Orthogonal diretion on the Minkowski diagram

[mersecske]

I have a trivial question:

Let assume a world sheet of a time-like spherical shell in Minkowski space-time.
On the 2D-Minkowski diagram (R,T), where R is the radius and T is the time,
the world line is represented by a time-like curve.
Let assume that the shell collapse and its 4-velocity is

[tex]u^\alpha=(\dot{T},\dot{R},0,0) [/tex]

where [tex]\dot{T}=\mathrm{d}T/\mathrm{d}\tau >0[/tex] and [tex]\dot{R}=\mathrm{d}R/\mathrm{d}\tau <0[/tex].

Let [tex]n^\alpha[/tex] be an orthogonal vector to the 4-velocity and the surface.
With the metric diag(-1,1,1,1) we get:

[tex]n_\alpha=s (-\dot{R},\dot{T},0,0) [/tex]
[tex]n^\alpha=s (\dot{R},\dot{T},0,0) [/tex]

where s is the sign depending on the orientation.

I expect that [tex]n^\alpha[/tex] is orthogonal to [tex]u^\alpha[/tex] on the Minkowski diagram, and its space-like. For example, I expect that the outward oriented normal vector has positive coordinates. But since [tex]\dot{T}>0[/tex] and [tex]\dot{R}<0[/tex] this is not true, and for example for s=+1, [tex]n^\alpha[/tex] is a past directed vector on the diagram!?

 

[DrGreg]

I have a trivial question:

Let assume a world sheet of a time-like spherical shell in Minkowski space-time.
On the 2D-Minkowski diagram (R,T), where R is the radius and T is the time,
the world line is represented by a time-like curve.
Let assume that the shell collapse and its 4-velocity is

[tex]u^\alpha=(\dot{T},\dot{R},0,0) [/tex]

where [tex]\dot{T}=\mathrm{d}T/\mathrm{d}\tau >0[/tex] and [tex]\dot{R}=\mathrm{d}R/\mathrm{d}\tau <0[/tex].

Let [tex]n^\alpha[/tex] be an orthogonal vector to the 4-velocity and the surface.
With the metric diag(-1,1,1,1) we get:

[tex]n_\alpha=s (-\dot{R},\dot{T},0,0) [/tex]
[tex]n^\alpha=s (\dot{R},\dot{T},0,0) [/tex]

where s is the sign depending on the orientation.

I expect that [tex]n^\alpha[/tex] is orthogonal to [tex]u^\alpha[/tex] on the Minkowski diagram, and its space-like. For example, I expect that the outward oriented normal vector has positive coordinates. But since [tex]\dot{T}>0[/tex] and [tex]\dot{R}<0[/tex] this is not true, and for example for s=+1, [tex]n^\alpha[/tex] is a past directed vector on the diagram!?

Orthogonality in Minkowski space does not "look like" Euclidean orthogonality on a diagram. Two vectors drawn on a diagram which are Minkowski-orthogonal won't (usually) have an angle of 90° on your Euclidean piece of paper.

In fact, because of the way that lowering of indexes works, you will find that the covector [itex]n_\alpha[/itex] is "Euclidean-orthogonal" to the vector [itex]u^\alpha[/itex], if you draw both on the same diagram, since

[tex]n_\alpha u^\alpha = 0[/tex]​

(Of course, strictly speaking vectors and covectors live in different spaces, so you shouldn't really draw them on the same diagram, but you can get away with it in Minkowski coordinates.)

 

[mersecske]

If somehow you draw them on the same diagram, and they are Euclidean-orthogonal, this means that you suppose the basis unit vectors are the same (representations are the same) in the two kind of vector space, which is not a good choice, I think!