A Physics Question

#1

A chemist friend of mine sent me this, and it’s beyond my ability to figure out a satisfying answer:

Suppose we have a long container filled with an ideal gas- one end of this container is near a massive object and the other is far away, causing time to run much slower on one end than the other. What happens to the gas? (ignoring the influence of the mass just pulling the gas particles, which would obviously make them gravitate to the slow end).

Intuitively, I’d say that the gas will still move to the slow end, but can this be shown mathematically?

#2

Ignoring gravity, the gas will take the shape of its container with an equalizing pressure. In a static fluid, Pressure is proportional to Temperature and inversely proportional to Volume. The total Volume is fixed, of course, so that leaves only Pressure being directly proportional to Temperature.

Temperature is a measure of the average random kinetic energy of molecules within a gas. In a region of great time-dilation, the average speed of gas molecules will be reduced (with respect to gas in an area without significant time dilation). Therefore, when we observe the temperature of this gas in a region of time dilation, it will be colder. The kinetic energy its molecules carry will seem reduced (or, rather, the area unaffected by time dilation will appear to have greater molecular excitation).

The gas, with this interpretation, will shift toward the dilated side in response to a perceived local reduction in pressure, until that pressure appears to be equalized. The gas will be more dense on that end.

…

But gravity will have an absolutely dominant role in this. Heck, when you start to notice time dilation, you’re already dealing with significant tidal forces. So the above conversation remains purely theoretical.

Disclaimer: I don’t know if time dilation will affect the actual kinetic energy of a substance. It will be moving apparently slower, yes, but the energy doesn’t just disappear. Perhaps it’s conserved with an increase in the molecule’s momentum or inertia. And how should the unaffected gas interact with the dilated gas? By the time they interact, they will have moved along the gradient to become roughly equal in speed. So it makes no sense for the “higher-speed / higher-temperature” gas to push its way toward the dilated end.
As you see, I can think of a million reasons to discredit my original discussion.

#3

According to this rather dense paper, temperature is apparently constant through time dilation.

#4

I’m just going to come out and say I’ve never been able to understand the concept of time dilation. I should say, I know what the definition is, but I don’t understand how it works. How is time defined in physics?

#5

Speed of light is same for everyone, regardless you are moving or not. So light beam you see when you stay still, or light beam you see when you are driving a car, has to have same speed.

Now consider this situation. You are driving a car, look at a light pole, you see light beam is ejected from a light pole (the pole is in front of your car here), your car whiz by, and now the pole is located at the back of your car (since you moved). Then from your perspective, you did not move - light pole did. It’s gone from front of your car to behind you. Here, light has traveled diagonally, which is greater than straight.

Now a pedestrian observe you driving a car, whizzing by a light pole. From his perspective, light pole did not move - you did. And light beam ejected from light pole has just gone straight down crashing into the ground. This is shorter path than what you observed.

Speed of light has to be same for both you and this pedestrian. You observed light to be moving at extra distances than the ped, so the time that it took for it to travel from your perspective has to be slower, so if you had your watch on your wrist, it would have to have tick slower. Time is dilated.

In this thread we are talking about what is known as gravitational time dilation. What I talked about above is the time dilation caused by moving object. Time dilation also happens when there are strong gravity, because gravity bends spacetime, so light has to move more distance when there are stronger gravity. So it’s almost the same idea.

So that’s how it works - light takes longer path, slower time passage.

Time is a consequence of causality, a series of events that fills our timeline. Without “events”, time is not registered. It’s not that “happening of an event” requires “passage of time” - a mere existence of, “event”, has literally, caused the time to pass. So for example, what happened before the Big Bang? Nothing. So no time has passed before Big Bang. Time started ticking after Big Bang, since events started happening.

#6

So what you are saying, is that time is defined as the perception of events, thus when gravitation interferes with perceptions such as light, “time” is interfered with, or dilated?

So here is a scenario.
A star explodes light years away from two different points. One point, there is no gravitational interference. The second point, while the same distance away from the star, has significant gravitational obstacles in the way. The first point will stop receiving light sooner than the second because it has taken the light directed through the gravitational obstacles longer, even though the distance is the same?

Time in this case doesn’t refer to when the actual event happens, but rather to when the information gets successfully passed to another location?

#7

That’s true. Gravity interferes bent-ness of space, and light, traveling in that space, is also affected.

Actually had hard time trying to know what you exactly mean by it, but if I understood correctly, yes. In fact, it is possible that the second point, being exerted with so much gravity, may not even never recieve star light. This is called gravitational lensing. When space is so bent, and light travels along this bent curve, it may bypass the point in space it originally would have reached if it were not for the warped space.

