Relativity Worries

Factual Questions
1

Yo,

Heres the deal, there are 2 stars around 4 lightyears away from each other that are moving at 0 relative speed to each other. 2 spaceships leave each and head toward each other. Well before they meet each other, they are both going 99% C relative the star they came from. The question is, how fast do the ships measure each other going?

Here’s another question, same 2 stars. One ship leaves from one star, goes 99%C towards the other. Relative both stars, it would take a little more than 4 years, but in the ship how much time would it seem like to them? Relative to the stars, the clocks in the ship would look like they were running much slower, but do they just ‘look like’ they are running slower, or to the ship would the trip seem much shorter, almost instantaneously? Also, since the apparent mass of the ship would increase, would that increase the gravity well around the ship relative to the 2 planets?

Final question, stars ‘A’ & ‘B’, each moving away from one central star ‘D’ in opposite directions at 99%C. If someone could magicly transport instantaneously from star D to star B, stay for a week of that stars time, then transport to star A, and stay for a week. How much time on D would have passed when they return to it?

Help Please!!!

-Blah

2

For your first question, you’ll need to know the formula for adding speeds in relativity. It’s easiest if you express V in terms of c. Then:

Opposite Directions: V[sub]Total[/sub] = (V[sub]1[/sub] + V[sub]2[/sub]) / (1 + V[sub]1[/sub]V[sub]2[/sub])
Same Direction: V[sub]Total[/sub] = (V[sub]1[/sub] - V[sub]2[/sub]) / (1 - V[sub]1[/sub]V[sub]2[/sub])

V[sub]1[/sub] = 0.99 and V[sub]2[/sub] = 0.99, and they’re approaching, so they’re going opposite directions. Using the first formula, we get V[sub]Total[/sub] = 0.99995, or 99.995% the speed of light. That’s how fast each ship measures the other as going.

For your second question, you’ll need the time dilation formula:

t’ = t × sqrt(1 - V[sup]2[/sup])

Where t is the rest-frame time it takes to happen (4 years) and t’ is the time it takes in the moving frame. If V = 0.99, then t’ = 0.564 years. Almost seven months. That’s how long the trip would feel. Not instantaneous, but shorter than four years. About one-seventh the time. The “mass” would increase by the same factor, so the ship would appear seven times as massive, and yes, it would have seven times the gravity.

Your third question is by far the trickiest, because of the - you guessed it - magic transporting. Without getting too bogged down, let’s just say the tranportations “happen” in the reference frame of star D. In that case, they would spend seven D-weeks at star B, and seven D-weeks at star A, so that when they returned to star D, 14 weeks would have passed total.

3

Achernar,

Acccck!!! Math!!! Hide me!!! Thanks for the answers, I figured the 2 ships approaching each other would strech out the decimal points. I had a bet with my dad who is a nuclear physicist and used to make parts for nuclear warheads…ME, a lowly Coder won :smiley:

One thing about the third answer. Shouldnt they still have time dilation b/c they are moving away from the D star at 99%C? I guess my question wasnt specific enough, when they magically transport to A or B star they would be moving at 0 V relative the A or B star. Wouldnt the answer then be: 3.3432 hours at each star relative the time at D, so the total time away relative star D’s time would be almost 7 hrs? I HATE math so heres how I did it…

168xsqrt(1 - .99^2)

168 = 1 week in hours.
If a ship accelerates to .99% C stays there for most of the trip then decelerates while approaching the dest star the time would be MUCH longer, both for the stars and the ship…To find the time out I assume it would have to do something with an equation that defines the curve of the Acc, dec of the ship, plugged into the dilation equ…but prolly something MUCH more complicated than that.
-Blah

4

Well, nothing wrong with giving it a shot. :smiley: Here goes:

I’m not sure what you mean exactly regarding the third question, but they do experience a time dilation. Since A and B are each moving at 0.99c, and D is still, then when one minute passes on A, seven minutes pass on D. And when one minute passes on B, seven minutes pass on D. In this case, D is the rest frame, and B (or A) is the moving frame. Remember that t corresponds to the rest frame and t’ corresponds to the moving frame. Since one week passes in the moving frame, then it’s t’ that is equal to 168, not t. So our equation looks like this:

168 = t × sqrt(1 - 0.99[sup]2[/sup])

If you solve for t, you should get 1191, which corresponds to 49 days, 15 hours. (Also, make sure you’re taking the square root there where it says sqrt.) That’s how much time passes at D. Then of course, you double this.

