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Whether a “reasonable” spacetime is geodesically connected depends on the definition of “reasonable”. If we consider globally hyperbolic spacetime as reasonable, then there is a theorem by Avez (1963) & Seifert (1967), that states that globally hyperbolic spacetimes are causally geodesically connected. Note, that the two points here must be causally ...


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There is, of course, no superluminal speed for photons in AdS spacetime, they travel at precisely the speed of light. Taking a look at the metric of AdS: $$ ds^{2}=-\left(1+\frac{r^2}{\ell^2}\right)\,dt^{2}+\frac {dr^2}{1+\frac{r^2}{\ell^2}}+r^{2}\,d\Omega _{n-2}^{2}, $$ one can notice that the $g_{tt}$ component diverges with growth of radial coordinate $r$....


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The metric is, if we set the angles to zero, $$ds^2 = c^2 dt^2 - a(t)^2 d\chi^2$$ where I use co-moving coordinates $\chi$. Setting $ds^2=0$ just gives the differential equation $$d\chi^2 = \frac{c^2}{a(t)^2}dt^2.$$ Taking the square root, $$d\chi = \pm \frac{c}{a(t)}dt.$$ Now we can integrate and get $$\int d\chi=\int \pm \frac{c}{a(t)}dt$$ which ...


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Singularities are defined properly by the following condition : For a spacetime $M$, such that every standard measurable quantity on it is defined (ie we won't allow the Riemann tensor to be divergent at a point and so forth), that spacetime is singular if there exists inextendible curves of bounded acceleration for which the curve's half-length (measured by ...


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It is not possible to have the topology you describe. In order for a gamma ray to create a particle antiparticle pair, it needs the electromagnetic field of a nucleus for energy and momentum conservation to work in the center of mass of the e+e- pair. In this feynman diagram Z is the nucleus in whose electromagnetic field the pair production can happen. ...


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There is a lot more than one curvature scalar out there. When considering scalars strictly linear in the curvature tensor, the only independent scalar is the Ricci scalar $R$. When considering scalars quadratic in the curvature tensor, one can choose the set of three independent scalars to be the Kretschmann scalar $K_1 = R^{\mu\nu\kappa\lambda}R_{\mu\nu\...


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There's a variety of metrics used for the Morris-Thorne metric. The two most common are the Schwarzschild coordinates and the proper radial coordinateS. The Schwarzschild coordinates are just done using the classic spherical symmetric coordinates, $$ds^2 = -e^{2\phi_{\pm}(r)}dt^2 + \frac{dr^2}{1 - b_\pm(r) / r} + r^2 d\Omega^2$$ This is done on two ...


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If we assume no torsion, and metric compatibility with the connection ($\nabla_{\mu}g_{\alpha\beta}=0$) there is the formula of: $$ \Gamma^{\mu}_{\alpha\beta}=\frac{1}{2}g^{\mu\rho}(\partial_{\alpha}g_{\rho\beta}+\partial_{\beta}g_{\rho\alpha}-\partial_{\rho}g_{\alpha\beta})$$


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