# Solutions to the space-time metric

• Mazulu

#### Mazulu

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

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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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.

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.

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.

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 ωi is the photon frequency along the radii . Such a solution would describe the redshift/blueshift caused by traversing a gravitational potential.