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For a null geodesic curve $X^i$, $$0=g_{ij}V^iV^j.$$ When we derive the geodesic equation from E-L equations, will this affine parametrization cause it to blow up? How is it justified to use the geodesic equation, which is derived from space/time-like parametrization, for null geodesics?

Qmechanic
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Shadumu
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1 Answers1

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Actually, you don't have to use proper time for a parametrisation of the Euler-Lagrange-Equations / geodesic equations in GR. Just take any parametrisation you want. However, if you solve the equations and use initial conditions for a time/light/space-like-path, that geodesic will stay time/light/space-like with $$ g_{ij}V^iV^j = \text{const.} $$ over the whole path $X^i$.

Proof in: Steven Weinberg - Gravitation and Cosmology (page: 76)

image357
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  • Of course, if you choose a parametrization of any curve satisfying $$ g_{ij}V^iV^j = \text{const.} $$. This remains true along the curve. I am asking if the constant is zero, which makes the curve null and any reparametrization does not change the zero, what is the affine parameter and what is the geodesic equation derived from it? – Shadumu May 04 '15 at 21:10
  • The affine parameter in GR is the proper time $\tau = \int d\tau = \int \sqrt{g_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}}~d\lambda$. The statements $d\tau > 0$, $d\tau < 0$ and $d\tau = 0$ are invariant under any coordinate transformation and parametrization. If you are talking about nullgeodesics proper time is no longer a good parametrization. You have to take a different parametrization $\lambda \ne \tau$ then. The geodesic equations for this parametrization is still the Euler-Lagrange Equation for $L = \sqrt{g_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}}$ – image357 May 04 '15 at 22:42
  • @user3229471: A solution to the geodesics equations that satisfies $g_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\mu}{d\lambda} = 0$ once on the path will satisfy it everywhere on the path and for any parametrization $\lambda$. In this sense, the statement that a path is a nullgeodesic is a parametrization-independent statement ( as it should be, since a path doesn't "care" about its parametrization). – image357 May 04 '15 at 22:53
  • So the real difference about time/light/space-like paths is just the initial conditions you impose on them. They are all solutions of the same geodesic equations (which are the Euler-Lagrange-Equations for a stationary proper time $\tau$ and a certain parametrization $\lambda$) – image357 May 04 '15 at 23:06
  • Acutally, because $d\tau > 0$ or $d\tau < 0$ is invariant over the whole path, is why one can still take $\tau$ as a (natural) parametrization of time/space-like geodesics in GR. – image357 May 04 '15 at 23:11
  • I just mean if it is the null case, it is hard to show how to extermize the Lagrangian and get the usual geodesic equation. and What is the affine parameter if it is not proper time or length – Shadumu May 05 '15 at 01:17
  • @user3229471: the fact that for null-geodesics $\tau = \int d\tau = 0$ doesn't mean you cannot apply calculus of variation to the functional (!) $\tau$. In usual euclidean manifolds, the affine parameter $\tau$ is the length along the path. It can be used as a parametrization because this length is a monotone function along the path. In GR this monotony holds for time/space-like paths too, such that it is still a valid parametrization. However, $\tau = 0$ along the path for null-geodesics. – image357 May 05 '15 at 10:46
  • This is why you have to choose a different parametrization. Still, the Euler-Lagrange-Equations are the ones to solve if you wanna get geodesics. You can express these equations with paramatizations $\lambda \ne \tau$. There is no difficulty to this. However, those equations are likely to transform differently under coordinate transformations since in general $\lambda$ won't be a scalar then. Still, this is no problem. – image357 May 05 '15 at 10:49
  • It might be interesting to think of this as a limiting process. We view a null geodesic as the limit of timelike geodesics. – Drake Marquis Aug 12 '15 at 19:48