Equation of motion in curved spacetime

In summary: The first two terms on the left-hand side are just the conserved quantity for $Q^k$ and the momentum conjugate to $p_k$. The third term is just the momentum of the particle along the chosen geodesic. The last two terms are the constraint that the Hamiltonian must vanish. Solving for $Q^t$ and $H^t$, you get\begin{equation}\begin{split}Q^t &= -\frac{1}{2}\left(\frac{1}{r^2}+\frac{2}{r}\right)\dot{t}^2 - \frac{1}{2}\frac{2r^3+1}{(
  • #1
Markus Kahn
112
14
Homework Statement
Consider the spacetime metric given by
$$d s^{2}=-r^{2}\left(1-\frac{1}{r^{3}}\right) d t^{2}+\frac{d r^{2}}{r^{2}\left(1-\frac{1}{r^{3}}\right)}+r^{2}\left(d x^{2}+d y^{2}\right)$$.

1) Identify three quantities which are constant along geodesics, corresponding to the Killing vectors ##\partial_k## for ##k\in\{t,x,y\}##.
2) Show that the geodesic equations can be reduced to an equation of the form
$$\dot r ^2 + V(r)=K^2,$$
for some potential function ##V(r)##. Fix ##K## in terms of the constants of motion from 1).

optional: 3) Sketch the potential ##V(r)## fro timelike and null-geodesics.
Relevant Equations
All given above.
1) We know that for a given Killing vector ##K^\mu## the quantity ##g_{\mu\nu}K^\mu \dot q^\nu## is conserved along the geodesic ##q^k##, ##k\in\{t,r,x,y\}## . Therefore we find, with the three given Killing vectors ##\delta^t_0, \delta^x_0## and ##\delta^y_0## the conserved quantities
$$Q^t := -r^2 (1-r^{-3})\dot q^0,\quad \text{and}\quad Q^k := r^2\dot q^k \quad \text{for}\quad k \in \{x,y\}.$$

2) We first construct a Lagragian
$$L=\frac{1}{2}g_{\mu\nu} \dot q^\mu \dot q^\nu = -\frac{1}{2}r^2\left(1-\frac{1}{r^3}\right)\dot t^2 + \frac{1}{2r^2(1-\frac{1}{r^3})}\dot r^2 +\frac{1}{2}r^2(\dot x^2 +\dot y^2). $$
We then find
$$\frac{\partial L}{\partial q^k} = 0 \quad \text{and}\quad \frac{\partial L }{\partial \dot q^k} = Q^k\quad\text{for}\quad k\in\{t,x,y\}$$
and
$$\frac{\partial L}{\partial r} = -\frac{1}{2}\left(\frac{1}{r^2}+2r\right)\dot t^2 - \frac{1}{2}\frac{2r^3+1}{(r^3-1)^2}\dot r^2 + r(\dot x^2 + \dot y^2)\quad \text{and}\quad \frac{\partial L }{\partial \dot r} = \frac{1}{2}\left(1-\frac{1}{r^3}\dot r\right).$$
Now, since ##Q^k## is conserved in time, three of the four geodesic equations are automatically satisfied:
$$\frac{d}{dt}\frac{\partial L}{\partial \dot q^k}-\frac{\partial L}{\partial q^k} =0 \quad\Longleftrightarrow \quad \frac{d Q^k}{dt}= 0 \quad \text{for}\quad k\in\{t,x,y\}.$$
The only non-trivial eom is the one for ##r##. The issue is, I can't get it into the desired form... What I get is:
$$\frac{d}{dt}\frac{\partial L}{\partial \dot r}-\frac{\partial L}{\partial r} =0 \quad\Longleftrightarrow \quad \ddot r - \frac{2r^3+1}{2r(r^3-1)}\dot r^2 -\frac{(Q^x)^2 + (Q^y)^2}{r}\left(1-\frac{1}{r^3}\right) -\frac{1}{2}\frac{2 r^3 + 1}{r - r^4}(Q^t)^2 =0$$
I tried to incorporate the different ##Q^k##, but I just don't see how this is supposed to reduce to the desired result...

3) Here I'm just curious how exactly the fact that the particle is spacelike, timelike or lightlike could influence how ##V(r)## looks like.. I mean, where exactly could that information enter?
 
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  • #2
A:For the second part of your question, it may be easier to solve using the Hamiltonian formulation. The canonical momentum conjugate to $r$ is given by\begin{equation}p_r = \frac{\partial L}{\partial \dot{r}} = \frac{1}{2}\left(1 - \frac{1}{r^3}\right) \dot{r}.\end{equation}The Hamiltonian is then given by\begin{equation}H = p_r \dot{r} - L = \frac{1}{4}\left(1 - \frac{1}{r^3}\right)\dot{r}^2 + \frac{1}{2}r^2 \left(\dot{x}^2 + \dot{y}^2\right) - \frac{1}{2}r^2 \left(1 - \frac{1}{r^3}\right)\dot{t}^2.\end{equation}Using $Q^k$ and $H$, you can rewrite the equation of motion as\begin{equation}\ddot{r} = -\frac{1}{2}\left(\frac{1}{r^2} + \frac{2}{r}\right)\dot{t}^2 - \frac{1}{2}\frac{2r^3 + 1}{(r^3 - 1)^2}\dot{r}^2 + \frac{1}{r}\left(1 - \frac{1}{r^3}\right)(Q^x)^2 + \frac{1}{r}\left(1 - \frac{1}{r^3}\right)(Q^y)^2 + \frac{1}{2}\frac{2r^3 + 1}{r^3 - 1}(Q^t)^2 - \frac{\partial H}{\partial r}.\end{equation}Now consider the constraint that the particle must be either spacelike, timelike or lightlike. This is equivalent to the constraint that the Hamiltonian must vanish, i.e.\begin{equation}H(q^k,p_k) = 0.\end{equ
 

Related to Equation of motion in curved spacetime

1. What is the equation of motion in curved spacetime?

The equation of motion in curved spacetime is a mathematical representation of how objects move through space and time in the presence of a gravitational field. It takes into account the curvature of spacetime caused by massive objects, as described by Einstein's theory of general relativity.

2. How is the equation of motion in curved spacetime different from Newton's second law?

The equation of motion in curved spacetime differs from Newton's second law in that it accounts for the effects of gravity on the motion of objects. Newton's second law only applies in flat, non-accelerating spacetime, whereas the equation of motion in curved spacetime takes into account the curvature of spacetime caused by massive objects.

3. What is the significance of the equation of motion in curved spacetime?

The equation of motion in curved spacetime is significant because it allows us to accurately describe the motion of objects in the presence of a gravitational field. It is a fundamental concept in understanding how gravity works, and it has been confirmed by numerous experiments and observations.

4. Can the equation of motion in curved spacetime be applied to all objects?

Yes, the equation of motion in curved spacetime can be applied to all objects, regardless of their mass or size. However, its effects are only noticeable on a large scale, such as with planets, stars, and galaxies.

5. How does the equation of motion in curved spacetime affect our understanding of the universe?

The equation of motion in curved spacetime has greatly impacted our understanding of the universe by providing a more accurate and comprehensive description of gravity. It has also allowed us to make predictions and observations that have been confirmed by experiments and observations, furthering our knowledge of the cosmos.

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