1PN metric vs. Schwarzchild

In summary, the conversation discusses the time-time component of the GR metric and the Schwarzschild metric, both of which have different expressions due to a difference in coordinate systems or gauges. While the Schwarzschild metric is an exact solution of the field equations, the 1PN metric has a quadratic term in the Newtonian potential. However, in the special case of a static spherically symmetric field, the quadratic term can be eliminated through a change of gauge.
  • #1
andert
12
0
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 Schwarzschild 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?
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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 Schwarzschild 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?
 

1. What is the difference between the 1PN metric and the Schwarzschild metric?

The 1PN metric is a more accurate description of spacetime in the vicinity of a spherical, non-rotating mass compared to the Schwarzschild metric. It incorporates effects of the mass' linear and angular momentum, while the Schwarzschild metric assumes a stationary mass with no angular momentum.

2. How is the 1PN metric derived from the Schwarzschild metric?

The 1PN metric is derived using the post-Newtonian approximation, which expands the Schwarzschild metric to include terms up to first-order in the ratio of the mass to a characteristic length scale, such as the radius of the mass. This results in a more precise description of the spacetime around the mass.

3. Which metric is more accurate for describing the spacetime near Earth?

The 1PN metric is a more accurate description of the spacetime near Earth, since Earth has both linear and angular momentum. The Schwarzschild metric is a good approximation for massive objects with no angular momentum, such as black holes.

4. Is the 1PN metric used in any practical applications?

Yes, the 1PN metric is used in precision tests of general relativity, such as the measurement of the perihelion precession of Mercury. It is also used in the calculation of gravitational waves from compact binary systems.

5. Can the 1PN metric be extended to include higher-order corrections?

Yes, the 1PN metric can be extended to include higher-order corrections, such as the 2PN metric which includes terms up to second-order in the ratio of the mass to a characteristic length scale. These higher-order corrections improve the precision of the metric and are important in understanding gravitational effects in extreme environments.

Similar threads

Replies
13
Views
522
  • Special and General Relativity
Replies
7
Views
2K
  • Special and General Relativity
2
Replies
43
Views
2K
Replies
12
Views
1K
  • Special and General Relativity
Replies
12
Views
2K
  • Special and General Relativity
Replies
18
Views
1K
  • Special and General Relativity
Replies
6
Views
1K
  • Special and General Relativity
Replies
8
Views
1K
  • Special and General Relativity
2
Replies
45
Views
2K
  • Special and General Relativity
Replies
8
Views
1K
Back
Top