Gravitational attraction thought experiment

[Waltr]

Hi all. New to physics forum and glad to be here.
I have been referencing the site for a while now and have finally come up with a question i haven't been able to find on here. So I guess i thank all of you who are already on here for the help.

Assumptions:
-Imagine an infinitely large plate (just so the field lines are all going in the same direction)
-This plate is super dense
-The plate is fixed in space and cannot bend
-It is arbitrarily far away (in a vacuum duh)
-There is a ball of relatively small mass (or rest mass because this will change)

Question:
The ball is released and begins its long travel to the plate accelerating towards the plate. After a while it is really moving quickly. As it approaches the speed of light it gains mass (that is what I have always been told). But the force between the two should be growing proportional to the increased mass (that is using the general theory of gravity). As this force increases, the acceleration should stay the same. If that were true there shouldn't be anything stopping the particle from reaching the speed of light. Unless there is a force that (i don't know about) inhibits masses from reaching it.

 

[jedishrfu]

Welcome to PF!

You don't actually need to specify an infinitely large plate you could use an enormous black hole for your thought experiment as the object will be gravitationally attracted to it.

Special Relativity says a material object can approach the speed of light but never reach the speed of light as the energy required to do so becomes infinitely large.

http://en.wikipedia.org/wiki/Special_relativity

In your case, as the object falls its converting the potential energy of the gravitational field into kinetic energy but it will never be able to reach the speed of light. Instead an observer sitting outside the black hole will observe it approaching but never falling into the black hole.

From the perspective of the observer on the falling object they will observe passing the event horizon of the black hole in a finite time and will then be stretched apart like taffy by tidal forces.

http://en.wikipedia.org/wiki/Black_hole

 

[PeterDonis]

Hi, Waltr, and welcome to PF!

As it approaches the speed of light it gains mass (that is what I have always been told).

Many sources do phrase it this way, but that phrasing is misleading. This concept of "mass" is more properly called "relativistic mass", and it is not the right thing to use when determining how gravity acts on the ball.

Also, it's important to remember that speed in relativity is frame-dependent; the ball is only approaching the speed of light in a particular frame, the one in which the plate is at rest. One can always choose a frame, at least locally (in space and time), in which the ball is at rest and its mass is just its rest mass. This is an important reason why "relativistic mass" is not the right thing to use when determining how gravity acts on the ball: any physical effect like that can only depend on things that are invariant, i.e., not dependent on your choice of frame.

the force between the two should be growing proportional to the increased mass (that is using the general theory of gravity).

If by "general theory of gravity" you mean General Relativity, no, this is not correct. In GR, gravity is not a force; it's spacetime curvature. So to determine how the ball moves, you need to consider two cases:

(1) The ball's effect on the spacetime curvature is negligible. In this case, the ball is referred to as a "test object". Formulating this criterion in a mathematically precise way has some subtleties, but a good approximate way of thinking of it is that the ball's rest mass is very, very small compared to the mass of the source of gravity (in this case, the plate). In this case, the ball's motion is determined entirely by the spacetime curvature produced by the plate; the ball's mass plays no role.

(2) The ball's effect on the spacetime curvature is not negligible. In this case, you can't just think of what is happening as "the ball falls towards the plate"; what you have are two sources of gravity (spacetime curvature), and you have to take both into account from the start. In general, cases like this can only be solved numerically; there are very, very few closed-form analytical solutions known. But the key thing to remember is that, for any source of gravity, the spacetime curvature it produces can't depend on your choice of frame. That means we have to find a way of describing the source that is frame-independent; the mathematical object that does that is called the stress-energy tensor, and in the general case you would have to determine the stress-energy tensor of both the plate and the ball, and then use the Einstein Field Equation to determine the spacetime curvature.

As this force increases, the acceleration should stay the same.

It's also worth noting that the "acceleration" you refer to here also depends on your choice of frame; in relativity, the term "coordinate acceleration" is used to emphasize this. The ball *feels* no acceleration at all: it is weightless. The term for felt acceleration, which is invariant, is "proper acceleration", and if you're trying to understand a problem in relativity, it's usually much better to focus on proper acceleration than on coordinate acceleration.

there shouldn't be anything stopping the particle from reaching the speed of light.

