# Two questions regarding special relativity and acceleration

#### jedidia

##### shoemaker without legs
It's been a while since I played with special relativity, I seem to have forgotten most of it. And right now, I'm having a problem that has acceleration thrown in, and it's doing weird things to my brain.

The first question - constant acceleration. Let's assume a spacecraft constantly accelerates with 1G until it reaches 0.9c. Obviously it's the guy in the spaceship that wants to accelerate with one G, for comfort and stuff. But... as time slows down for him, those 9.81 m/s^2 wouldn't be 9.81 m/s^2 to the observer, because his time is running slower. Except there's the lorentz contraction from the spaceship's point of view, which I rather unsure how to calculate after all this time. Do the length contraction and the time dilation cancel each other out, so the acceleration is determined to be the same to both the observer and the observed, or do they actually measure different accelerations?

Second question - and a rather troubling one, I'm afraid: How do I calculate how much time passes during the acceleration phase on the space ship? It's easy enough for constant velocity, but for an acceleration phase, it seems to be rather difficult. I found this paper here, but I can't handle integrals: https://www2.math.uconn.edu/~bridgeman/posts/acceleration.pdf
Can somebody give me a function t=f(v), where v is a given velocity that was reached by accelerating with 1G from 0, and t is the relative time elapsed on the ship?
That is, if my above assumption is correct that both measure the same acceleration. If not, this is becoming nightmarish...

Huh... Looks like the atomic rockets website has quite a comprehensive collection of equations for this. Because of course it has, why didn't I check that first?

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#### Linguofreak

##### Well-known member
It's been a while since I played with special relativity, I seem to have forgotten most of it. And right now, I'm having a problem that has acceleration thrown in, and it's doing weird things to my brain.

The first question - constant acceleration. Let's assume a spacecraft constantly accelerates with 1G until it reaches 0.9c. Obviously it's the guy in the spaceship that wants to accelerate with one G, for comfort and stuff. But... as time slows down for him, those 9.81 m/s^2 wouldn't be 9.81 m/s^2 to the observer, because his time is running slower. Except there's the lorentz contraction from the spaceship's point of view, which I rather unsure how to calculate after all this time. Do the length contraction and the time dilation cancel each other out, so the acceleration is determined to be the same to both the observer and the observed, or do they actually measure different accelerations?

They measure different accelerations (at least if they're calculating acceleration in terms of velocity rather than rapidity). On an x vs. t plot, the spaceship, as seen by the observer, traces a hyperbola that has the speed of light as an asymptote.

Second question - and a rather troubling one, I'm afraid: How do I calculate how much time passes during the acceleration phase on the space ship? It's easy enough for constant velocity, but for an acceleration phase, it seems to be rather difficult. I found this paper here, but I can't handle integrals: https://www2.math.uconn.edu/~bridgeman/posts/acceleration.pdf
Can somebody give me a function t=f(v), where v is a given velocity that was reached by accelerating with 1G from 0, and t is the relative time elapsed on the ship?
That is, if my above assumption is correct that both measure the same acceleration. If not, this is becoming nightmarish...

Huh... Looks like the atomic rockets website has quite a comprehensive collection of equations for this. Because of course it has, why didn't I check that first?

I'm not much good with integrals either, but there's a helpful thing to know about special relativity: all of the length contraction and time dilation and stuff falls out of the Pythagorean theorem. The only difference is that the t^2 term has the opposite sign to the spatial terms. So if in some reference frame we have the origin at t=0 and x,y,z =0, and an event E occurs at time t and spatial coordinates x, y, z, we can define a quantity s:

s^2 = -t^2 + x^2 + y^2 + z^2

You'll also see the other sign convention:

s^2 = t^2 - x^2 - y^2 - z^2

The two are equivalent, as long as time has the opposite sign from the spatial dimensions, and as long as you use one convention consistently in any given set of calculations.

"s" is called the spacetime interval, and is either positive, zero, or imaginary. It's equivalent to the distance from the origin for the normal Pythagorean theorem. For the first sign convention (with the squares of the spatial coordinates positive), if s is positive, it indicates the proper distance between event E and the origin (that is, the distance in whatever reference frame the distance is longest in (no length contraction)). If it's imaginary, then s indicates the proper time between E and the origin (the time in whatever frame has no time dilation). If it's zero, then either event E is the origin itself, or to travel from the origin to E (or vice versa, if E is earlier in time than the origin) requires traveling at exactly the speed of light. This is why lengths contract toward zero as v approaches c, and why a clock on a fast-moving spaceship appears to slow down towards zero tick rate for an observer at rest.

If you're using the other sign convention, flip the "positive s" and "imaginary s" cases above.

