Given a Metric, find the constants of motion

[MathematicalPhysicist]

Main Question or Discussion Point

I have another question, if I am given a metric and I want to find the constants of motion of that system, then how do I do it?

Thanks!

 

[MathematicalPhysicist]

To be more specific I have the metric: $ds^2=-dt^2+a(t)^2(dx^2+dy^2+dz^2)+2dt(f(t)dx+h(t)dy+g(t)dz)$

and I want to find the constants of motions, relate $U^\mu = dx^\mu/d\tau$ to these constants, and find $t(\tau)$ with respect to these constants?

I just need guidelines that hopefully will work for any metric not just this!

 

[MathematicalPhysicist]

OK, I think I found them for the Lagrangian is $L=-\dot{t}^2+a(t)^2(\dot{x}^2+\dot{y}^2+\dot{z}^2)+2\dot{t}\dot{x}f(t)+2h(t)\dot{t}\dot{y}+2g(t)\dot{t}\dot{z}$ and then to use the Euler Lagrange equation with respect to cyclic variable which are x,y,z and t.

I am not sure with the second part of my question.

 

[Orodruin]

In your expression, t is certainly not cyclic, since the integrand depends on it. You do not get $\partial L/\partial\dot t = const.$

The general way of finding the constants of motion is to look for the Killing vectors and Killing tensors of your manifold, ie, its symmetries. However, there is no guarantee that this is easy to do and no general recipe.

Also, you cannot guarantee to find all constants of motion by looking at what coordinates are cyclic. Your coordinates may be chosen in a way that obscures this.

Also, you have yet to complete the original task in this thread so I am not sure why you put another question in here (that should probably go in its own thread).

 

[MathematicalPhysicist]

In your expression, t is certainly not cyclic, since the integrand depends on it. You do not get $\partial L/\partial\dot t = const.$

The general way of finding the constants of motion is to look for the Killing vectors and Killing tensors of your manifold, ie, its symmetries. However, there is no guarantee that this is easy to do and no general recipe.

Also, you cannot guarantee to find all constants of motion by looking at what coordinates are cyclic. Your coordinates may be chosen in a way that obscures this.

Also, you have yet to complete the original task in this thread so I am not sure why you put another question in here (that should probably go in its own thread).

Then, for the example I gave I get that the constants of motion are:

$$p_x=2a^2(t)\dot{x}+2f(t)\dot{t}$$
$$p_y=2a^2(t)\dot{y}+2h(t)\dot{t}$$
$$p_z= 2a^2(t)\dot{z}+2g(t)\dot{t}$$

How to find $U^\mu = dx^\mu/d\tau$ and $t(\tau)$?

 

[berkeman]

should probably go in its own thread

Done, posts broken out of the previous thread into this new thread.

 

[MathematicalPhysicist]

@Orodruin or anyone else, I still didn't get an answer as to how to find in my example $U^\mu$?

 

[MathematicalPhysicist]

ANyone?

 

[PeterDonis]

How to find $U^\mu = dx^\mu/d\tau$ and $t(\tau)$?

You can't find those out from just the metric. You need some information about how the particular object whose 4-velocity $U^\mu$ you want to obtain is moving; 4-velocity is a property of a particular object, not the spacetime geometry.

 

[stevendaryl]

Then, for the example I gave I get that the constants of motion are:

$$p_x=2a^2(t)\dot{x}+2f(t)\dot{t}$$
$$p_y=2a^2(t)\dot{y}+2h(t)\dot{t}$$
$$p_z= 2a^2(t)\dot{z}+2g(t)\dot{t}$$

How to find $U^\mu = dx^\mu/d\tau$ and $t(\tau)$?

You have those 3 constants. So you can solve for $\dot{x}$ in terms of $p_x$ and $\dot{t}$ and similarly for $\dot{y}$ and $\dot{z}$. Then you can plug those back into the formula for $L$. $L$ is also a constant (it has value 1). So you get an equation involving $\dot{t}$. You can solve that for $t$ (presumably) and use the equation for $\dot{x}$ to get a differential equation for $x$. Similarly, you can solve for $y$ and $z$.

 

[PeterDonis]

$L$ is also a constant (it has value 1).

Actually it's $-1$ with the signature convention the OP is using.

So you get an equation involving $\dot{t}$. You can solve that for $t$ (presumably) and use the equation for $\dot{x}$ to get a differential equation for $x$. Similarly, you can solve for $y$ and $z$.

This won't give a unique solution since there will be constants of integration involved.

Also, this method assumes that the object in question is traveling on a geodesic. It won't work for objects on non-geodesic worldlines.

 

[stevendaryl]

Actually it's $-1$ with the signature convention the OP is using.



This won't give a unique solution since there will be constants of integration involved.

Also, this method assumes that the object in question is traveling on a geodesic. It won't work for objects on non-geodesic worldlines.

I thought he was asking for how to find the geodesics.

 

[PeterDonis]

I thought he was asking for how to find the geodesics.

Could be, but the description in posts #1 and #2 of this thread is leaving a lot out if that's the case.