View Full Version : Question about Special Relativity

CaptainToonces

2010-Jul-12, 08:06 AM

If there is a speed limit on the velocity of an object at c, is there a limit on the rate something can accellerate? and does it work in the same lorentz transform fashion?

Sorry if this is a dumb question and the answer is something obvious like square root of c.

grant hutchison

2010-Jul-12, 08:14 AM

There's no limit to the acceleration an object can experience in its own frame of reference, beyond the purely practical aspects of generating and withstanding that acceleration.

But the Lorentz transformations mean that an object which is experiencing constant acceleration in its "own frame" (actually, a succession of instantaneous rest frames) will be seen to have a steadily decreasing acceleration by outside observers, so that it can never reach c relative to any observer: it just approaches lightspeed ever more gradually.

Grant Hutchison

AriAstronomer

2010-Jul-12, 09:51 AM

Also something to note, is that since acceleration is velocity/time = distance/time^2, since we all know time slows as you approach c, it can appear you are accelerating very quickly due to the slowing of time. If you are able to travel 100km in one stretched out second, it will seem like a much greater acceleration than 100km in an hour. In your own frame of reference of course. As Grant says, to outside observers, you will appear to be decelerating.

grant hutchison

2010-Jul-12, 10:00 AM

It all gets a bit complicated, and one has to be careful not to mix frames. It's true that a second in the traveller's frame may carry him a very large distance in his original rest frame. However, in the traveller's series of instantaneous rest frames, his departure point falls astern at an ever-decreasing rate while he maintains his constant acceleration.

Grant Hutchison

AriAstronomer

2010-Jul-12, 10:14 AM

so you are saying that as he approaches c (in his frame on his ship), his spot of departure (aka earth) seems to approach an asymptote, always getting farther away, but at a reduced rate per second.

grant hutchison

2010-Jul-12, 10:44 AM

so you are saying that as he approaches c (in his frame on his ship), his spot of departure (aka earth) seems to approach an asymptote, always getting farther away, but at a reduced rate per second.Yes.

Here (http://www.ghutchison.pwp.blueyonder.co.uk/relativity/starflight.jpg) is a little diagram I made a while ago, illustrating the traveller's point of view when making a 98-light-year journey at 1g acceleration, with a midpoint turnaround: accelerate for half the journey, decelerate for the other half. Along the bottom is the traveller's proper time in years, showing that it takes him only nine years of his own time to travel 98 light-years. The red line, on the negative axis, shows how far behind "home" is; the blue line on the positive axis shows how far ahead the destination is. Both are measured according to the coordinates of special relativity in the traveller's instantaneous rest frame: they're the distances the traveller would measure if he triangulated and then compensated for aberration.

No matter how long he accelerates, the traveller never gets more than a light-year from his departure point, according to this measurement; only once he turns around and starts to decelerate does he see his departure point fall farther astern. At turnaround he measures his destination as being just a light-year ahead, but it then takes him the entire deceleration phase of the journey (half his travel time) to reach it.

Grant Hutchison

AriAstronomer

2010-Jul-12, 11:02 AM

very useful and interesting. Thanks Grant.

Ken G

2010-Jul-12, 02:23 PM

CaptainToonces, you have received excellent answers to your question, that demonstrate the important point that the rules that apply to "acceleration" depend a bit on whether you are talking about your own acceleration or the apparent acceleration of something else. Would you like additional information about this difference?

grant hutchison

2010-Jul-12, 03:49 PM

CaptainToonces, you have received excellent answers to your question, that demonstrate the important point that the rules that apply to "acceleration" depend a bit on whether you are talking about your own acceleration or the apparent acceleration of something else. Would you like additional information about this difference?You're becoming quite the scaffolding salesman. :lol:

Grant Hutchison

CaptainToonces

2010-Jul-12, 06:37 PM

CaptainToonces, you have received excellent answers to your question, that demonstrate the important point that the rules that apply to "acceleration" depend a bit on whether you are talking about your own acceleration or the apparent acceleration of something else. Would you like additional information about this difference?

What I am asking is if the rules that apply to speed, specifically the "cosmic speed limit" c, would also apply to acceleration, since acceleration is the speed of speed. To put it in terms of an SAT analogy, speed is to location as acceleration is to speed, therefore would the rules governing speed's relationship to location also apply in the same way to acceleration's relationship to speed.

Ken G

2010-Jul-12, 06:47 PM

You're becoming quite the scaffolding salesman. :lol:

:) Is that a Grantism? (I'm hoping I will eventually be asked not to any more. Won't that be a hoot.)

grant hutchison

2010-Jul-12, 06:54 PM

What I am asking is if the rules that apply to speed, specifically the "cosmic speed limit" c, would also apply to acceleration, since acceleration is the speed of speed. To put it in terms of an SAT analogy, speed is to location as acceleration is to speed, therefore would the rules governing speed's relationship to location also apply in the same way to acceleration's relationship to speed.No, there's no cosmic acceleration limit. But the cosmic speed limit is manifested in the way acceleration seems to reduce at higher relative velocities. No matter how hard the spaceship accelerates in its own frame (no matter how many g's the astronauts are pulling), the spaceship can only ever edge towards light-speed ever more gradually for outside observers.

