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Double potential energy paradox resolved by Komar mass

  1. Nov 14, 2008 #1

    Jonathan Scott

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    In February 2007, I was trying to understand why a pair of masses being brought together appear to lose energy by twice the change in gravitational potential energy when analyzed using a semi-Newtonian linear approximation to GR.

    I was pointed to the "Komar mass" expression in GR by Pervect (thank you, Pervect), but found it difficult to understand how it could apply. However, I now realize that it applies in a very simple way and solves the original problem, although it then gives rise to a new question.

    Basically, in Newtonian terms, the Komar mass density includes not only ordinary energy density but also components of pressure in the x, y and z direction. It is however only valid in a static, unchanging system.

    Consider two small masses of proper mass [itex]m_1[/itex] and [itex]m_2[/itex] separated by a distance [itex]r[/itex]. The time dilation at mass [itex]m_1[/itex] due to mass [itex]m_2[/itex] is then [itex]-Gm_2/rc^2[/itex] so the potential energy decrease caused by this time dilation is [itex]-G m_1 m_2/r[/itex]. Similarly, the potential energy decrease of mass [itex]m_2[/itex] due to mass [itex]m_1[/itex] is [itex]-G m_2 m_1/r[/itex] which is of course the same.

    This appears to mean that the total potential energy decrease is twice the Newtonian decrease. However, this configuration is not static unless some framework is added to keep the masses in place. If we add a massless rigid rod of cross-section [itex]A[/itex] and length [itex]r[/itex] to hold the masses apart, the pressure in the plane perpendicular to the length of the rod will be equal to the gravitational force between the particles divided by the cross-section, [itex]Gm_1m_2/r^2A[/itex], and the volume of the rod will be [itex]rA[/itex], so the contribution to the Komar mass of the pressure in the rod will be the product of these, [itex]Gm_1m_2/r[/itex]. This exactly cancels with one copy of the potential energy, leaving the total Komar mass as equal to the original total rest mass minus the Newtonian potential energy.

    This scheme describing the interaction between any two point masses also applies to any set of masses being held statically in place by pressure, as the pressure contributions add up. If there is any additional mechanical pressure within the system, it must be arranged in such a way that the forces cancel over any plane, which means that its overall contribution to the Komar mass is zero.

    So in a Newtonian approximation for a static system, the Komar mass describes gravitational energy in a way which is completely localized (although this isn't exactly "energy" in the usual Newtonian sense).

    However, now suppose we have the same two masses initially at rest, but we do not have a supporting rod between them. This configuration is obviously only momentarily static before the masses start accelerating towards each other, but one might expect it to have the same energy. However, the Komar mass contribution of the rod is missing. This implies that in a situation where positions appear to be momentarily static but the situation is dynamically changing, the Komar mass doesn't even provide an approximate answer to the location of gravitational energy. I'm now wondering whether there's some alternative mass expression (involving acceleration terms or higher) which can also cope with the dynamic case at least in the semi-Newtonian linear approximation situation, and I'm going to think about that when I have time. Anyone heard of such a thing?
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