Yes, there is; the same thing that always stops that from happening in relativity, namely, the geometry of spacetime itself. No matter how fast, relative to a given set of coordinates, the ball approaches the plate, a light beam released from the same point as the ball will get there first. It's possible to set up coordinates in which it looks like the ball is falling "faster than light", i.e., faster than ##c## in those coordinates; but in any such coordinates, an ingoing beam of light will also be moving "faster than light", i.e., faster than ##c## in those coordinates, and also faster than the ball. This is, as I said just now, because of the geometry of spacetime: there's no way to beat it by choosing coordinates.

 

[Waltr]

super helpful thank you both

Many sources do phrase it this way, but that phrasing is misleading. This concept of "mass" is more properly called "relativistic mass", and it is not the right thing to use when determining how gravity acts on the ball.

You are right. That is very misleading. I am glad you told me what to look for in the distinction.
That was one of the main assumptions i was making for the premise of the question. Without that everything kind of falls apart and becomes just another you can't go faster than the speed of light answers.

If by "general theory of gravity" you mean General Relativity, no, this is not correct. In GR, gravity is not a force; it's spacetime curvature. So to determine how the ball moves, you need to consider two cases:

I was actually referring to Newtonian gravitation. Where it is considered a force. i was thinking more F=mg if m were to increase with speed so should the force. But as I said earlier without the mass increasing, gravity does not act differently and that argument has no leg to stand on.

as for the reference point, I am sorry i didnt make it clearer but i was trying to refer to one at the initial position of the ball. though one on the plate would have been i think more interesting=)



I was also wondering (and this might be for another post but) how fast the effect of gravity moves, like if the sun were to just vanish. Would we see that first or feel it first? or would it be simultaneous? in which case they would move at the same speed. and if that were true wouldn't the orbits of the planets be super unstable and volatile? if Earth were being pulled toward the place where the sun was a few minutes ago?

 

[jedishrfu]

For a vanishing sun:

http://io9.com/what-would-happen-if-the-sun-just-disappeared-464769645

 

[PeterDonis]

I was actually referring to Newtonian gravitation. Where it is considered a force. i was thinking more F=mg if m were to increase with speed so should the force.

You can't combine Newtonian gravitation with relativity; the two are inconsistent. And since the idea of mass increasing with speed only arises within relativity, you can't combine that idea with Newtonian gravity in a consistent way either.

how fast the effect of gravity moves, like if the sun were to just vanish.

This probably does deserve a separate thread if you want to dig into it further, but I'll go ahead and make a long-winded post here anyway.

The "vanishing Sun" is a common scenario, but it has a fundamental problem: the Sun can't just vanish, because that would violate conservation of mass-energy. (In GR, this conservation law is a constraint that the stress-energy tensor must satisfy: if the Sun were to just vanish, the stress-energy tensor that describes the Sun would violate the constraint.) So you can't formulate the "vanishing Sun" scenario in a consistent way.

In fact, it turns out to be very difficult to formulate *any* kind of scenario like this for testing "the speed of gravity", precisely because of that conservation law; since the stress-energy tensor is the source of gravity, and since it's conserved, all sorts of obvious thought experiments about suddenly changing the source of gravity and watching the change propagate are actually impossible. For example, suppose we fire a big laser at the Sun in order to push it in some particular direction, so we can watch what happens to the orbits of the planets. The problem is that the laser beam itself carries energy, as does whatever is firing the laser, and if there's enough energy in the laser to push the Sun, there's enough energy in it to already be affecting the orbits of the planets way before it ever hits the Sun.

The best we can do (at least so far) at addressing the "speed of gravity" question is to attack it indirectly. One way is simply to ask GR, as a theory, what it says the speed of gravity is. The answer to that is unequivocal: GR says the speed of gravity is equal to the speed of light, in the following precise sense: Take any event in spacetime and ask what information you need to have in order to precisely determine the effects of gravity at that event. The answer is that all you need is information about the event's past light cone, i.e., about events in spacetime from which information can propagate to your chosen event at a speed less than or equal to the speed of light. You never need to know anything outside the past light cone.

