Solutions to the space-time metric

[Mazulu]

Can someone direct me to the solution to the space-time metric,
$ds^2 = -dt^2 + dx^2 + dy^2 + dt^2$? Thanks.

 

[Matterwave]

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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[HallsofIvy]

I think you mean $ds^2= dt^2- dx^2- dy^2- dz^2$. 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.

 

[Mazulu]

I think you mean $ds^2= dt^2- dx^2- dy^2- dz^2$. 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 $s(t,x,y,z)$ 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 $e^{i(kx+ky+kz-\omega t)}$} 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.

 

[Matterwave]

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:

$$S=\sqrt{(t-t_0)^2-(x-x_0)^2-(y-y_0)^2-(z-z_0)^2}$$

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.

 

[Mazulu]

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:

$$S=\sqrt{(t-t_0)^2-(x-x_0)^2-(y-y_0)^2-(z-z_0)^2}$$

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 $(\vec{k}\vec{r}-\omega_i t)$, where $\vec{r}$ is along the radii of a black hole and ω_i is the photon frequency along the radii . Such a solution would describe the redshift/blueshift caused by traversing a gravitational potential.