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Jonathan Scott

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Consider a system containing multiple massive objects, where the field is weak everywhere and velocities are non-relativistic. In a linearized GR approximation (as for example described in Wald "General Relativity" section 4.4) described relative to a flat background coordinate system, we find that the time component of the metric causes a fractional decrease in clock rate (that is, time dilation) by the sum of the Newtonian potentials due to each source.

When an object (not necessarily of negligible mass) is within a gravitational potential, all time rates and rest energy values relating to local measurements are decreased by the time dilation when observed in the background coordinate system. This for example explains the red-shift of photons produced at a lower potential when compared with photons produced by a similar reaction at a higher potential, as in the Pound-Rebka experiment.

When the potential is assumed to be created by a fixed central mass, the difference in the time-dilated rest energy at different potentials is exactly the same as the difference in potential energy (and for a free-falling body is the negative of the difference in kinetic energy, preserving the total energy). This suggests that these two quantities could be identified with one another.

However, this model only takes one side of the picture into account. If we consider for example a two-body Newtonian case, where two masses m_1 and m_2 are orbiting about their center of mass with at distances r_1 and r_2, then in the time dilation model, both of these masses would experience a decrease in rest energy due to the potential of the other one.

In the pure Newtonian model, the total potential energy, relative to when the masses are at infinite separation, is

-G m_1 m_2/(r_1 + r_2).

In this case the potential energy can be unambiguously shared between the two masses proportionally to their radial distances from the center of mass (or equivalently inversely proportionally to the magnitudes of the masses).

The potential energy of each mass is then given by:

-G m_1 m_2/(r_1+r_2) * r_1/(r_1+r_2)

-G m_1 m_2/(r_1+r_2) * r_2/(r_1+r_2)

or using the mass ratio instead as

-G m_1 m_2/(r_1+r_2) * m_2/(m_1+m_2)

-G m_1 m_2/(r_1+r_2) * m_1/(m_1+m_2).

However, in the time dilation model, the potential due to each of the two masses still gives rise to the full time dilation at the location of the other. For example, the potential due to the second mass at the location of the first is

-G m_2/(r_1+r_2)

and the energy loss is therefore given by multiplying this by m_1, which gives the same as the total potential energy. For the other mass, the terms m_1 and m_2 are switched round, but the result is the same. This says that the total energy decrease due to the time dilation effect is exactly twice the change in potential energy.

In fact, the same factor of two appears in all cases where both sides of the time dilation effect are considered, including the case of a single central mass built up from smaller masses. This means that the change to the energy due to the time dilation cannot after all be identified with the location of the potential energy, as there is always a factor of two mismatch.

So what is the explanation for this mismatch?

1. Could there be a self-consistent way that the time dilation could be "shared out" (so that for example two equal masses would each affect each other's time rate by only half the usual amount)? I couldn't find one, and this idea seems to conflict directly with the linearized GR expression for the scalar potential.

2. Could it be that although the locally energy is simply multiplied by the time dilation in the context of the Pound-Rebka experiment, there is some additional first-order factor which applies when considering a system of multiple masses, which for example would decrease the effect by as much as a half when two equal masses were involved?

3. Could there be some other positive internal form of energy created when a mass is in a potential which has the effect of compensating for the double decrease due to time dilation?

For example, in a static central mass situation, the GR "Komar mass" expression adds to the energy another term which consists of the integral of three times the pressure over the volume of the central mass. This gives the right mathematical result for the central mass case, but doesn't extend to the two-body dynamic case. This pressure term can be loosely likened to twice the thermal kinetic energy of a monatomic ideal gas, as in the virial theorem. However, it doesn't seem to make a lot of sense as a real physical quantity unless one adds a restriction that the "Komar mass" expression only applies to a monatomic idea gas.

In the case of two bodies in free fall in each other's potential, it seems that any pressure induced in the other body could only be a tidal effect, and would therefore be of the wrong order to have a first order effect on the energy change due to time dilation.

4. Finally, it could perhaps be that the above factor of two is actually correct, and that the missing energy (equal in magnitude to the potential energy) has gone somewhere else in the system. In that case, the mathematically obvious solution is that the missing energy has gone into the gravitational field, and is identical in form to the Maxwell energy density of an electrostatic field, as this gives exactly the correct amount of energy for any combination of sources (at least in the weak-field non-relativistic approximation). This solution would also provide a specific explanation as to where gravitational energy is located in the linear approximation.

I've previously been offered various other suggestions which all essentially say that it's obviously not valid to apply the time dilation to the local rest energy to get the effective rest energy in the background coordinates, because, as I've shown, it gives the wrong answer. This doesn't seem to me to be a very scientific approach. If there's a problem extending the GR time dilation effect from the Pound-Rebka situation to the Newtonian two-body system, I'd like to know specifically why the problem arises and what the corrected solution should be.