General relativity and time dilation

In summary, the concept of time dilation in general relativity can be simplified by using the approximation T=T0/(sqrt(1-2Gm/Rc^2)) for weak gravitational fields. This formula is an exact solution for the Schwarzschild metric, which describes a single massive body. However, for multiple massive objects, one must solve Einstein's field equations to find the metric, which is a difficult task. Therefore, the approximation formula is commonly used in practical, solar-system situations. It is important to note that GR does not have a potential in the same sense as Newtonian gravity, and therefore, talking about "the potential" in GR can be misleading.
  • #1
Joe1
11
0
Hello,
I have a question about general relativity and time dilation. I found an expression for the time dilation around a sphere of radius R and mass m.
T=T0/(sqrt(1-2Gm/Rc^2))
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html
Is there a way to calculate the time dilation for a different structure directly from the potential? Can I just substitute my potential in for Gm/R in the equation above?

Thanks,
Joe
 
Physics news on Phys.org
  • #2
Basically, yes, as long as you are in a "weak field" approximation. "Weak field" would include anything in the solar system.

Note that in the weak field, T0/sqrt(1-2GM/rc^2) is essentially equal to
T0*(1+GM/rc^2)
 
  • #3
Joe1 said:
Hello,
I have a question about general relativity and time dilation. I found an expression for the time dilation around a sphere of radius R and mass m.
T=T0/(sqrt(1-2Gm/Rc^2))
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html
Is there a way to calculate the time dilation for a different structure directly from the potential? Can I just substitute my potential in for Gm/R in the equation above?

Thanks,
Joe
In general relativity the term "potential" has a different expanded usage. The 10 independant components of the metric tensor are the potentials of the gravitational field. That's why the metric is sometimes referred to as the tensor potential. The general expression for time dilation is

d[tex]tau[/tex] = sqrt(g00)dt

Pete
 
  • #4
Pete's response is at cross purposes from mine.

While some people do call the metric coefficients the "gravitational potential", I find that it is less confusing to just call the metric coefficients the metric coefficients.

The potential I was talking about was the Newtonian potential.

In the weak field approximation, time dilation is just 1+U/c^2, where U is the negative of the Newtonian potential,

i.e U = GM1/R1+GM2/R[2]...+GMn/Rn

This will be the same as 1/sqrt(1-2U) in any situation in which the approximation is valid.

Here the Mi are the masses, and the Ri are the distances that a point P is away from the masses.

This is also called the PPN approximation.

The PPN approximation requires non-relativistic velocities as well as a weak field, but is routinely used for problems in the solar system.

If you have a general metric, you can always use sqrt(|g00|) or it's reciprocal as the time dilation factor. (The reciprocal factor depends on whether one wants a number lower than 1 to represent a slower clock or a number larger than 1 to represent a slower clock - the first form uses the first convention, the reciprocal form uses the second convention). Stationary clocks in a gravity well always tick more slowly than stationary clocks at infinity.

The problem with this approach in general is that finding the metric for an arbitrary configuration is not an easy task. Fortunately strong field situations are rare, and the PPN approximation works fine for solar-system problems.
 
Last edited:
  • #5
ok, so T=T0/(sqrt(1-2Gm/Rc^2)) is an approximation for weak gravitational fields, but for massive objects it is necessary to use the equation that Pete mentioned. How would I find g00? Can this be determined directly from the potential?

Thanks for your help,
Joe
 
  • #6
g00 is a component of the metric tensor. The metric tensor satisfies the gravitational field equation; one must therefore solve the field equation for the 00 component. This is, in general, quite a difficult thing to do. However, for some simple systems people have solved it (i.e. Schwarzschild solution etc.)
 
  • #7
Joe1 said:
ok, so T=T0/(sqrt(1-2Gm/Rc^2)) is an approximation for weak gravitational fields, but for massive objects it is necessary to use the equation that Pete mentioned. How would I find g00? Can this be determined directly from the potential?

Thanks for your help,
Joe

T=T0/(sqrt(1-2Gm/Rc^2))

is actually exact, because it's in the form T = T0 / sqrt(|g00|).

Look at for instance
http://en.wikipedia.org/wiki/Schwarzschild_metric

and see that g00 for the Schwarzschild metric is just the coefficient of c^2 dt^2, i.e.

g00 = -1 + 2GM/Rc^2

Thus the formula you cite is exact for the Schwarzschild metric, which is the exact solution for a single massive body.

If you have a single massive object, this above formula is good. If you have more than one massive object, to get an exact solution you'd have to solve Einstein's field equations to find the metric. Analytic solutions are only known for 1 body, so you'd have to solve Einstein's equations numerically. This is an extremely difficult task. Therfore people use the approximation I mentioned.

Talking about "the potential" in GR is just going to confuse the issue, I'm afraid.

Talking about the Newtonian potenital makes sense in the Newtonian approximation, which is fine for practical, solar-system purposes. Talking about "the potential" in GR is making a conceptual error. GR doesn't have a potential in the same sense that Newtonian gravity does.
 

1. What is general relativity?

General relativity is a theory of gravitation developed by Albert Einstein in the early 20th century. It describes the relationship between mass, energy, and the curvature of space and time.

2. How does general relativity explain time dilation?

According to general relativity, time is relative and can be affected by gravity. This means that the closer an object is to a massive body, the slower time will pass for that object. This effect is known as time dilation.

3. Can time dilation be observed in everyday life?

Yes, time dilation can be observed in everyday life. For example, GPS satellites have to account for time dilation due to their high speed and distance from Earth's surface in order to provide accurate location data.

4. What are the implications of time dilation for space travel?

Time dilation has significant implications for space travel. As a spacecraft approaches the speed of light, time will appear to slow down for the travelers on board. This means that they will experience less time passing compared to those on Earth, potentially allowing for longer space missions.

5. Is time dilation the same as time travel?

No, time dilation is not the same as time travel. While time dilation can cause time to appear to pass at different rates for different observers, it does not allow for jumping to different points in time. Time travel, on the other hand, refers to the concept of moving backwards or forwards in time to a specific point.

Similar threads

  • Special and General Relativity
Replies
2
Views
1K
  • Special and General Relativity
2
Replies
36
Views
2K
  • Special and General Relativity
Replies
19
Views
1K
  • Special and General Relativity
Replies
16
Views
2K
  • Special and General Relativity
2
Replies
62
Views
3K
  • Special and General Relativity
Replies
25
Views
827
  • Special and General Relativity
Replies
16
Views
956
  • Special and General Relativity
2
Replies
58
Views
3K
  • Special and General Relativity
Replies
1
Views
701
  • Special and General Relativity
2
Replies
37
Views
3K
Back
Top