Gravitational Energy in GR: Energy Conservation Explained

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Discussion Overview

The discussion revolves around the concept of gravitational energy in General Relativity (GR) and the implications for energy conservation in different frames of reference. Participants explore the nature of energy in free fall versus a fixed frame on Earth, as well as the relationship between velocity, momentum, and mass in these contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how energy is conserved when viewing free fall as an inertial frame, noting that items on the Earth's surface appear to gain velocity and kinetic energy relative to this frame.
  • Another participant confirms that energy is frame-dependent, stating that items gain kinetic energy relative to the free-fall frame but not to a frame fixed to the Earth.
  • A follow-up question is posed regarding the implications of increasing velocity and momentum on the mass of an item on Earth, referencing the equation E^2 = (pc)^2 + (mc^2)^2.
  • In response, a participant clarifies that while velocity increases in the freely falling frame, energy also increases, and that mass remains constant in both frames.
  • A participant notes the complexity of energy in GR, mentioning that locally energy is frame variant but conserved, while globally it is not well-defined or conserved due to the challenges of adding vectors in curved space.

Areas of Agreement / Disagreement

Participants express differing views on the conservation of energy in various frames, with some agreeing on the frame-dependence of energy while others highlight the complexities and limitations of defining energy in GR. The discussion remains unresolved regarding the implications of these concepts on mass and energy conservation.

Contextual Notes

Participants acknowledge that energy in GR is a nuanced topic, with local conservation differing from global definitions. There are also references to the challenges of understanding energy in curved space, which may limit the discussion's scope.

david316
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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.
 
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david316 said:
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.
 
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.
 
david316 said:
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.
 
That makes sense. Thanks a lot.
 
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
 
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