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Jonathan Scott
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I was recently checking out a paper about gravitational energy and found a unexpected minor inconsistency when I did the calculation entirely in isotropic coordinates. I later tracked down what caused it, and it surprised me a bit, so I'm wondering if others were aware of this. Basically, I'd forgotten to allow for the fact that when all other values are expressed as coordinate values, ##G## (or more accurately ##G\hbar/c^4##) has to vary slightly as well.
The problem is with terms of the form ##Gm/rc^2## which must have a dimensionless numerical value.
This expression can be split up into the following three parts, consisting of an adjusted form of the gravitational constant with additional constants to get the dimensions right, a frequency (which is purely affected by the time part of the metric) and an inverse distance (which is purely affected by the space part of the metric): $$\frac{G \hbar}{c^4} \frac{m c^2}{\hbar} \frac{1}{r}$$ When studying equivalence between GR and Newtonian terms, it is often useful to look at quantities expressed entirely in isotropic coordinates, including replacing the standard ##c## with the coordinate speed of light (which is only meaningful for isotropic coordinates as otherwise it can be different in different directions at a given point).
In the weak field approximation for the Schwarzschild solution, the scale factor between coordinate and local frequency values cancels with the scale factor between coordinate and local rulers, so the overall term ##G\hbar/c^4## has approximately the same effective value both for local calculations and for calculations in isotropic coordinates.
However, the space and time factors in the metric are not actually exactly the same, so overall the product of the second and third terms varies very slightly with potential. The time factor in the Schwarzschild metric in isotropic coordinates is $$\frac{1-Gm/2rc^2}{1+Gm/2rc^2}$$ and the space factor is $$(1+Gm/2rc^2)^2$$ To get the view from some other potential, we have to divide the frequency and distance values by the metric factors for that location, as follows:
$$\frac{1+Gm/2rc^2}{1-Gm/2rc^2} \frac{1}{(1+Gm/2rc^2)^2} = \frac{1}{1-(Gm/2rc^2)^2}$$
In order for this overall to remain a dimensionless value which is unaffected by the potential at the relevant location, this means that the ##G\hbar/c^4## term must vary with observer potential by the inverse of this factor.
This also highlights a side-effect of using geometric units for gravity, where ##G=1## and ##c=1##. I have always disliked this convention because these are quantities with dimensions which can change from their standard values in different frames of reference, but if they are omitted, it is not clear which dimensions apply in a given case. With geometric units, a term of the form ##m/r## has to be dimensionless, which means that the value of ##m## must behave like a length, not like a frequency as would normally be expected for an energy value, so a mass term expressed in these units would have a local value determined not by the time factor of the metric but rather by the space factor. This seems very confusing to me!
The problem is with terms of the form ##Gm/rc^2## which must have a dimensionless numerical value.
This expression can be split up into the following three parts, consisting of an adjusted form of the gravitational constant with additional constants to get the dimensions right, a frequency (which is purely affected by the time part of the metric) and an inverse distance (which is purely affected by the space part of the metric): $$\frac{G \hbar}{c^4} \frac{m c^2}{\hbar} \frac{1}{r}$$ When studying equivalence between GR and Newtonian terms, it is often useful to look at quantities expressed entirely in isotropic coordinates, including replacing the standard ##c## with the coordinate speed of light (which is only meaningful for isotropic coordinates as otherwise it can be different in different directions at a given point).
In the weak field approximation for the Schwarzschild solution, the scale factor between coordinate and local frequency values cancels with the scale factor between coordinate and local rulers, so the overall term ##G\hbar/c^4## has approximately the same effective value both for local calculations and for calculations in isotropic coordinates.
However, the space and time factors in the metric are not actually exactly the same, so overall the product of the second and third terms varies very slightly with potential. The time factor in the Schwarzschild metric in isotropic coordinates is $$\frac{1-Gm/2rc^2}{1+Gm/2rc^2}$$ and the space factor is $$(1+Gm/2rc^2)^2$$ To get the view from some other potential, we have to divide the frequency and distance values by the metric factors for that location, as follows:
$$\frac{1+Gm/2rc^2}{1-Gm/2rc^2} \frac{1}{(1+Gm/2rc^2)^2} = \frac{1}{1-(Gm/2rc^2)^2}$$
In order for this overall to remain a dimensionless value which is unaffected by the potential at the relevant location, this means that the ##G\hbar/c^4## term must vary with observer potential by the inverse of this factor.
This also highlights a side-effect of using geometric units for gravity, where ##G=1## and ##c=1##. I have always disliked this convention because these are quantities with dimensions which can change from their standard values in different frames of reference, but if they are omitted, it is not clear which dimensions apply in a given case. With geometric units, a term of the form ##m/r## has to be dimensionless, which means that the value of ##m## must behave like a length, not like a frequency as would normally be expected for an energy value, so a mass term expressed in these units would have a local value determined not by the time factor of the metric but rather by the space factor. This seems very confusing to me!