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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.
[itex]ds^2 = -dt^2 + dx^2 + dy^2 + dt^2[/itex]? Thanks.
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.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 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.