Recent content by adsquestion

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

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...

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...

The Gibbons Hawking boundary term is given as ##S_{GHY} = -\frac{1}{8 \pi G} \int_{\partial M} d^dx \sqrt{-\gamma} \Theta##. I want to calculate its variation with respect to the induced boundary metric, ##h_{\mu \nu}##. The answer (given in eqns 6&7 of...

I want to try and see the intersection between the hyperboloid and the 2-plane giving an ellipse. So far I have the following: I'm going to work with ##AdS_3## for simplicity which is the hyperboloid given by the surface (see eqn 10 in above notes for reason) ##X_0^2-X_1^2-X_2^2+X_3^2=L^2## If...

Actually, let me rephrase/simplify the questions I was asking in post #9 to the following: 1, Why does intersecting a surface with a plane give the geodesics? 2, Looking at Figure 2.3 of the link I provided in post #1, we see the Penrose diagram for AdS including the aforementioned sinusoidal...

Yep I think this calculation assumes they all have initial position ##\rho=0## and then the initial velocity will depend only on energy E. So I'm wondering about how, given that these sine curves are the ellipses we get by intersecting the AdS hyperbolic by planes, why does intersecting by...

Sorry, I thought we agreed that timelike geodesics were sinusoidal when calculated with the full, uncompactified AdS metric?

Thanks very much again. The fact that the timelike geodesics are sinusoidal curves agrees with the fact that they are expected to be ellipses (which are parametrically sine curves) since the intersection of the AdS hyperboloid by a plane gives an ellipse as seen in Figure 11 of these nice...

I believe I've read that null geodesics can reach the boundary of AdS space within finite affine parameter and that this allows for a causal connection between the bulk AdS spacetime and the boundary on which the CFT lives and that this is very important for AdS/CFT. I can't find a reference...

Thanks for your reply. This kind of makes sense. But what is the reason for drawing the uncompactified geodesics on the compactified metric diagram? Is it just because it's "the best we can do" in terms of visualisation? Or is there a deeper reason relating to the fact that my calculation which...

I'm trying to understand why timelike geodesics in Anti de-Sitter space are plotted as sinusoidal waves on a Penrose diagram (a nice example of the Penrose diagram for AdS is given in Figure 2.3 of this thesis: http://www.nbi.dk/~obers/MSc_PhD_files/MortenHolm_Christensen_MSc.pdf). Bearing in...

Right and we can show it's possible in the black string case since ##\partial_T## is hypersurface orthogonal for all r. Still though, for Kerr, suppose I take the metric in Boyer-Lindquist coordinates and show that ##\partial_t## doesn't satisfy the Frobenius equation - what does that tell me...

To include the hole would require what? A global coordinate system in which ##g_{t \phi}=0 \forall r##?

Earlier we said that there was no frame in Kerr in whcih the black hole was not rotating. But we know that an observer at infinity sees no rotation since ##g_{t \phi} \rightarrow 0##. Is the point then that this is only true locally and as soon as we move away from infinity, we would observe...