Another way of attacking the question indirectly is to ask if there are other ways of observing the speed of gravity, given that we can't conduct the obvious sorts of experiments I described above. In situations where gravity is weak and all motions are slow, it turns out that the description of gravity given by GR is very close to the Newtonian description: in these situations, gravity can be considered to be a "force" in the usual Newtonian sense, and the speed of gravity question becomes a question about how fast this force propagates. Then we can use our Newtonian intuitions in the following way: a perfectly Newtonian force propagates instantaneously, so at any instant, the force on an object (such as a planet) due to a gravitating source (such as the Sun) should point directly at the source. But a force that propagates at a finite speed will have a time delay, so at a given instant, at the object, the force it feels should point, not at where the source is "now", but where the source was some finite time ago (the time it takes the force to travel the distance). The difference between these two directions (where the source is "now" and where it was a travel time ago) is called "aberration".

A number of people have used this intuition to argue that gravity must propagate much faster than light, because we do not, in fact, observe aberration of gravity--at least, we don't in almost all cases (but there are exceptions, which I'll get to in a minute). This contrasts sharply with light, for which we *do* observe aberration--for example, the direction in which we see light coming from stars changes as the Earth changes direction in its orbit around the Sun. However, when you look into the details, you find that there is another relativistic correction to the Newtonian behavior that is very important: the "force" of gravity in relativity does not just depend on the distance to the source, as it does in Newtonian theory. It also depends on the velocity of the object relative to the source.

It turns out that this velocity dependence, in the case of gravity, cancels out almost all of the aberration due to the finite speed of gravity, for cases where the velocity is only changing slowly (as it is for a planet orbiting the Sun). The small amount of aberration that remains shows up as a shift in the perihelion of the planet (the point where it comes closest to the Sun in its orbit), and just such a perihelion shift has been observed--first with Mercury (this was actually known well before Einstein developed GR, and was one of the first tests applied to the theory), but now, IIRC, it has been observed with other planets as well. So this is another indirect indication that gravity does in fact propagate at the speed of light.

The definitive treatment of all this is the paper by Carlip on Aberration and the Speed of Gravity:

http://arxiv.org/abs/gr-qc/9909087

It's somewhat technical but still very readable.

 

[PeterDonis]

For a vanishing sun:

http://io9.com/what-would-happen-if-the-sun-just-disappeared-464769645

This is not a good treatment, since it makes the error I pointed out in the post I made just now in response to the OP: it assumes that the "vanishing Sun" scenario can be formulated consistently in the first place, when in fact it can't because it violates conservation of energy.

 

[A.T.]

This probably does deserve a separate thread if you want to dig into it further,

There are plenty separate threads on this.
https://www.physicsforums.com/showthread.php?t=699522
https://www.physicsforums.com/showthread.php?t=667553
https://www.physicsforums.com/showthread.php?t=658001
https://www.physicsforums.com/showthread.php?t=634813
https://www.physicsforums.com/showthread.php?t=604975

 

[PeterDonis]

There are plenty separate threads on this.

And, of course, in accordance with Murphy's Law of Forum Posts, one of these links to the FAQ entry in this forum that I forgot was there:

https://www.physicsforums.com/showthread.php?t=635645

 

[Dale]

Doesn't the Rindler metric describe the vacuum outside an infinite plane of mass?

 

[pervect]

Hi all. New to physics forum and glad to be here.
I have been referencing the site for a while now and have finally come up with a question i haven't been able to find on here. So I guess i thank all of you who are already on here for the help.

Assumptions:
-Imagine an infinitely large plate (just so the field lines are all going in the same direction)
-This plate is super dense
-The plate is fixed in space and cannot bend
-It is arbitrarily far away (in a vacuum duh)
-There is a ball of relatively small mass (or rest mass because this will change)

Question:
The ball is released and begins its long travel to the plate accelerating towards the plate. After a while it is really moving quickly. As it approaches the speed of light it gains mass (that is what I have always been told). But the force between the two should be growing proportional to the increased mass (that is using the general theory of gravity). As this force increases, the acceleration should stay the same. If that were true there shouldn't be anything stopping the particle from reaching the speed of light. Unless there is a force that (i don't know about) inhibits masses from reaching it.

Lets time-reverse the problem around, and ask if light (or a sufficiently speedy particle approaching the speed of light) can escape your plate.

If the answer is no, then light speed is never exceeded when we re-reverse time and go back to your original scenario.

