# Given a Metric, find the constants of motion

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• MathematicalPhysicist
I would assume that a geodesic curve is desired, but that is not stated explicitly.Agreed. The question is not clear at all. In summary, the conversation discusses the process of finding the constants of motion for a given metric, using the Euler-Lagrange equation and looking for Killing vectors and tensors. However, this method may not always be applicable and there is no general recipe for finding the constants of motion. In the specific example given, the constants of motion are found to be related to the momentum and position variables. To find the 4-velocity and proper time, more information about the object's motion is needed, such as whether it is following a geodesic.

#### MathematicalPhysicist

Gold Member
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!

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!

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.

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).

Orodruin said:
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)##?

Orodruin said:
should probably go in its own thread
Done, posts broken out of the previous thread into this new thread.

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

ANyone?

MathematicalPhysicist said:
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.

MathematicalPhysicist said:
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##.

stevendaryl said:
##L## is also a constant (it has value 1).

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

stevendaryl said:
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.

PeterDonis said:
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.

stevendaryl said:
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.