### Science and Astronomy Questions

Posted:

**12 May 2018 19:56**Mr. Abner has the correct answer, with (almost) the correct explanation. An object with constant acceleration in its frame of reference (what we call "proper acceleration") does not have constant acceleration as we see in the rest frame. The observed acceleration decreases as its speed approaches the speed of light, so it never actually reaches or exceeds the speed of light.

It is tempting to say that this is because the object's mass increases, so that the same force provides less acceleration. But I recommend avoiding that temptation. In fact, applying it naively will lead to the wrong answers. Also, if there is an observer fixed to this accelerating object, that observer will say that the acceleration

What's actually happening is that space and time behave differently between these two frames. The observer fixed to the object will say that distances in the direction of motion become shorter (length contraction). They say the distance between the portals decreases. Observers in the rest frame have another explanation as well: they see that the accelerating clock ticks more slowly (time dilation).

How exactly does the acceleration seen in the rest frame change as the object gets faster? To find out we can use the relativistic velocity addition formula. If an object is moving with a velocity

This is very different than what you would expect from intuitions. You might think that the velocities simply add together, like

Now let's find the acceleration. If the object has constant proper acceleration

Now if we take the time derivative of both sides (this is a great calculus exercise), we'll get the acceleration observed in the rest frame of an object that is experiencing constant acceleration in its frame. The result is:

Where gamma is the Lorentz factor,

Ok, enough math. Let's make a graph to get some easier insight. This is the acceleration (a) that we measure, as a function of the velocity (v) that we measure, in the rest frame, assuming that the acceleration of the object in its frame (α) is constant at 1m/s

As its speed increases, the acceleration decreases, eventually to zero. The curve describing how this happens is interesting. From intuition I would have expected most of the decrease in acceleration to happen when v got very close to c, but it's more of a steady decline through intermediate speeds.

What about the object's observed velocity, with respect to time? How long must we wait for the object to get really close to the speed of light? If we integrate the previous result with respect to time, we get

And here's the plot:

The velocity asymptotes toward c, getting "pretty close" to it after about a human lifetime. How close is "pretty close"? Let's plot the difference between the speed of light and the object's speed. We'll also take this one out to 100,000 years:

After 100 years it is moving about a million m/s slower than light (about 99% of c).

After 1000 years it is moving ten thousand m/s slower than light (99.99% of c).

After 100,000 years it is moving barely 1m/s slower than light (99.999999% of c).

So it takes a long time for it to asymptote to the speed of light (1m/s

Length contraction is also interesting in this case. The distance between the portals decreases. Eventually, it would even be shorter than the size of the object. This is problematic, as it implies the object eventually no longer fits between them. This is also weird, because in our frame we say that the distance between the portals is fine, and it is the object itself getting length contracted, so it should have no problem fitting. That's another classic "paradox" of special relativity, but resolving that one will have to wait for another time. Or if you want to go ahead and read about it, google the "Pole-Barn Paradox".

It is tempting to say that this is because the object's mass increases, so that the same force provides less acceleration. But I recommend avoiding that temptation. In fact, applying it naively will lead to the wrong answers. Also, if there is an observer fixed to this accelerating object, that observer will say that the acceleration

*is*constant. So this "relativistic mass" description does not actually explain what's happening between the two reference frames. It is better to say that the mass is invariant (it does not change with speed).What's actually happening is that space and time behave differently between these two frames. The observer fixed to the object will say that distances in the direction of motion become shorter (length contraction). They say the distance between the portals decreases. Observers in the rest frame have another explanation as well: they see that the accelerating clock ticks more slowly (time dilation).

How exactly does the acceleration seen in the rest frame change as the object gets faster? To find out we can use the relativistic velocity addition formula. If an object is moving with a velocity

**v**relative to our rest frame, and its speed relative to that moving frame is increased by_{rel}**u'**, then the new velocity**v**that we observe isThis is very different than what you would expect from intuitions. You might think that the velocities simply add together, like

**v****=****v**( this is the 'Galilean' view of relativity). But they don't. They can't add in such a straightforward way, especially for large velocities, because the sum cannot exceed the speed of light. This is a situation where 1+1 does_{rel }+ u'*not*equal 2.**Example: If a spaceship is moving at 0.5c relative to us, and then it fires a bullet ahead of it, with a speed of 0.5c relative to the ship, then we observe the velocity of that bullet to be 0.8c. Not 1.0c.**Now let's find the acceleration. If the object has constant proper acceleration

**α**in its frame, then in a short time period**dt'**(also measured in its frame), it will change its velocity (seen in the initial moving frame) by**du' = αdt'**. Then we can rewrite the velocity addition formula asNow if we take the time derivative of both sides (this is a great calculus exercise), we'll get the acceleration observed in the rest frame of an object that is experiencing constant acceleration in its frame. The result is:

Where gamma is the Lorentz factor,

Ok, enough math. Let's make a graph to get some easier insight. This is the acceleration (a) that we measure, as a function of the velocity (v) that we measure, in the rest frame, assuming that the acceleration of the object in its frame (α) is constant at 1m/s

^{2}.As its speed increases, the acceleration decreases, eventually to zero. The curve describing how this happens is interesting. From intuition I would have expected most of the decrease in acceleration to happen when v got very close to c, but it's more of a steady decline through intermediate speeds.

What about the object's observed velocity, with respect to time? How long must we wait for the object to get really close to the speed of light? If we integrate the previous result with respect to time, we get

And here's the plot:

The velocity asymptotes toward c, getting "pretty close" to it after about a human lifetime. How close is "pretty close"? Let's plot the difference between the speed of light and the object's speed. We'll also take this one out to 100,000 years:

After 100 years it is moving about a million m/s slower than light (about 99% of c).

After 1000 years it is moving ten thousand m/s slower than light (99.99% of c).

After 100,000 years it is moving barely 1m/s slower than light (99.999999% of c).

So it takes a long time for it to asymptote to the speed of light (1m/s

^{2}proper acceleration isn't too quick). Yet the object does say that its acceleration is constant the whole time. I think this is pretty mind-bendy. Even when it is moving 1m/s slower than light, it says it is still speeding up by 1m/s per second. Yet it never reaches the speed of light. The speed of light is somehow always faster.Length contraction is also interesting in this case. The distance between the portals decreases. Eventually, it would even be shorter than the size of the object. This is problematic, as it implies the object eventually no longer fits between them. This is also weird, because in our frame we say that the distance between the portals is fine, and it is the object itself getting length contracted, so it should have no problem fitting. That's another classic "paradox" of special relativity, but resolving that one will have to wait for another time. Or if you want to go ahead and read about it, google the "Pole-Barn Paradox".