As for the acceleration thing, yes, you would need to do something complicated to take it into account. However, if the majority (say 95%) of the trip was spent at a constant speed, with no acceleration or deceleration, you can probably ignore any effects it would have.

5

eek… :o

I just saw my 2 mistakes, solved for the wrong variable and misread the line of your post “14 weeks” as 14 days :o

sorry

one more thing…If someone was on B then went to A for a week then back to B, how long would pass there? (trying my terrible math skills again…

(V1 + V2) / (1 + V1V2) = VTotal

(.99 + .99) / (1+.99 x .99) = approx 0.9999494975

168 = t × sqrt(1 - 0.9999494975^2)

t = 168 / sqrt(1 - 0.9999494975^2)

hmmm I got 16716.422 hrs…hmmm obviously my algebra is wrong…its been a LONG time…

Or do use a different equation for 2 moving FoRs?
-Blah

6

Just a small nitpick (which I am probably not qualified to make, and therefore will likely regret). I don’t think this relativistic mass is actually “real”, and therefore will not affect the gravity of the object. This is probably why physicists avoid the term “relativistic mass”. I believe they prefer to think of the energy of the system instead.

A thought experiment to demonstrate why gravity is probably not affected: The speed of light is an asymptotic limit, and you can add an infinite amount of energy to a system before reaching it. After some amount of energy, the “mass” of your ship would’ve increased so much that it should collapsed into a black hole. However, from the ship’s reference frame it is stationary and the planet (or whatever it is moving towards) is the one moving towards it at some huge speed. Therefore, according to the ship, the planet should collapse into the black hole. Since both reference frames are equally valid, there’s no way to determine which one has gained sufficient mass to collapse. In fact, since from some reference frames the mass of any object is sufficient to collapse, everything in the universe has this problem – i.e., every object looks to have a huge “relativistic” mass from some point of view, and those observers should expect to see it as a black hole. This result seems ridiculous, so it would logically seem then that gravity only affects “rest mass”.

I think. Hopefully ring or someone will be along shortly to tell me if I’ve flubbed it.

-b

7

well, bryanmcc, you’ve ALMOST got it.

You’re thought experiment is a clever one, but unfortunately your conclusion isn’t right.

Mass, per se is actually not what determines gravity. This is one of the most difficult concepts to explain; it seems like physicists are contradicting themselves left and right.

First things first, when dealing with gravity in its full glory it is something called the matter stress-energy tensory that dictates the so-called “amount” of gravity present. One way to look at it is to consider an object that gets all the measurements of the energy just right so there is no problem with different frames of reference when seeing how gravity is generated. The tensor defines the shape of space for that particular frame of reference perfectly, and for another frame of reference the same curvature will apply. How can this be? Well, although time and space are relative, the spacetime structure itself is invariant. This can be seen as a direct consequence of the speed of light being the speed limit.

In short, to be perfectly honest, one needs to define the stress-energy tensor when talking about gravity and not just the mass. This is where confusing definitions of “mass” come into play and why physicists simply refer to “mass” as the rest mass and forget all the rest of the nonsense. This is rather confusing to those not interested in the goriest of technical details and not at all intuitive, but we have yet to see a case where thinking of it this way hasn’t worked.

When someone talks about “mass increasing” as we go to higher gravity they are sort of right and wrong at the same time. Rest mass does not increase, however in a certain frame of reference it will appear that there is more curvature since there is actually stuff added to the stress-energy tensor. Indeed, you add enough energy to a system, whether it be in the form of mass or not, you will end up with an event horizon, but it’s a bit more subtle than that. If you add an infinite amount of energy to a system then, you will end up collapsing. Moreover, it doesn’t matter what frame of reference you’re in… the event horizon will have been created and you will have left the (observable) universe.

