How is energy conserved in General Relativity?

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Discussion Overview

The discussion centers on the conservation of energy within the framework of General Relativity (GR), exploring theoretical implications, definitions, and interpretations of energy conservation in curved spacetime. Participants examine specific scenarios involving light and gravitational fields, as well as the mathematical formulations that underpin these concepts.

Discussion Character

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

Main Points Raised

  • Some participants question how energy conservation applies in GR, particularly in non-trivial manifolds where four-momentum components may change.
  • There is a discussion about the implications of the Schwarzschild metric and its lack of explicit time dependence, suggesting a conserved quantity may exist but its correspondence to traditional energy is uncertain.
  • One participant expresses interest in how GR denies conservation of energy, particularly in relation to perpetual motion machines.
  • Participants debate the definitions of conservation in GR, contrasting the ordinary coordinate derivative with the covariant derivative of the stress-energy tensor.
  • Some participants assert that while energy conservation holds approximately in curved spacetime, it is not as straightforward as in flat spacetime.
  • There is a suggestion that GR could reinterpret conservation relations without relying on gravitational potential energy, which is not well-defined in GR.
  • Others note that in stationary spacetimes, a well-defined notion of gravitational potential energy can exist, and the OP's scenario could be modeled accordingly.
  • One participant introduces the concept of "energy at infinity" as a conserved quantity for free-falling test objects in stationary spacetimes.
  • There is a request for clarification on how gravitational potential energy is defined in stationary spacetimes and its relation to conserved quantities.

Areas of Agreement / Disagreement

Participants express a range of views on the conservation of energy in GR, with no consensus reached. Some agree on the existence of conservation laws in specific contexts, while others highlight the complexities and limitations of these concepts in curved spacetime.

Contextual Notes

Participants note that the definitions of conservation may depend on the context and mathematical formulations used, with distinctions between ordinary and covariant derivatives. The discussion also highlights the challenges in defining gravitational potential energy in GR compared to classical mechanics.

  • #61
sweet springs said:
is wrong?

It's ok as far as it goes. The errors in your previous posts came after that equation.

sweet springs said:
Normality of 4-velosity

The 4-velocity is a unit vector in GR, yes.
 
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  • #62
sweet springs said:
tells us about speed, doesn't it ?

It's true that ##u^1## tells you something about speed; but it's not equal to speed. (Also, you need to be careful how you're defining "speed". Speed relative to what?)
 
  • #63
sweet springs said:
Why this top position in trajectory was chosen to give the value ?
Because at the top v=0 so it is easier to calculate. Remember, the energy is a constant over the whole trajectory, so we can pick any point to calculate it. We may as well pick a point that makes the math easier
 
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  • #64
stevendaryl said:
Second, using the effective Lagrangian above, we could form the corresponding Hamiltonian, which is a conserved quantity. However, as I said, as a conserved quantity, it's kind of boring, since its value is just 12mc2\frac{1}{2} mc^2. So it's independent of the motion of the test particle.
It's independent of the motion and the position. Even ##\gamma## at infinity is disregarded. It is equal to rest mass in local Minkowsky space where the body is at rest. I am afraid that might be all that GR says about energy rigorously.
 
  • #65
sweet springs said:
I am afraid that might be all that GR says about energy rigorously.

No, it isn't. An object's energy at infinity is not in general the same as its rest mass. That is obvious from the equation for energy at infinity that we have already derived.

Also, as I have already pointed out, the term "energy" has multiple meanings. Several of them have been covered in this thread, and energy at infinity is only one of them.

At this point I am closing the thread. I think you need to consider more specifically what concept of energy you want to ask about, so that future threads can have a more focused and useful discussion.
 

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