# Upper Limit of Acceleration?

1. Nov 13, 2007

### Paranoia

It is my understanding that as you travel faster, the distance in front of you contracts. This makes sense. Here is the question: Can that contraction exceed the speed of light?

Same question, different way of asking.
Muons have a shorter half-life than the time it takes light to travel from the upper atmosphere to the ground, yet muons can still travel that distance and make it to the earth's surface. This is because, from the muons perspective, the distance between the muon and the earth contracts, reducing the time for it to reach the ground. That's what I get. My question is, is it possible for something to accelerate so fast, that from its point of view, the distance between it and an object in front of it contracts faster than the speed of light?

Thanks,
Para

2. Nov 14, 2007

### Shooting Star

If what you are saying is true, then you will catch up with the object in front of you faster than light, which is not possible.

But there should not be any upper limit of acceleration of a point mass wrt an IFR. Accn = delta v/delta t, and the denominator can be made as small as desired, as long as we don’t make the final speed more than c.

3. Nov 14, 2007

### RandallB

Of course the answer is yes but because the “distance between it and an object” is not something REAL that is changing lengths but just an observation that changes perspective as the muon observer changes reference frame. NO different than a shadow or laser spot on the moon can move across the surface FTL.

Pick something with a long half-life that lasts accelerated up to 0.8c in one minute. Much slower and longer than a muon. The distance one light-minute away will change to 1/2 Light-minute away – a rate change in one minute of only 0.5c
But pick an object one light year away the “distance” observed changes to half a light year change in only one minute, a rate change very much FTL. But it doesn’t mean anything, the only way to correctly measure a real speed for that distant object approaching in the traveler’s view, is by using the clock or ‘time’ of the traveler. Which is much slower than the original starting frame and gives speeds of 0.8c or just as expected.

Last edited: Nov 14, 2007
4. Nov 14, 2007

### Chris Hillman

5. Nov 14, 2007

### Paranoia

Thank you. I would still call that distance real, and very much different than a shadow; but if I understand what you're saying correctly, since it is the dimension contracting, rather than an object moving, the speed of the contraction is not limited.

Thanks again,
Para

6. Nov 14, 2007

### Staff: Mentor

Hi Paranoia,

Don't forget, the rest frame of your accelerating observer is, by definition, a non-inertial frame. If you look at other non-inertial frames you can see that things like this happen routinely. For example, if a frame is rotating at about 1 revolution per second then the moon is "travelling" faster than c, and distant stars are "traveling" many many times c. So as randall says, the answer to your question is "yes" but it doesn't mean anything more significant or real than the moon "traveling" faster than c in a modestly rotating reference frame. None of this poses a problem for physics.

7. Nov 14, 2007

### pervect

Staff Emeritus
To avoid confusion and to get standard answers, velocities in SR should be measured in inertial frames.

If one doesn't do this, it's possible to get answers that are not only many multiples of the speed of light, but answers that even point in the wrong direction. Suppose you are traveling at a high relativistic velocity with a high gamma factor towards a distant star, and "turn on the brakes" and start to deaccelerate. As your velocity decreases, "the" distance (which was originally Lorentz contracted) can actually increase increase, even though in an inertial frame you are still moving towards your destination. So in this case your velocity as measured in an inertial frame has a different sign than the rate of change of distance with respect to time measured in a non-inertial frame.

I worked this out in some old post, but I don't recall exactly where this post is anymore.

Note that in order to work this problem out I had to make some assumptions about what "distance" in a non-inertial frame means.

Distance in an inertial frame is the length of a straight line. "Distance" in a non-inertial frame is the length of some non-straight line, defined by some notion of "simulaneity" adopted by the non-inertial observer.

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