How can one show from General Relativity that gravity is attractive force, and under which conditions it becomes repulsive, also why positive energy vacuum drives repulsive gravity?

# [Physics] Vacuum and repulsive gravity

cosmological-constantgeneral-relativitygravityvacuum

#### Related Solutions

Energy conservation in GR only holds approximately in spacetime regions that are very small, compared to curvature radii. But in general, since parallel transport is dependent on the trajectory in spacetime, an observer cannot uniquely define the energy of another distant observer.

The so-called Universal Time in the FLRW cosmology is a convention. We choose to define a set of synchronized observers, so that the metric can be separated in two factors, one of them with only spatial coordinates. But that doesn't mean that the universal time can be used as an analogous concept to a newtonian $t$ that would allow to define conservation laws. You could slice spacetime in another way, defining a different set of synchronized observers.

Anyway, physics is all about mathematical models, and you are free to think in terms of the one called Newtonian Cosmology in astrophysics books. It has mainly didactical purposes, but it leads to some correct results (and it is what many people secretly have in mind when they talk about cosmology). In that framework, you may define a total energy and see if it is conserved. The problem is that Newtonian Cosmology doesn't have dark energy. How would you model the summand of the dark potential energy?

The conserved quantity in GR, in problems that do NOT deal with the cosmological-scale dark energy (for instance when studying neutron stars and black holes) is the energy momentum tensor of normal matter. I think that perhaps an equivalent statement to your question may be: Is there any way to include the dark energy as part of the energy-momentum tensor, so that its conservation law still holds? There is nowadays a very active theoretical research on how to model dark energy, so your question is eventually a very interesting one, and is still open.

Welcome to physics.se .

The luminiferus aether was postulated because of the existence of light as electromagnetic waves. Physicists at the time had studied waves in various media and could not conceive of a wave existing without a medium to carry it, similar to water for waves on the sea, or air for sound waves. So this *hypothesized* medium was an inertial frame against which everything had some velocity.

There exists the Michelson Morley experiment which first disproved the existence of the luminiferous aether and several later and recent ones. The disproof says that there is no basic inertial frame for electromagnetic waves, since they always move with velocity c in any frame.

The behaviour of matter and electromagnetic waves is consistent with this when described in terms of special relativity which ensures that the velocity of light is c. Any physics model that is Lorenz invariant assures that this is true. In fact field theoretical models for particle physics populate the vacuum with virtual pairs of particles, BUT, the whole model is Lorenz invariant so this teeming population of the vacuum is not the luminiferous aether of old.

General relativity equations also respect Lorenz invariance so even if it is hard to visualize space as an underlying dynamical system, since it is Lorenz invariant it cannot be thought as a substitute of the ancient luminiferous aether, either.

It is Lorenz invariance found to be always validated experimentally that disproved the existence of ether .

## Best Answer

The Einstein field equations actually don't say anything at all about the nature of matter. Their structure is that they relate a certain measure of spacetime curvature G to the stress-energy tensor T: $G_{ab}=8\pi T_{ab}$. The stress-energy tensor describes any matter that is present; it's zero in a vacuum. Trivially, you can write down any equations you like describing an arbitrary spacetime that you've made up, and then by calculating G you can find the T that is required in order to allow the existence of that spacetime. However, that T may have properties that are different from those of any known type of matter. T has a very specific structure for certain types of matter, such as electromagnetic radiation or "dust" (meaning a perfect fluid made of particles that have velocities $\ll c$ relative to one another). There are various conjectures, called energy conditions, about what types of stress-energy tensors are physically possible for realistic types of matter. They have names like the weak energy condition (WEC), strong energy condition (SEC), etc. The WEC amounts to a statement that the energy density is never negative in any frame of reference. If it was violated, then you could get repulsive gravity. Basically all energy conditions are known to be violated under some circumstances. Here is a nice discussion of that: http://arxiv.org/abs/gr-qc/0205066

The cosmological constant is sometimes taken as a separate term in the Einstein field equations, but it can also be treated as a type of matter with a certain contribution to the stress-energy tensor. A spacetime with only a cosmological constant, and nothing else in it, violates various energy conditions.