I have seen it stated that GR doesn’t always maintain the axiom of the conservation of energy. However, I haven’t been able to find any general explanation of why this is the case without getting drawn into the complexity of Killing vectors etc. So my initial questions are:(adsbygoogle = window.adsbygoogle || []).push({});

- Is it true that GR doesn’t maintain the conservation of energy?

- Is there a simple illustrative example?

- Is it a fringe issue under the extremes of spacetime curvature, i.e. gravity?

As a related, but somewhat tangential issue, it is sometimes pointed out that potential energy is not a concept used by general relativity. However, most of us will probably have been introduced to the concept of gravity using the Newtonian concept of potential energy and gravitational force. Equally, the classical concept of the conservation of energy is often described in terms of just 2 fundamental forms of energy, i.e. kinetic and potential. So my next question is:

- Can the GR theory of gravity still be accurately transposed into the ‘concept` of potential energy and gravitational force, e.g. [tex] F=GMm/r^2[/tex], even though this may not now be the preferred approach under GR?

By way of an example, classic effective potential is defined in the context of the conservation of total energy in the form of kinetic and potential energy, i.e.

[tex]E_T=1/2m(v_r^2+v_o^2) - \frac{GMm}{r}[/tex]

Where [tex][v_r][/tex] is the radial velocity, while [tex][v_o][/tex] is the orbital velocity. If we only consider a circular orbit with no radial component of velocity, the effective potential is stated to be:

[tex]Veff=1/2mv_o^2 - \frac{GMm}{r}[/tex]

However, a very similar form can be derived from the Schwarzschild metric, which is an accepted GR solution of Einstein’s field equations given the usual caveats. While the derivation is not reproduced here, the equivalent solution is given as:

[tex]Veff=1/2mv_o^2 - \frac{GMm}{r} \left( 1 + \frac{v^2}{c^2} \right)[/tex]

The implication seems to be that as the orbital velocity increases towards [c], an additional relativistic factor comes into play to maintain the orbit, i.e. as [tex][v_o \rightarrow c][/tex] the factor approaches 2. It is not clear whether this factor of 2 explains the anomaly in the classical and GR equations for light deflection?

Classical: [tex]\theta = \frac{2GM}{rc^2}[/tex] as opposed to GR: [tex]\theta = \frac{4GM}{rc^2}[/tex]

However, the only point of all this was simply to ask whether the GR theory of gravity can still be accurately presented in terms of potential energy?

P.S. PF Admin, please note that the thread title keeps resetting after preview!

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# Conservation of Energy

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