702×531

Look at the massive ball star between Earth and the constallation. Originally speaking, you wouldn’t be able to see constallation from Earth - this giant ball is blocking the view. But the ball warps space (gravity), light travels along the deformed space, thus you can see the constallation even though you shouldn’t be able to.

“Simultaneity”, that two different event can happen at same time, is intrinsically impossible. So even if assume there was no gravity obstacle so two points mentioned above did recieve exploding light at the same time, you cannot say for definite that they really did recieve light at the same time.

#8

Yay. That makes a good amount of sense.

I think of normal “time” as the average rate at which electrons move around nuclei. Electrons are responsible for chemical and physical interactions between matter. When a nucleus is accelerated / in motion, electron speed is still capped at the speed of light. So, picture an electron in orbit around a moving nucleus (an outdated model, but the concept holds true): It’s moving in the same direction as the nucleus half of its orbit. And it’s moving “against the grain” for the other half of its orbit.

With a nucleus at speed N, and electrons moving at the speed of light C, the electrons can only surpass the nucleus with a relative speed of C-N. On the return journey, they are allowed to travel at a relative speed of C+N. Since the distance to travel each way is the about same, the electron will spend a greater amount of time on the slower-half of its orbit than the faster-half.

For an analogous every-day situation: If you take a boat up-river a mile and then down-river for a mile, at a maximum speed relative to water, at first you’re fighting the current, but then the current will aid you. However, because you go slower while fighting the current, it hinders you for longer. And on your return trip, it aids you for less time. So the greater the river current, the slower your average speed will be. And the fewer round-trips you will make in a day.

As the nucleus N reaches the speed of light C, electrons become trapped, unable to make a return trip from the rear of the nucleus (their relative speed being capped at C-N = C-C = 0), effectively freezing the matter in place.

And to turn this into boats?
If your boat’s maximum speed through water is equal to that of the river’s current, you will never be able to make progress upstream.

#9

My mind is blown. Thanks for explaining the process.

#10

Ignoring time dilation for a moment, we can examine the movement of any point within the tube via Newton’s second law, i.e. that the acceleration that some object or particle experiences is the force acted on it divided by it’s mass (a = F / m), and that the Force is the sum of all forces acting on it.

In a situation such as this, there will be three main forces acting on the particle: the force of gravity exerted by the mass, the pressure exerted by the gas above some point, and the pressure exerted by the gas below some point. For the latter two, above will be furthest from the mass, while below will be closest to the mass. This gives us a somewhat general F = F_below - F_above - F_Grav. This leads to a natural equilibrium at F_Grav = Δp. Now, an ideal gas from your problem follows the equation pV=nRT, where p = pressure, V = volume, n = number of moles of a the gas, R = gas constant of 8.314 j/mol*k, and T is temperature.

Let’s take some infinitesimally thin slice of the tube. since we want a slice where F_Grav = Δp. Derivation of the ideal gas law gives us that ΔpV+pΔV=nRΔT (or, in its infinitesimal form, dpV+pdV=nRdT). Since ΔV = 0, we are left with ΔpV=nRΔT => Δp*V / nR = ΔT. This implies that the ideal gas that is closer to the mass is at a higher temperature than the gas further from the mass. Mind you, only parts of the tube within an area of high gradient of gravity will see any appreciable differences in temperature.

As to the exact relationship that we’ll see, gravity falls of as 1 / r[sup]2[/sup], so the exact conditions in the tube will depend on several factors; where the tube starts and ends and how much gas is within the tube. Of course, this is with an ideal gas.

Now, bringing time dilation into play, as people above have said, based on how fast particles are traveling and how close they are to the mass, they’ll experience varying amounts of time dilation. To put it simply, the closer to the mass or the faster you are going, the more dilation there is. The full equation near a non-rotating body is as thus: link because black text on black background

Just assume that math is happening in the above equation.

The momentum is conserved, but the energy is affected. At least, based on how fast it’s moving, via special relatively. At relativistic speeds, kinetic energy is subject to the lorentz transformation. K = mc[sup]2[/sup] / (1 - v[sup]2[/sup]/c[sup]2[/sup])[sup]1/2[/sup] - mc[sup]2[/sup]. Indeed, energy is affected by velocity, as we should expect from special relativity. However, it’s important to note that this is not a consequence of time dilation.

Short version: There’s a temperature gradient that’s related to the force of gravity at a point, are time dilation exists for particles close to the mass and are are traveling fast, and YES, it can be shown mathematically.

EDIT: And no, time dilation does changes only the passage of time. Temperature and energy are unaffected.

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