Gravitational Energy in GR: Energy Conservation Explained

[david316]

If you view free fall as an inertial frame and therefore items at "rest" on the Earth's surface are accelerating away from the centre of mass I do not understand how energy is conserved. Taking this view, relative to the free fall frame the items will be gaining velocity which implies that the kinetic energy will be increasing. Can someone explain to me why this wrong. Thanks a lot.

 

[PeterDonis]

Taking this view, relative to the free fall frame the items will be gaining velocity which implies that the kinetic energy will be increasing.

Yes, that's true. Energy is frame-dependent. The items are gaining kinetic energy relative to the free-fall frame, but not relative to a frame that is fixed to the Earth.

 

[david316]

That makes sense... I think. It makes a little sense to me if you consider there is no absolute frame of reference in the universe. Followup question, if I use E^2 = (pc)^2 + (mc^2)^2 and since velocity and hence momentum are increasing relative to freefall does that mean the mass of the item on Earth will be getting lighter.

 

[PeterDonis]

if I use E^2 = (pc)^2 + (mc^2)^2 and since velocity and hence momentum are increasing relative to freefall does that mean the mass of the item on Earth will be getting lighter.

No. Velocity increases in the freely falling frame, but so does energy. In a frame fixed to the Earth, velocity is zero and energy is constant. In both cases, $m$ remains constant.

 

[david316]

That makes sense. Thanks a lot.

 

[Dale]

Note also that energy itself is a bit of a tricky concept in GR.

Locally, it is frame variant, as mentioned by Peter Donis (energy and momentum form a four-vector). However, locally at least it is conserved (stress energy tensor has no divergence).

Globally, it is not even defined in general, let alone conserved. This is due to the difficulty in adding different vectors in different locations in a curved space.

Here is a good overview of the issues.
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html