Derivation of energy and angular momentum (Schwarzschild metric)

[Lambda96]

Homework Statement

Derive the equations $E= mc^2 \bigl( 1- \frac{r_s}{r} \bigr) \dot{t}$ and $l=mr^2 \dot{\varphi}$

Relevant Equations

none

Hi,

Unfortunately, I can't quite work out the terms in the task

The expression in the integral (4) corresponds to the Lagrangian and since $t$ and $\varphi$ are cyclic variables, the following follows:

$$-\frac{d}{dt} \frac{d L}{d \dot{t}}=0 \quad \rightarrow \quad -\frac{d L}{d \dot{t}}=const=E$$

$$-\frac{d}{dt} \frac{d L}{d \dot{\varphi}}=0 \quad \rightarrow \quad -\frac{d L}{d \dot{\varphi}}=const=l$$

Then I received the following:

$$\frac{d}{d \dot{t}} \Bigl( m \bigl( 1- \frac{r_s}{r} \bigr) c^2 \dot{t}^2 \Bigr)= 2m \bigl( 1- \frac{r_s}{r} \bigr) c^2 \dot{t} =E$$

$$\frac{d}{d \dot{\varphi}} \Bigl( m r^2 \dot{\varphi}^2 \Bigr)= 2m r^2 \dot{\varphi}=l$$

I get the same results except for the 2, just my question, did I do something wrong or was the 2 forgotten in the task sheet?

 

[Orodruin]

You simply called what is usually called $E$ by $2E$ and the same for angular momentum. If $E$ is constant, then obviously $2E$ is constant as well.

 

[Ibix]

As Orodruin says, you can simply take the 2 into the definition of the constant, or not. It makes no difference to the orbits.

However, the reason for the book not having a 2 is that they've chosen their constants to match up to customary definitions. For example, instead of defining $-dL/d\dot{t}=E$ you could have used $-dL/d\dot{t}=A$ where $A$ is an unknown constant. Then you'd get to $2m\left(1-\frac{r_s}r\right)c^2\dot{t}=A$. Now consider an orbit where the particle comes instantaneously to rest at infinity. What happens to the LHS? And hence how would you define $A$?

Essentially, you jumped in a bit early with assuming the first constant you saw was $E$. If you don't assume that and instead use a generic unknown constant, then you can find a case where you can write that constant in terms of what you know.

 

[Lambda96]

Thanks Orodruin for your help , I thought I had done something wrong I hadn't thought of this simple observation

Thanks also Ibix for your help