I Energy conditions and non-physical phenomena

[accdd]

Is the inability to exceed the speed of light a consequence of general relativity?
Is the fact that no energy is created from empty space a consequence of general relativity?
Or are they both constructions deriving from the energy conditions imposed to have solutions to Einstein's equations that are compatible with observations?

 

[Orodruin]

Is the inability to exceed the speed of light a consequence of general relativity?

You need to define what you mean by "exceeding the speed of light".

Is the fact that no energy is created from empty space a consequence of general relativity?

You need to define what you mean by "no energy is created from empty space".

 

[accdd]

Locally, nothing can exceed the speed of light.
If I take a small volume I don't expect it to generate stuff out of a vacuum.

 

[Orodruin]

Locally, nothing can exceed the speed of light.

This in essence follows from 4-momentum being non-spacelike.

If I take a small volume I don't expect it to generate stuff out of a vacuum.

This, in the form $\nabla_\mu T^{\mu\nu}$ is a direct consequence of varying the Einstein-Hilbert action with an additional term to describe the matter fields (and thereby generating the stress-energy tensor). The Einstein field equations resulting from varying the Einstein-Hilbert action are on the form $G_{\mu\nu} = C T_{\mu\nu}$, where $C$ is a constant and the divergence of the Einstein tensor $G_{\mu\nu}$ is equal to zero.

However, "global" energy is generally not conserved in general relativity as demonstrated, e.g., by FLRW cosmologies.

 

[accdd]

Sean Carroll in Spacetime and Geometry writes (4.6, last section):
[Energy conditions ... serve to prevent other properties that we think of as "unphysical", such as energy propagating faster than the speed of light...]
What does this means?

 

[Orodruin]

I suggest looking at the basic descriptions of different energy conditions in relativity. They are all concerned with the stress-energy tensor and are at varying degrees of strictness. For example, look at https://en.wikipedia.org/wiki/Energy_condition under "Mathematical statement".

The statement that relates to the flow of energy is the dominant energy condition which relates to $T_{ab} Y^b$ where $Y$ is a time- or light-like vector field. The resulting 4-vector describes energy density and flow.