8

ACK! I hit post instead of preview… now I need to correct myself…

The first sentence in the third paragraph should refer to the matter stress-energy tensor, not tensory.

The first sentence in the last paragraph should read…

There are other nitpicky style-issues in my post that are slightly annoying, but I’m just going to have to let them be as they aren’t detrimental to the ultimate explanation.

9

One not as minor nitpicky issue… it’s more than just the massive part of the stress-energy tensor that gravitates, is it not? This doesn’t matter, obviously, when we’re talking about pure matter which doesn’t contribute in any other way, but on a fairly regular basis we run into the misconception on these boards that massless objects don’t gravitate because they don’t have mass, and we ought to nip that in the bud here.

10

Thanks for the clarification, JS. But I fail to see how my conclusion is invalid.

First off, rest mass is definitely not alone in determining gravity, because photons have and are affected by gravity, and they have no rest mass. I don’t have the terminology to describe what is producing and affected by gravity, but your “stress-energy tensor” sounds good to me. :wink:

What I can’t figure out, though, is how you could say that something could gain enough velocity to collapse. If two trains approach each other at some velocity arbitrarily close to light, both will think the other will collapse, no? Both of their reference frames are equally valid, so the only conclusion I can see is that neither can collapse, as from some reference frame, any object in the universe could be seen to have enough velocity to collapse.

You said: “you add enough energy to a system, whether it be in the form of mass or not, you will end up with an event horizon,” but from whose point of view? The rocket that blasts itself up to .99999c may look like it should collapse from the point of view of an observer on a planet, but the observer looks like he is the one travelling at .99999c according to someone on the rocket. It is my understanding that the point of Relativity that both points of view are equally valid, so neither one of them could logically collapse.

Where’s the fault in this argument?

-b

11

BlahMan, you asked a question about leaving B and going to A for a week. Well, it’s difficult to understand, because of what I said earlier, the magical instant teleporting. Rather than trying to explain it correctly, though, let me just tell you why it’s difficult to understand.

You seem to have a fair grasp on the idea of frames of reference. That’s good. In your second question in the OP, the two stars are in the same frame of reference because they’re not moving with respect to each other. Okay, great. Now then, Special Relativity says that, there are no such things as simultaneous events. Or more specifically, if two events happen in different places, you can not say whether or not they were simultaneous, unless you specify the reference frame. Suppose that in your second question, you were stationary with respect to the stars, and you detected that the two stars went supernova at the same time. For someone on the ship traveling between them, they did not go supernova at the same time - one went off before the other.

So, how does this affect your third question? Because implicit in the teleporting idea is the idea of simultaneous events. You leave one place and arrive at another place, and these two events (the departure and the arrival) are simultaneous. SR tells us that you can’t just say that - you also have to say what reference frame the two events are simultaneous in. For the purposes of answering your question, I assumed it was the reference frame of star D, because that seemed to make sense. But now you’ve rephrased the question, which is fine, but you’ve rephrased it in such a way that the teleporting events are simultaneous in the reference frame of star B! It subtly changes the mathematics behind it. But, if you make the change, then yes, the answer is about 100 weeks. That is, provided I did the Algebra right. I’m probably just as bad as you, but I don’t have the “It’s been a while.” excuse. :smiley:

12

Adding energy to a system changes your inertial rest frame and is subtly dependent on how exactly energy is transferred through spacetime (quickly or slowly, concentrated or diluted). An interesting side-note, because we’re dealing with the interactions of space and time not only is energy density considered in the stress-energy tensor but also pressure. This may seem a bit misleading tangent at first, but the whole point of the way gravity works is that there has to be something to measure against. That something is the stress-energy tensor and it works out so that everybody agrees on the gravity it produces.

When you put a whole bunch of energy into a rocketship, you are changing the values of its stress-energy tensor. In theory, you are not changing the values for tensor for Earth unless you consider the fact that the rocketship was once part of Earth’s tensor values. So, we’ll stear clear of that complication and stay very theoretical and ignore this effect. In reality you need to get your energy from SOMEWHERE… and right now the somewhere is Earth.

