Solutions to the space-time metric

In summary, the conversation is about a solution to a metric that might look like a Fourier series with arguments (\vec{k}\vec{r}-\omega_i t), where \vec{r} is along the radii of a black hole and ωi is the photon frequency. If this solution exists, it would describe the redshift/blueshift caused by traversing a gravitational potential.
  • #1
Mazulu
26
0
Can someone direct me to the solution to the space-time metric,
[itex]ds^2 = -dt^2 + dx^2 + dy^2 + dt^2[/itex]? Thanks.
 
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  • #2
You have 2 dt terms and they would cancel. What do you mean by solution to this metric? A metric is a metric...what do you want to solve for? It's like asking "what's the solution to 4?"

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  • #3
I think you mean [itex]ds^2= dt^2- dx^2- dy^2- dz^2[/itex]. But, as Matterwave asked, what do you mean by a "solution" to a metric? The geodesics? That is the metric for Euclidean space-time so the curvature tensor is 0 and all geodesics are straight lines.
 
  • #4
HallsofIvy said:
I think you mean [itex]ds^2= dt^2- dx^2- dy^2- dz^2[/itex]. But, as Matterwave asked, what do you mean by a "solution" to a metric? The geodesics? That is the metric for Euclidean space-time so the curvature tensor is 0 and all geodesics are straight lines.
It's a differential equation. Shouldn't there be an equation of the form [itex]s(t,x,y,z)[/itex] that when differentiated, will satisfy the equation. I actually wanted to solve some more difficult metrics, but I started with something easy (flat space-time). I was hoping that a solution of the form [itex]e^{i(kx+ky+kz-\omega t)}[/itex]} might pop out of it; or something that looks like light or a Poynting vector.

By the way, thank you for the heads up that a metric is used to find the curvature tensor of a geodesic. I just thought that Maxwell's equations should pop out of it as well.
 
  • #5
It's a differential line element. It tells you how "far" apart two events are. I guess if you wanted a "solution" in the form of S=S(t,x,y,z), it would be:

[tex]S=\sqrt{(t-t_0)^2-(x-x_0)^2-(y-y_0)^2-(z-z_0)^2}[/tex]

That's just the non-differential form of the equation. In general, doing something like this is not possible for general metric, but because of the niceness of the Minkowski metric, you can do this.
 
  • #6
Matterwave said:
It's a differential line element. It tells you how "far" apart two events are. I guess if you wanted a "solution" in the form of S=S(t,x,y,z), it would be:

[tex]S=\sqrt{(t-t_0)^2-(x-x_0)^2-(y-y_0)^2-(z-z_0)^2}[/tex]

That's just the non-differential form of the equation. In general, doing something like this is not possible for general metric, but because of the niceness of the Minkowski metric, you can do this.

I don't think the Minkowski metric is what I'm looking for. The AdS/CFT correspondence model contains both gravity and quantum mechanics. I think I need to look there. It would be amazing if I actually found what I'm looking for. There should be a solution that looks like a Fourier series with arguments [itex](\vec{k}\vec{r}-\omega_i t)[/itex], where [itex]\vec{r}[/itex] is along the radii of a black hole and ωi is the photon frequency along the radii . Such a solution would describe the redshift/blueshift caused by traversing a gravitational potential.
 

1. What is the space-time metric?

The space-time metric is a mathematical representation of the relationship between space and time in Einstein's theory of general relativity. It describes how the geometry of space and time is affected by the presence of massive objects, such as planets and stars.

2. How is the space-time metric calculated?

The space-time metric is calculated using the Einstein field equations, which relate the curvature of space-time to the distribution of matter and energy within it. This involves solving a set of complex differential equations.

3. What is the significance of the space-time metric?

The space-time metric is significant because it provides a framework for understanding the behavior of objects in the presence of massive objects. It allows us to accurately predict the motion of planets, stars, and other celestial bodies, and has been confirmed by numerous experiments and observations.

4. Can the space-time metric change?

According to Einstein's theory of general relativity, the space-time metric can be affected by the presence of massive objects and the distribution of matter and energy. This means that it can change over time as these factors change. However, the overall structure and principles of the metric remain the same.

5. Are there alternative solutions to the space-time metric?

There have been attempts to develop alternative theories of gravity that offer different solutions to the space-time metric, such as modified Newtonian dynamics and string theory. However, these theories have not yet been proven and the current understanding of the space-time metric remains the most widely accepted explanation for the behavior of objects in the universe.

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