Relativistic Resistance against Acceleration?

[Bjarne]

Let’s say there was a stable Acceleration Due to gravity at 1m/s^2 towards a certain direction, and a stone would keep accelerating that direction.

The Stone wood not reaches the speed of light, because it requires more and more energy to reach a diminishing speed increment.

How fast would the max speed be the stone could travel, relative to an observer at rest?
And how can it be calculated?

 

[chill_factor]

Let’s say there was a stable Acceleration Due to gravity at 1m/s^2 towards a certain direction, and a stone would keep accelerating that direction.

The Stone wood not reaches the speed of light, because it requires more and more energy to reach a diminishing speed increment.

How fast would the max speed be the stone could travel, relative to an observer at rest?
And how can it be calculated?

it can travel asymptotically close to the speed of light.

however i think, eventually, it could appear to exceed the speed of light if it travels outside the Hubble radius, and it will take a few tens of billion years for that to happen.

 

[Bjarne]

.

however i think, eventually, it could appear to exceed the speed of light if it travels outside the Hubble radius, and it will take a few tens of billion years for that to happen.

I cannot believe that, - the limit must be much before, - an equation must exist

 

[Doc Al]

I cannot believe that, - the limit must be much before, - an equation must exist

Why do you think so?

 

[chill_factor]

I cannot believe that, - the limit must be much before, - an equation must exist

why not? locally, that is to say, if you were racing with it, it can never exceed the speed of light but can get arbitrarily close depending on how much energy you put into it.

 

[bcrowell]

Let’s say there was a stable Acceleration Due to gravity at 1m/s^2 towards a certain direction, and a stone would keep accelerating that direction.

The Stone wood not reaches the speed of light, because it requires more and more energy to reach a diminishing speed increment.

How fast would the max speed be the stone could travel, relative to an observer at rest?
And how can it be calculated?

There are a couple of possible ways to interpret your question.

(1) There is a region in which the gravitational field is uniform, as measured by a certain observer. Anywhere in this region, this observer sees that objects released from rest have an initial acceleration of 1 m/s2.

(2) The gravitational field is such that the stone has a constant acceleration of 1 m/s2 relative to the observer.

If the question is #1, then the answer is that the stone's velocity will approach the speed of light, and it will get pretty close to the speed of light within a few years, after having traveled a few light-years.

If the question is #2, then by definition the stone will reach the speed of light in some finite time, which turns out to be about 9.5 years. After 10 years it will be moving faster than c. During this time, it will have traveled a few light-years. The observer will *not* see this gravitational field as being uniform like the one in interpretation #1. This velocity >c is not necessarily inconsistent with relativity, since the speed limit of c is a local speed limit, and in fact we can't define the speed of an object as seen by a distant observer in relativity. We have a FAQ about this: https://www.physicsforums.com/showthread.php?t=508610

 

[DrGreg]

Bjarne, is gravity an essential part of your question, or are you really just interested in constant acceleration, whatever causes it?

If gravity is important to you, we will have to give you an answer using general relativity, which is quite complicated and may be difficult to explain to you if you don't already understand general relativity.

On the other hand, if you are not really interested in gravity, there is a much simpler explanation using special relativity only.

 

[bcrowell]

If gravity is important to you, we will have to give you an answer using general relativity, which is quite complicated and may be difficult to explain to you if you don't already understand general relativity.

I don't actually think GR is completely necessary here. He just wants a constant gravitational field, which we can have in flat spacetime.

I assume the reason he posed it in terms of gravity is that it appears to lead to a paradox, but I think the paradox can be resolved without depending crucially on GR.

The place where GR might come in handy would be in understanding why you may not be able to unambiguously define the velocity of the rock when it's very distant from the observer; in SR, we can always define global frames of reference.

 

[PAllen]

If we're sticking to SR (flat spacetime, whatever the 'source' of the acceleration):

- the growth in proper distance between two world lines using some family of simultaneity surfaces, divided by proper time for one of the observers can certainly exceed c. This, however, is not a relative velocity - it is a separation rate. Even in in the trivial case of two bodies moving inertially in opposite directions in an inertial frame it can approach 2c.

- but in SR, parallel transport is path independent, thus there is a unique definition of relative velocity. It will never exceed c.

In GR, IMO, the situation is similar except that relative velocity does not definable in any unique way. However, I still prefer to think of the thing that can exceed c and is very much coordinate dependent as a separation rate, not a relative velocity - which just doesn't exit.

 

[bcrowell]

@PAllen - I think your #9 does a better job than my #6 of clearly working this out as a problem in SR. But:

- but in SR, parallel transport is path independent, thus there is a unique definition of relative velocity. It will never exceed c.

I have two quibbles with this.

It will certainly exceed c if the OP decrees it by saying that the acceleration is constant as measured by this observer. It's just that in this interpretation of the OP's question, the question is asking us to start with assumptions that violate the laws of physics.

Your analysis of the relative velocity is right, and mine was wrong. However, at the level of the OP's question, we need to be careful not to leave the OP with the impression that parallel transport is what the observer "sees" for the velocity of a distant object. One way to point up this difference is to note that there is no uniquely well-defined notion of simultaneity in this situation, and the observer can't "see" things instantaneously. Before we can take a velocity vector of the rock and parallel transport it back to the observer, we need to decide *which* velocity vector we want (at what time?). Since both the observer and the rock are noninertial, Einstein synchronization doesn't really work; it's non-transitive, so if it's carried out via a sequence of intermediate clocks, the result depends on what you choose for the intermediate clocks. So the path-independence of velocity works fine, but it doesn't necessarily help us since we have a kind of path-dependence of simultaneity.

 

[PAllen]

One way to point up this difference is to note that there is no uniquely well-defined notion of simultaneity in this situation, and the observer can't "see" things instantaneously. Before we can take a velocity vector of the rock and parallel transport it back to the observer, we need to decide *which* velocity vector we want (at what time?).

Excellent point. If you want to talk about physics rather than mathematical definition, you need to propose how to measure it. For example, by redshift you are measuring the velocity of source at time of emission of some light relative to velocity of target at time of reception. In SR this is unambiguously a relative velocity. In GR, though there is no ambiguity in the actual Doppler measured, it isn't valid to talk about the relative velocity because the same source 4-velocity transported on different paths to the same reception event will lead to different results when the transported vector is compared to the target vector.

 

[Bjarne]

Thank's for all the answers