But for argument’s sake, let’s say you had some source of energy outside Earth and you wanted to accelerate that bad boy rocket up as close to the speed of light as possible. If there is enough energy confined to the spacetime frame so that we end up with an event horizon, the rocketship will find itself in a black hole. After the change occurs to this rocketship, there would be no way for the Earthling to see this rocketship anymore (no photons could escape from it). There is no relativity symmetry here because we are changing reference frames.

To wit, there is no collapse in the other direction because nothing was added to Earth to change its reference frame. Therefore Earth does not find itself in a black hole, nor do observers watching it see Earth becoming a black hole.

Another way to look at this is from the idea that acceleration and gravity are indistinguishable. If you get enough acceleration on a rocketship so that light cannot escape, you are by definition in a black hole.

Hope this makes sense.

13

I’m sorry, but no, it doesn’t. :slight_smile:

I still can’t see how this could work for a train going north at .99999c and a train going south at .99999c. Which would be a preferred reference frame, and which would collapse, and why?

And your point about the rocket ship blasting its engines sounds like special pleading. I don’t see any reason why it should matter how the relative differences in velocities were obtained, the fact is just that they are now large relative to each other. In fact, we should be able to ignore the accelerations completely – it shouldn’t matter how the velocity was generated, just that the velocity is present. Body A sees body B traveling at speed X, and B thinks it is A traveling at speed X in the opposite direction. It doesn’t matter how this speed was acquired (I think), just that it is there. As long as there is no acceleration, both views are valid, so the velocity cannot possibly cause one of the objects to be a black hole.

How this is wrong I don’t know. You’ll have to give the “For Dummies” explanation, I’m afraid. :slight_smile:

Of course, I’m assuming no acceleration; just a velocity between two objects. Obviously, if a rocket accelerates fast enough, something wonky will happen, since acceleration is indistinguishable from gravity. I wouldn’t be surprised if an object accelerating fast enough would appear as a black hole, although I don’t have the ability to figure out the details. But just because one object is traveling blisteringly fast and another is standing still doesn’t mean that the fast one is a black hole, because according to Relativity you can’t say that the reference frames aren’t the other way around.

-b

14

Ah ha! We’ve tracked it down…

The answer is, of course, neither of them! If they are in inertial reference frames there is no problem. From my understanding, this was not really the issue here. Both of these trains can be going at whatever speed they please, for the purposes of our discussion it IS how they get there that matters. That is the ONLY way to get to the point where the event horizon is created. We see highly relativistic massive particles (cosmic rays) go whizzing by us all the time. We’re not exactly sure how they got to be that way, but there are any number of ways for us to get to a preposterously high relative velocity given enough spacetime. You could sit in a rocketship at 1 g acceleartion and get arbitarily near to the speed of light wrt Earth. It’ll just take a lot of spacetime. Just because you get arbitrarily close to c doesn’t mean you’re going to collapse into a black hole. Again, this goes back to our friend the stress-energy tensor and the relationship between energies and spacetime.

Here’s another way to think of the relationship: Black holes can have any mass they please, right? It just matters that the things are in the right concentration. I can have a whole lot of mass, but if it isn’t in a small amount of space there is no collapse. Likewise I can have a small amount of space, but if there isn’t enough mass there is no collapse. A semi-equivalent way of thinking about your thought experiment would be to ask why all these galaxies aren’t in black holes? After all, there masses are very large and in the intertial frame of a photon there doesn’t seem to be that much to them. The whole point of the matter is you have to take into account the energy as it relates to spacetime. Right now your trains’ relationships to their spacetime geometries aren’t changing because we are sitting on objects that are going at constant velocity.

That’s the whole thing. It DOES matter how the differences are arrived. Gravity deals in NON-inertial frames. That means we’re changing to a different inertial frame (in this case, a different speed). HOW the energy changes makes all the difference.

Maybe you are confused by what we are talking about. In your last post, we are talking about inertial frames, but I was more concerned about your thought experiment and the posts previous to it… this is what you say,

Herein lies the problem. You cannot add an infinite amount of energy and expect simply to not get any gravitation. The whole problem is we are CHANGING reference frames.

