- #1
Haorong Wu
- 420
- 90
- 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!
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!