How to Change Between Unit Systems w/ ##c \ne 1, \hbar \ne 1, G \ne 1##

[Haorong Wu]

TL;DR Summary

How to change a result from a special unit system back to the normal unit system?

If we use the nature units or the geometrized units, how could we transfer the final result back to the normal units with $c \ne 1, \hbar \ne 1, G \ne 1$? I know I coule analysis its dimension to multiply them back.

But what about other units system? For example, in a recent paper, the Lagrangian density is given by $$ \mathcal{L}=-(1-\frac m r) \dot t^2+(1- \frac m r)^{-1} \dot r ^2 + r^2 (\dot \theta ^2 +\sin ^2 \theta \dot \phi ^2).$$and $m=2GM$ is the Schwarzschild radius, $G$ is the gravitational constant, and the units are such that $m=1$.

In the end, the paper give a result that $O=- \frac {8i a^2} {z_R} (x^2-y^2)$ where $a$ is a scaling parameter, $z_R$ is the rayleigh range.

But following this equation, the author gives another equation $O=- \frac {i9ma^2} {z_R} (x^2-y^2)$ where $m$ is the schwarzschild radius of the sun. It seems that the author have transferred the result back to the normal units and there is an extra factor $\frac {9m} {8}$. But I could not see how to derive it.

I have sent a email to the author, and havn't got replies. I hope maybe some friends here would give me some advice. Thanks!

 

[PeterDonis]

in a recent paper

Please give a reference.

 

[Haorong Wu]

Please give a reference.

Hi, @PeterDonis . It is

Exirifard, Qasem, Eric Culf, and Ebrahim Karimi. "Towards Communication in a Curved Spacetime Geometry." arXiv preprint arXiv:2009.04217 (2020).

It can be downloaded from https://arxiv.org/pdf/2009.04217.pdf

In page 11, after Eq. (S76), it mentions that the units are chosen such that $m=1$.

Then in page 18, Eq. (S120) gives the expression for operator $O$. This equation together with Eq. (S112) do not match with the following equation (S123). There is some extra factor where the schwarzschild radius of the sun pops out.

 

[PeterDonis]

Exirifard, Qasem, Eric Culf, and Ebrahim Karimi. "Towards Communication in a Curved Spacetime Geometry." arXiv preprint arXiv:2009.04217 (2020).

Thanks for the reference. On an initial look, it does seem like quite a few steps in their logic are left out. I have not had time to read through it closely enough to work out the missing steps and see whether they match the conclusions the paper is claiming.

 

[Haorong Wu]

Thanks for the reference. On an initial look, it does seem like quite a few steps in their logic are left out. I have not had time to read through it closely enough to work out the missing steps and see whether they match the conclusions the paper is claiming.

Thanks for your time, @PeterDonis .

In my second look, I analyze the dimensions of the equations. The first one gives the dimension of $[m]^{-1}$ while the second one is $1$, so maybe it just insert the schwarzschild radius with dimension of $[m]$ to match the dimensions of the two equations.

As for the factor, I think the author may have some errors in the calculations.

However, I am not sure whether this method of dimension analysis is correct.