Light Cone Analogue in Minkowski Space: Exploring Null Rays

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in Minkowksi, the set of all possible null rays from a point defines a cone (light cone).

Now imagine I change the signature of Minkowski from (-,+,+,+) to (-,-,+,+) i.e. a space with two timelike directions and a metric ##ds^2=-dx_1^2-dx_2^2+dx_3^2+dx_4^2##. What kind of surface would the set of null rays form? Is it still a cone? Or is it something else?

Thanks

 

[Simon Bridge]

How would you go about showing that the set of possible null-rays forms a cone in the regular metric?

 

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How would you go about showing that the set of possible null-rays forms a cone in the regular metric?

Probably by noting that possible null trajectories have ##ds^2=0## and by differentiating with respect to the affine parameter, we see this corresponds to ##x_1^2=x_2^2+x_3^2+x_4^2## (working with Minkowski at the moment). This can be recognised as the equation of a cone (really a 4d hypercone I suppose).

Now for the case at hand we'd arrive at something like ##x_1^2+x_2^2=x_3^2+x_4^2## and I'm not sure how to interpret this? Would there be an apex in two of the directions?

 

[fresh_42]

Writing it ##x_1^2 = - x_2^2 + x_3^2+x_4^2## may help.

 

[adsquestion]

Writing it ##x_1^2 = - x_2^2 + x_3^2+x_4^2## may help.

so it's like a "hyper-hyperboloid"? it looks like each choice of ##x_1## gives a different shaped hyperboloid.

 

[fresh_42]

I'm not quite sure for my imagination of 4-d objects isn't very well. But to illustrate the light cone you already contracted two space dimensions to one, i.e. the light cone is actually the shape of emerging circles ##x_1^2 =x_2^2+x_3^2## of radius ##x_1 ∈ [0,∞[ ##. Applying the same here would give us ##x_1^2 = - x_2^2 + x_3^2## with ##x_1 ∈ [0,∞[ ##, a hyperboloid. At least this is my understanding of the situation.