If the answer is yes, you have, by defintion, a black hole, because you have a trapped surface that light can't escape. Can a black hole really be plate-shaped, rather than a singularity? Probably not, the reason you assumed it had a plate shape was that you assumed the plate was strong enough to hold itself together against gravity. Once you realize that the plate must be a black hole, you can start to examine this assumption, and come to the conclusion that you probably can't really build a plate so strong.

Note that the relative velocity between the event horizon of a black hole- any event horizon of any black hole - is by defintion always equal to c. This leads to the impression that the "velocity" of the infalling object is equal to "c" but the actual situation is that it is the event horizon that is moving at "c" (this is a loose way of putting things, the more precise way is to say that the event horizon is lightlike, or is following a null trajectory).

Thiw, while the velocity of a material object relative to the event horizon can be, and must be, equal to "c", it's the event horizon that is lightlike, and the infalling object that is timelike. You can basically regard the object as being stationary (in its own frame), and the light (which does not have a frame of its own) is approaching it at "c".

What about the "frame" of the event horizon? THe event horizon does not have a "frame" , just as a photon doesn't have a prame. Sometimes people think light "should" have a frame, but it's mathematically inconsistent when you try. There's a FAQ on this if you care to look it up in the FAQ section.

 

[PeterDonis]

Doesn't the Rindler metric describe the vacuum outside an infinite plane of mass?

I'm not sure how it can since the Rindler metric is a metric on Minkowski spacetime, which is flat. The vacuum outside an infinite plane of mass is, IIRC, not flat.

 

[Dale]

I thought it was flat, but I could be wrong. I cannot remember the source, which is never a good indication.

 

[Waltr]

Lets time-reverse the problem around, and ask if light (or a sufficiently speedy particle approaching the speed of light) can escape your plate.

If the answer is no, then light speed is never exceeded when we re-reverse time and go back to your original scenario.

If the answer is yes, you have, by defintion, a black hole, because you have a trapped surface that light can't escape. Can a black hole really be plate-shaped, rather than a singularity? Probably not, the reason you assumed it had a plate shape was that you assumed the plate was strong enough to hold itself together against gravity. Once you realize that the plate must be a black hole, you can start to examine this assumption, and come to the conclusion that you probably can't really build a plate so strong.

...

What about the "frame" of the event horizon? THe event horizon does not have a "frame" , just as a photon doesn't have a prame. Sometimes people think light "should" have a frame, but it's mathematically inconsistent when you try. There's a FAQ on this if you care to look it up in the FAQ section.

Thank you for the response. i really am grateful people are willing to spend their time and share their knowledge.

I did not pick a black hole for a reason and it is a thought experiment the so obviously a plate could not be infinite and could not exist in real life but i wanted the field lines to be equidistant and uniform across the piece of space.
i didnt want the distance the ball to the plate to matter when it came to the force between the two

If you want to make this a bit more realistic the plate could be a very flat planet (far from infinite) spinning very fast along its axis like a ball of pizza dough. It could also be a galaxy with nearly equal density of solar systems packed in tightly together. Sure there is a big ole black hole in the center that would throw off the gravitational field lines a bit.


I never really thought about light being able to escape from the plate but that in itself is also interesting.

if light were able to escape the reference point from the plate would be possible. The location of the light source i suppose would matter as well. If it were radiating from the plate itself it the light would have to go out and hit the ball and bounce right back. If the ball is moving it would appear that the ball is farther away than it actually is.

if the ball had its own light source the light would only have to travel to the plate and it would give a better representation of where the ball is in space but would depend on the speed and distance of the ball as to how accurate that observation that would be.

 

[PeterDonis]

it is a thought experiment the so obviously a plate could not be infinite and could not exist in real life but i wanted the field lines to be equidistant and uniform across the piece of space.
i didnt want the distance the ball to the plate to matter when it came to the force between the two

The thought experiment part is not a problem; we do idealized thought experiments in physics all the time.

However, there are issues in trying to construct a self-consistent solution in GR that has the property you want, namely that the "force" on an object is independent of distance from the plate. There is a fairly detailed discussion on MathPages here:

http://www.mathpages.com/home/kmath530/kmath530.htm

I thought it was flat, but I could be wrong. I cannot remember the source, which is never a good indication.

The MathPages article I just linked to actually discusses the Rindler metric in this connection. However, I don't know that it resolves the question of whether that metric can actually describe the field of an infinite flat plate, because the article never discusses the plate itself, only the vacuum region, and if the vacuum region is flat (as it would have to be if the Rindler metric is going to describe it), I don't see how the junction conditions could be met between at the boundary between the vacuum region and the plate itself. As far as I can tell, the article doesn't address this.

