravity (tensor-scalar theories, etc.) that disagrees with GR on this issue?

Of course extrapolating all the way to the intense fields of a highly charged black hole is a very big extrapolation. Clearly astrophysical black holes with these properties don't exist naturally. Even if the human race avoids extinction for another thousand years, I would be surprised if the Reissner–Nordström metric is ever tested directly against experiment. Maybe if large extra dimensions exist, in which case you might be able to make microscopic black holes in particle accelerators.

Last edited: Aug 16, 2010
3. Aug 16, 2010

### Passionflower

There are, mostly weak field, experiments that show that GTR is correct in Ricci flat situations. Which means that a big part of the theory remains untested, it is implied to be correct, for pretty good reasons, but that is not identical to being experimentally verified. The binary pulsar behavior also proves GTR correct in stronger field solutions.

Light deflection and the perihelion shift of Mercury are good tests. The pound-Rebka experiment is rather limited, it for instance does not prove the correctness of the Schwarzschild solution.

Last edited: Aug 16, 2010
4. Aug 16, 2010

### petergreat

Sure GR has lots of experimental tests, even strong field tests in binary neutron stars. But in almost every single one of them, the stress-energy tensor has the form $$T^{\mu \nu} = \rho U^\mu U^\nu$$, which is to say it origins from macroscopic masses. EM field? No way.

Thanks for pointing at CMB. That makes me more convinced, though the only difference here is the additional pressure term.

Still there's another direct consequence of Einstein field equation which doesn't convince me - the equivalence of vacuum-energy to cosmological constant. In quantum theory, if every energy level in shifted up by the same amount, the theory doesn't change at all, i.e. the absolute ground energy level is arbitrary. But with GR, such shifting becomes disastrous and would rip our universe apart.

Unless one day we can calculate accurately the parameters of inflation by identifying the underlying phase transition and the amount of vacuum energy involved in this phase transition, I would be reluctant to believe that the coupling between gravity and vacuum energy must be correct. Then again, I have no better suggestion...

5. Aug 17, 2010

### yuiop

I think it is generally acknowledged that when the vacuum energy is calculated using quantum theories, the resultant cosmological parameter in terms of dark energy or acceleration of the expansion, is many orders of magnitude greater than what is actually observered. As far as I know this is an "open problem" and one reflection of the conflict between GR and quantum theories. I guess it is possible the conflict is more a problem with quantum theories that do not yet have good desciption of quantum gravity, than with GR.