A change between two reference frames with the same origin and different velocities is simply a rotation in a plane that includes the time axis. Acceleration of a physical object over time is just the curvature of the object's worldline. But when you rotate in a plane defined by two axes whose terms in the Pythagorean theorem for the space you're working in have opposite sign (such as time and any spatial axis), you get a hyperbolic rotation instead of a circular rotation. This is why you get time dilation and length contraction instead of length dilation and time contraction, which would be the case if spacetime was just normal 4D Euclidean space, and why you can't accelerate to infinite speed after a finite amount of acceleration (equivalent to turning 90 degrees in the Euclidean case), and then do it again and find yourself traveling backwards in time (two 90 degree turns is a 180 degree turn), and then accelerate to infinite speed twice more and get back to zero velocity with your original time direction.

• jedidia

#### Linguofreak

##### Well-known member
Just stumbled across a youtube video that does a good job of explaining the geometry of special relativity:

• jedidia and N_Molson

#### Ajaja

##### Active member
Second question - and a rather troubling one, I'm afraid: How do I calculate how much time passes during the acceleration phase on the space ship?
And don't forget that you can compare clocks of two observers only in the same space-time point, so you can calculate "how much time passes for each observer" only when they meet again after first encounter. It's always a round trip.

• jedidia

#### MontBlanc2012

##### Member
I would be a little careful when combining accelerations with special relativity. Special relativity is built around the notion of inertial (i.e., non-accelerating) reference frames and it is an ongoing topic of debate as to whether (or how) accelerations should be accommodated within the framework of special relativity. E,g., http://www.ptep-online.com/2017/PP-51-07.PDF.

One view has the accelerations don’t really matter, that accelerated motion consists of just a series of snapshots of inertial motion; the other view is that accelerations do matter, as in time dilations due to gravitational effects (i.e., time runs slower on the Earth’s surface than it does in space.)

In short, there’s no easy or quick answer to your two questions - but a blithe manipulation of the Lorentz transformation probably isn’t the best way forward.

• jedidia

#### steph

##### Well-known member
I think that, while it would be hard to calculate, there would be a point in the curve where it really picks up as time dilation really manifests itself, as it's not exactly linear. 1 year to 0.77c on the ship would be 1.19 years on Earth. So that first year as seen fron Earth would be just slightly less than a year on the ship, and then it keeps getting better/worse. Two years on the ship mean 0.97c,, but 3.75 years back home etc.

I wonder if turning around wouldn't pose a problem at those speeds, you would have one end of the ship having a higher contraction factor than the other one, and that may manifest itself as structural stress

• jedidia

#### francisdrake

When analyzing special relativity you often encounter the "relativistic square-root" function: In a diagram it shows a curve like this: The x-axis is the speed in km/s. Note the highest value is cut off at 0.95 c.

The relativistic value gamma increases very little up to 2/3 of lightspeed,
but then starts to climb quickly.

• jedidia

#### Linguofreak

##### Well-known member
I would be a little careful when combining accelerations with special relativity. Special relativity is built around the notion of inertial (i.e., non-accelerating) reference frames and it is an ongoing topic of debate as to whether (or how) accelerations should be accommodated within the framework of special relativity. E,g., http://www.ptep-online.com/2017/PP-51-07.PDF.

This is a common misconception: In special relativity, it's only in inertial reference frames that you're guaranteed to be able to formulate the laws of physics identically in every frame, but that doesn't mean that you can't formulate the laws of physics at all in accelerated frames.

Additionally, special relativity is the no-gravity limit of general relativity, which does guarantee that the laws of physics can be formulated identically in any coordinates whatsoever, so as long as gravitational effects are negligible, you don't even need to reformulate the laws of physics for accelerated frames (that only is an issue if you're trying to mix gravity and special relativity without using general relativity).

• jedidia

#### jedidia

##### shoemaker without legs
I played around with a few things, and realised that I can't even assume constant acceleration at relativistic velocities. I mean, I would expect the addition of velocities to apply to acceleration, so accelerating with 1G for one second at 0.9c would not actually make you 9.81 m/s faster. Which seems weird, because a local accelerometer should still be able to measure the acceleration as 9.81m/s^2, and you'd still feel the physical effects of that acceleration, but when measuring your new speed against an external reference, you'd find that it did not in fact increase by 9.81m/s^2, only by... erm... 1.864 m/s, if I calculated that correctly. So something simple like calculating how much (proper) time a ship at one G would require tor reach 0.9c becomes devilishly complicated without iteration. Oh dear...

What I'd need in the end is a reliable way to calculate how long a trip lasts from point A to point B (which are, for simplicity, assumed to be in the same frame), in ships proper time, assuming a proper acceleration of 1G, and an arbitrary maximum proper velocity. And a way to calculate the distance of B (or any arbitrary point in the same frame of reference) to the ship, as observed from the ship, and then how long the entire trip apparently took in the reference frame of a and b. Woe is me!

