# Gravitational effects on particles orbit in a box

• FunkyDwarf
In summary, the conversation discusses the calculation of the period for a classical particle in a box in Minkowski space and in a curved space with a gravitational potential. The question is how to calculate the period in the curved space, with one suggestion being to use the geodesic equations with the Killing vectors. The conversation also touches on calculating a non-radial geodesic path and the use of a spacelike Killing vector to find the conserved angular momentum.
FunkyDwarf
Hi all,

If one were to consider a classical particle in a box of length L in Minkowski space then clearly the time taken to go between the walls of the box is L/v with v velocity.

In the GR case with some metric coefficients A(r) and B(r) (spherically symmetric system, say) it's clear that the distance traveled is longer and so the period should be longer as well, my question is how to calculate this? Let's say m = 0 so v = c.

Do I take dt = 0 and integrate the line element ds from 0 to L over dr? I thought of trying to take the already computed flat space period and applying the time dilation method to it but that contains a radial coordinate component (obviously as it must be position dependent) but I'm unsure as to how to 'get rid' of this part, i.e. would r here simply represent the end points or again is some integral/averaging needed?

I know this is probably kind of a trivial question but still it's doing my head in a bit =P
Cheers

FunkyDwarf said:
In the GR case with some metric coefficients A(r) and B(r) (spherically symmetric system, say)
Could you explain your notation and give the metric explicitly?

FunkyDwarf said:
it's clear that the distance traveled is longer and so the period should be longer as well, my question is how to calculate this?

Longer than what? Longer why?

Sure, sorry.

Lets say i have a metric
$$ds^2 = -A(r) dt^2 + B(r) dr^2 +r^2 d\Omega^2$$

I would expect that if i were to designate length L as the size of the box in Minkowski space then the geodesic path length traveled to go between ends of the box in a curved space, say that for an attractive gravitational potential, would be longer.

So, you want to find a radial geodesic path for the above metric? The easy way to do is to note that none of the coordinates depends on t. So our system has a time-like Killing vector. And the dot product of the time-like killing vector with the four velocity of the particle remains constant

In your example, if we assume that the particle has some 4-velocity r(tau), t(tau), then we can write

g_{ij} [(dt/dtau), (dr/dtau) ] dot [1,0] = constant,which can be simplified to

-A(r) dt/dtau = constant

This constant can be thought of as the energy of the particle.

Add one more constraint, that the magnitude of the four-velocity is always -1 (with your sign convention), and you've got enough equations to solve for a radial geodesic.

IT's not that much harder to do a non-radial path. You could also write out the Christoffel symbols and use the geodesic equation, but that's MUCH harder - it's a lot easier if you can find the Killing vectors. If you're interested in the concpets, though, it's worth working it out with the Killiing vector approach first, then confirming that your solution satisfies the geodesic equations.

[add]...the geodesic equations are well known, but I'll jot them down anyway

$$\frac {d^2 x^i}{d \tau^2} + \Gamma^{i}{}_{jk} \: \frac{dx^j}{d\tau} \: \frac{dx^k}{d\tau} = 0$$Maybe you want a non-radial geodesic. That's a little harder, but not much. You can assume that the motion occurs in the equatorial plane (theta=0), and then you have another Killing vector (a spacelike one) due to the fact that the metric doesn't depend on phi, giving you the other equation you need. This space-like killing vector corrseponds to a conserved angular momentum, because position invariance -> a conserved momentum, angular momentum because your coordinate is an angular coordinate.

Most textbooks should go through this in more detial if my run-through has been too fast.

Last edited:

## What is the concept of "Gravitational effects on particles orbit in a box"?

The concept refers to the study of how gravitational forces affect the motion of particles in a confined space or box, such as a planetary orbit or a particle accelerator.

## How do gravitational effects impact the behavior of particles in a box?

Gravitational forces can cause particles to orbit around a central point, collide with each other, or be deflected from their original path. The strength and direction of these effects depend on the mass and distance of the particles from each other.

## What is the significance of studying gravitational effects on particles in a box?

Understanding how gravitational forces influence the behavior of particles in a confined space is crucial in many fields of science, including astrophysics, particle physics, and engineering. It helps us predict the behavior of objects in space and develop technologies, such as spacecraft and particle accelerators.

## What factors affect the gravitational effects on particles in a box?

The main factors that influence gravitational effects are the mass of the particles, their distance from each other, and the strength of the gravitational force between them. Other factors, such as the shape and size of the box, can also play a role in the particles' behavior.

## How do scientists study gravitational effects on particles in a box?

Scientists use mathematical models and simulations to study the behavior of particles in a box under the influence of gravitational forces. They also conduct experiments in controlled environments, such as particle accelerators, to observe and measure the effects of gravity on particles.

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