Can massless particles reach the boundary at z=0?

[Poirot]

Homework Statement

We're working in 2-d Anti-de Sitter space with metric: \begin{eqnarray*}ds^2 = \frac{1}{z^2}(-dt^2 + dz^2)\end{eqnarray*} with 0<=z.

The solution is: \begin{eqnarray*}z^2 = (t+c)^2 + B\end{eqnarray*} And we've been asked to plot this (I think its a parabola with minima at t=-c, z=(B)^1/2 (told that B>0). (I'm not sure if the c in this case is the speed of light or just a constant, it hasn't been said)
The last part asks if massless particles can reach the boundary z=0?

Homework Equations

The Attempt at a Solution

I don't really understand this to be honest. I know that the geodesic L=0 for massless particles but I don't really know the physics implications for a massless particle. I assume it's to do with the fact that massless particles travel at the speed of light so perhaps there's a consequence at the boundary?

Thanks for any help, I'm a little confused!

 

[mark726]

Hello,

First of all, let's clarify what the metric and solution mean in this context. The metric given is the 2-dimensional Anti-de Sitter space, which is a type of curved spacetime. The solution given is an equation that describes the shape of a geodesic (the path that a particle follows) in this space. The constant c in the solution represents the starting point of the geodesic and B represents the curvature of the space.

Now, to answer your question about massless particles reaching the boundary z=0, we need to consider the implications of this solution. As you correctly stated, the geodesic for a massless particle has L=0, which means that the particle travels at the speed of light. In this case, the solution tells us that the particle will follow a parabola shape, with its minimum at t=-c and z=(B)^1/2.

If we try to plug in z=0 into the solution, we get a problem because the term (B)^1/2 becomes undefined. This means that a massless particle cannot reach the boundary z=0, as it would require an infinite amount of time to do so. This is a consequence of the curvature of the space, which prevents massless particles from reaching certain points.

I hope this helps to clarify the situation. Let me know if you have any other questions.