Let’s go back further in time. Here’s what bBlahMan** asks:

It’s that “leaving” bit that we are concerned with here. Forget the 99%C, all that does is tell us how much we are accelerating to. Once we arrive at that inertial frame, we’re home free. What we are concerned with here is the acceleration. So Achernar is technically on the ball when he talks about the increase in the gravity for the rocketship (assuming this weirdness about getting energy from some unknown source), though “mass” is a whole different matter.

You’re right as rain with this one. Bob’s your uncle.

Well, then, how are you EVER gonna get this rocketship off the ground? The whole point of the question we were responding to was that there was a reason to change inertial frames. If your initial conditions are two different frames, then you’re absolutely correct, you don’t ever have to worry about this problem. However, then you can only be in the same part of the universe once and only once over the lifetime of the universe (assuming the most basic of topologies and that the universe isn’t closed).

I hope this is nipped in the bud, but if not, we’ll have another go.

15

bryanmcc you are 100% correct. Ignore anyone who says otherwise.

16

i think there is a lot of confusion over the term “relative”
For instance a shp accelerating to near light speed does not in fact become more massive, it becomes relatively more massive as a function of its speed/kinetic energy. In other words it does not grow or become heavier or actually add mass, but the energy of its motion means that should it strike something it would do so as if it were a much more massive object. The difference is important because inertia and gravity are NOT the same.

For that matter the whole idea of time dilation seems to always come up as a rewsult of near light speed and faster than light speed travel. Again, the fact that something is travelling at that speed does not change the time functions of the rest of the universe it changes ONLY for the person travelling, And only relative to a non traveller. There is also the idea of an observer that again is, i think, commonly misunderstood. An onserver sees the light that has left the spaceship and what he sees is a function of the lights behavior. The “relatavistic” effects observed are not necessarily what is experienced by the traveller. What we see as observers is the effect of lights limits not the true happenings with the travelling body.

17

the ability to edited your own posts would sure be a welcome addition to this board…

18

Ignore, Ring. He’s one of those people who don’t take kindly when their favorite interpretations aren’t the focus of attention.

Err… what I meant to say was, Ring, would you care to comment instead of shooting straight from the hip?

I get the feeling you didn’t read my posts closely at all, because (as I put my prognosticator hat on) I think you’re insinuating is that relativistic mass doesn’t matter and that the only thing contributing to gravity is rest mass. All of this is generally true, but only up to a point. As you well know you can be in a frame where things with zero-restmasses have E-p nonzero. The problem is when you add infinite amounts of energy to a system, you’re going to end up with this very issue. This is the reason I said bryanmcc ALMOSt got it. He did ALMOST get it, it’s just his thought experiment isn’t quite on the money.[sup]*[/sup] Adding infinite energies and arbitrary accelerations does have gravitational implications whether you care to admit it or not.

Now, of course, is your cue to ask we what the outside world sees, in which case we will have to deal with the nature of this energy source we’ve been so glib about.

[sup]*[/sup][sub]By the way, I know Achernar’s comment about the gravity increasing seven-fold is wrong quantitatively (as relativistic mass is solely inertial), but the fact of the matter remains that there is a qualitative difference between the gravity of an accelerating object and one that is in an inertial frame, dependent upon the actual mechanisms for the acceleartion.[/sub]

19

Just in case anyone is interested in some of the gruesome details:

There is no space-time curvature whatsoever in a Rindler space-time (i.e. in an accelerating frame of reference), because the Ricci curvature scalar is zero.

20

Anyway, why is it so difficult to figure out the 2 moving FoRs question Achernar? You can just ignore the instanaeous travel part…right?

Also, will someone tell me what I did wrong with my algebra? (Im sure I solved for variable t wrong)

BTW, Im going to take precalc over again, but this time im actually going to learn everything…I keep running into troubles in my proggramming because of that…I had trouble with quadratics for bezier curves…Im STILL trying to find a point on a sphere n degrees from another point…I even had trouble with intersecting lines (for trimming out of bounds lines)…

-Blah