Grant Hutchison

grant hutchison

2010-Jul-12, 06:56 PM

:) Is that a Grantism?I can't make them Grantisms. I just put them out there and other people decide. :lol:

Grant Hutchison

Ken G

2010-Jul-12, 07:21 PM

What I am asking is if the rules that apply to speed, specifically the "cosmic speed limit" c, would also apply to acceleration, since acceleration is the speed of speed. To put it in terms of an SAT analogy, speed is to location as acceleration is to speed, therefore would the rules governing speed's relationship to location also apply in the same way to acceleration's relationship to speed.

I think your question touches on insights that our current physics gets pretty sketchy about (Unruh radiation, Mach's principle, the lot). But you seem interested in digging deeper than the answers you've received, so if you'll indulge me, I think the place you have to start is in understanding that the very meaning of acceleration must first be clarified.

This goal is to a large extent met by grant hutchison's answer, where he distinguished limits that might apply to your own acceleration, versus what happens to the apparent acceleration of other things. Your own acceleration is called your "proper acceleration", and it is what an accelerometer in your pocket would measure. As far as we know, there is no limit that isn't purely technological on that acceleration. (Except there might be an issue with the Unruh radiation you'll get from your connections to your surroundings.) But an even more important point is, acceleration limits in your own isolated little system have nothing to do with relativity, so we have not yet even made contact with anything analogous to a "universal speed limit".

I think the key point to get here is that physics divides into two really fundamentally different types of questions-- one type is questions about what happen to objects, according to observers (probably hypothetical) moving with the object or on the object, as various things happen to them (like forces act on them), and the other type is about what appears to happen to objects according to observers who are not moving with the objects, and who have to invoke a bit of indirect inference to piece together a coherent story about that object.

The first type of question is about what the accelerometer in your pocket will read when the rocket engine under your chair ignites, and that question has nothing whatsoever to do with relativity or universal speed limits. Newton's laws work just fine there, thank you very much, and continue to work for all time. That's why you can have any acceleration on that accelerometer in your pocket-- there's no relativity involved, and no contact with anything like a universal speed limit, so there's certainly no contact with a universal acceleration limit.

So what is relativity then? It is the means to connect what is happening to the object itself, on its own accelerometer and its own rocket engine, to what is happening to other observers (including, what they measure about that object). Questions like, "what speed is it going", or "what rate is that speed changing" fall into this second category-- they are results of measurements done by observers who are not moving with the objects in question. In particular, the accelerometers carried by those observers are not relevant, and normally we'd stick to inertial observers (whenever possible) whose accelerometers read zero. What I'm saying is, when those observers talk about the acceleration of that other object, they are not talking about anything that reads on any accelerometers. They are talking about coordinate acceleration, as you were-- coordinate acceleration is the speed of speed, like you said. But here's the thing about coordinate acceleration-- it depends on your choice of coordinates.

So when grant hutchison made those plots he linked to, he took great pains to describe the coordinates he was using, and the results come out pretty bizarre. Some other coordinates would give a different result, bizarre in some other way. All I'm saying is, you could get any answer you want, without limit, for either acceleration or speed, if you are talking about coordinate acceleration or coordinate speed. No limits at all, and indeed you can even get speeds (in the sense of dx/dt) larger than c, and do so in cosmology.

So what is this "universal speed limit" business? Well, in the absence of gravity, it turns out that Einstein found one particularly convenient coordinate system for analyzing the acceleration of other objects (the Einstein simultaneity convention and rigid-ruler distance reckoning by inertial observers), and in those coordinates, you can show that no two objects can crash into each other with a relative speed greater than c. Since the speed they crash into each other is an invariant (does not depend on coordinate choice, it's one of those truths of the first type above that you can ask the object itself about), anything you conclude in one coordinates must hold in all. Voila, objects cannot crash into each other faster than c, it's a universal law (nitpick: unless they started faster than c, which doesn't seem to happen). Is that a "speed limit"? No, not really-- speed is normally taken to mean coordinate speed, which can still be anything-- just not in those lovely convenient coordinates Einstein found (coordinates where light always travels at c).

What about when gravity is introduced? Things get more complicated, but the basic facts from the above continue to be true. Basically, you have to add a "third layer" of language onto the language of what is happening to the object itself, and what are the perceptions of observers not moving with the object but close to the object. You have to be able to talk about the perceptions of observers neither moving with the object, nor in its vicinity, and the machinery of GR is required to do that. I won't say more, both because it's OT, and because it stretches what I actually know about GR.

Bottom line: there is no limit to what the accelerometer in your pocket can read, unless there is a limit. Either way, it has little to do with relativity, not in your little closed accelerating system.

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