The article does, however, bring up another issue: if in fact the Rindler metric does describe the field in the vacuum region outside the plate, then the field does *not* have the property that Waltr would like it to have: the "acceleration due to gravity" is not independent of the distance from the plate (instead it goes like 1/x, where x is the distance). So if there is a solution that does have the property Waltr wants it to have, it can't be the Rindler metric.

 

[PAllen]

Going beyond mathpages, I find the following discussions of uniform gravity and infinite planes (respectively) in GR:

http://arxiv.org/abs/1003.3022
http://arxiv.org/abs/0708.2906

 

[jartsa]

Question:
The ball is released and begins its long travel to the plate accelerating towards the plate. After a while it is really moving quickly. As it approaches the speed of light it gains mass (that is what I have always been told). But the force between the two should be growing proportional to the increased mass (that is using the general theory of gravity). As this force increases, the acceleration should stay the same. If that were true there shouldn't be anything stopping the particle from reaching the speed of light. Unless there is a force that (i don't know about) inhibits masses from reaching it.

Let's say we are observing a proton at rest, and a proton moving at velocity 0.86 c.

Ok now we change our velocity by 10 m/s, in the direction of the motion of the fast proton.

We will observe that the velocity of the slow proton changes by 10 m/s, and the velocity of the fast proton changes by 5 m/s.

The relativistic longitudinal mass of the fast proton is eight times the mass of the slow proton.

Now we pretend that two forces caused the aforementioned velocity changes. The force that pushed the fast proton was four times the force that pushed the slow proton. (half the acceleration, eight times the mass)

Does a fast proton reach the speed of light, relative to us, if we keep on accelerating to the opposite direction? No. The imagined force pushing the proton becomes very large. The acceleration of the proton becomes very small, because the longitudinal mass increases rapidly as velocity increases. Longitudinal mass = γ³ * rest mass

If objects gravitate with their relativistic mass, do they gravitate with the transverse mass or the longitudinal mass? I would guess with the transverse mass.

 

[pervect]

I did not pick a black hole for a reason and it is a thought experiment the so obviously a plate could not be infinite and could not exist in real life but i wanted the field lines to be equidistant and uniform across the piece of space.

Let me expand on my point.

To oversimplify slightly, if the escape velocity of your object is less than "c", then it's not a black hole. However, a body falling from rest at infinity will reach the surface of the plate at escape velocity, which we've just said is less than c. So you won't exceed the speed of light by falling.

If your goal is to make something go (in some loose sense) faster than light, you will have to pick the case where the escape velocity is greater than c. But that will require you to consider the black hole case.

You didn't specify that you didn't want a black hole in your previous post. What I'm trying to point out to you is that while you didn't intend to specify that you wanted a black hole, when you look at your original requirements, those requirements DO imply that you need a black hole, whether that is what you "wanted" or not.

The remarks about the plate shape are remarks about some of the unintended consequences of accidentally specifying a black hole.

The case where you don't have the requirement that the escape velocity be greater than "c" does allow the plate solution you are looking for (in a general sense), and might be of some interest, though it's not what you originally asked. One of the papers posted addresses this in terms of the actual GR solution.

The actual solution given in the referenced paper will, however, not be in terms of the "field lines" that your thinking is in, but in terms of the metric. Unfortunately, I don't think you will find any textbook that will tell you how to convert a metric into "field lines".

The good news is that if you were sufficiently familiar with GR, and if you added in the concept of gravitational time dilation in a static metric (which hasn't played any part in your thinking that I can determine), you might be able to translate the metric solutions presented into textbooks into something that wasn't "field lines", exactly, but was close enough to being "field lines" that you could use your intuition about field lines to appreciate the solution.

The bad news is that this procedure isn't well documented in any textbook - only few words in some rather advanced texts hint how you might go about doing such a thing.

So it doesn't seem like anything appropriate to go into depth in here. Basically, if you really want to know what the GR solution is, you'll have to learn enough about metrics to understand solutions expressed in that form, which is what the papers will give you. If you stick with the picture of gravity as "field lines", I don't see how you'll be able to understand the solutions that GR gives you.