1PN metric vs. Schwarzchild


by andert
Tags: metric, schwarzchild
andert
andert is offline
#1
Mar18-10, 02:15 PM
P: 12
I'm sure there is a simple answer to this question, but I have been looking at the first Post-Newtonian (1PN) metric (for my own research) and noticed that the time-time component of the GR metric is:

[tex]g_{00} = -1 + 2U - 2 U^2[/tex]

where U is the Newtonian potential.

The time-time component of the Schwarzchild metric, however, is

[tex]g_{00} = -1 + 2M/r[/tex].

There is no quadratic in the Newtonian potential even though this metric is an exact solution of the field equations. Is this because it is in a different gauge?
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
bcrowell
bcrowell is offline
#2
Mar18-10, 05:36 PM
Emeritus
Sci Advisor
PF Gold
bcrowell's Avatar
P: 5,500
You can make g00 look like anything you like, just by a choice of coordinates. Suppose I set g00=f(r), where f is some arbitrary function, and I don't give you any other information about my coordinates. You can then use this equation to define an r coordinate. For instance, suppose g00 changes by a factor of 2 between r1 and r2. Then we've effectively defined r=r2 to be the location where gravitational time dilation differs by a factor of 2 from its value at r=r1. With this implicit definition of the r coordinate, we can now go ahead and infer the rest of the metric.
andert
andert is offline
#3
Mar19-10, 08:50 AM
P: 12
Alright, yes, coordinate system or gauge. I see that. Each of them is a in a specific gauge. Now, what is different about the gauges of the 1PN and Schwarzchild metrics specifically? The 1PN gauge is a harmonic one. So if I were to take a static spherically symmetric field, I would have the 1PN time-time component,

[tex] g_{00} = -1 + 2M/r - 2(M/r)^2[/tex]

Is the idea that, in this special case, we can make a change of gauge (coordinates) to eliminate the quadratic term but in the general case of many bodies (with a sum over masses) we cannot?


Register to reply

Related Discussions
Schwarzchild metric - rescaled coordinates Advanced Physics Homework 0
Schwarzchild spacetime singularity Special & General Relativity 5
Schwarzchild Metric Model Astrophysics 0
Schwarzchild mass Special & General Relativity 6
error in schwarzchild metric Special & General Relativity 4