I'll take some time to process all your responses and see what I come up with. Thanks everybody for chiming in!

#### francisdrake

If you want to travel, I think relativity will be no problem. To accelerate masses to high relativistic speeds is consuming energy beyond imagination.

For example, the fastest star in the known universe travels around the black hole in the center of our galaxy.
It reaches 8% of light speed. I doubt that any man-made vessel could beat that. At this speed the relativistic effects are very small.

#### jedidia

##### shoemaker without legs
If you want to travel, I think relativity will be no problem. To accelerate masses to high relativistic speeds is consuming energy beyond imagination.
I'm aware of that, but I decided to hell with it! • francisdrake

#### Linguofreak

##### Well-known member
I played around with a few things, and realised that I can't even assume constant acceleration at relativistic velocities. I mean, I would expect the addition of velocities to apply to acceleration, so accelerating with 1G for one second at 0.9c would not actually make you 9.81 m/s faster. Which seems weird, because a local accelerometer should still be able to measure the acceleration as 9.81m/s^2, and you'd still feel the physical effects of that acceleration, but when measuring your new speed against an external reference, you'd find that it did not in fact increase by 9.81m/s^2, only by... erm... 1.864 m/s, if I calculated that correctly.

Depends how fast the external reference is traveling: if you drop a wrench off the back of the ship, it will fall behind at 9.81 m/s^2

Velocities don't add, but rapidities do. The one slight wrinkle here is that the space of possible rapidities is hyperbolic, not Euclidean. This doesn't matter if you keep thrusting in one direction (and brake in the exact opposite direction), but if you accelerate up to a healthy fraction of c, turn 90 degrees, accelerate the same amount, turn another 90 degrees, and continue until you've completed four boosts, your final velocity will not be zero, as it would in Newtonian physics.

So something simple like calculating how much (proper) time a ship at one G would require tor reach 0.9c becomes devilishly complicated without iteration. Oh dear...

There's a conversion formula between velocity and rapidity (specifically rapidity = arctanh(v/c)), and rapidity is equal to velocity in the non-relativistic limit, so all you need to do is plug your velocity into the conversion formula, treat the resulting rapidity as a Newtonian velocity, and figure out how long it would take to reach that velocity under Newtonian acceleration. Again, this works for the colinear case, otherwise you run into the fact that the space of rapidities is hyperbolic.

See https://en.wikipedia.org/wiki/Rapidity

• jedidia

#### Ajaja

##### Active member
if I calculated that correctly
You can compare your calculations with

• jedidia

#### Ajaja

##### Active member
I mean, I would expect the addition of velocities to apply to acceleration, so accelerating with 1G for one second at 0.9c would not actually make you 9.81 m/s faster.
Faster relative to what? If you're measuring your speed/position relative to the position where you would have been without acceleration, then you are assuming that yours v0 a second ago was 0 m/s, not 0.9c. If you're an external observer who's observing some crazy spaceman flying around at 0.9c, then, of course, you're observing a completly different picture Last edited:

#### jedidia

##### shoemaker without legs
Faster relative to what?
Relative to the thing you're heading for, I should think. What I mean is that the closing velocity of the destination wouldn't have changed by 9.81 m/s...

#### Ajaja

##### Active member
Relative to the thing you're heading for, I should think. What I mean is that the closing velocity of the destination wouldn't have changed by 9.81 m/s...
Yes, of course. And the ship will be accelerating slower and slower according to an external observer at this destination.

#### MontBlanc2012

##### Member
This is a common misconception: In special relativity, it's only in inertial reference frames that you're guaranteed to be able to formulate the laws of physics identically in every frame, but that doesn't mean that you can't formulate the laws of physics at all in accelerated frames.

Additionally, special relativity is the no-gravity limit of general relativity, which does guarantee that the laws of physics can be formulated identically in any coordinates whatsoever, so as long as gravitational effects are negligible, you don't even need to reformulate the laws of physics for accelerated frames (that only is an issue if you're trying to mix gravity and special relativity without using general relativity).
My suggestion remains to treat this subject with a degree of caution.

In the realm of special relativity, one needs to consider the problematic issue of clock synchronisation. In the specific theory, Einstein proposed an explicit synchronisation procedure for clocks in different inertial reference frames, but the special theory does not describe a more elaborate procedure for clock synchronisation between, say, an inertial reference frame and an accelerating reference frame. I would suggest that anyone with an interest in the subject of incorporating accelerations in the special theory of relativity pay particularly close attention to what they mean by clock synchronisation in this particular instance.

It is true that Einstein’s general theory is more accommodating of gravitational accelerations. And we have a general procedure for carrying out clock synchronisation which involves two clocks originally located at (more or less) the same space-time point and synchronised when they are travelling with the same velocity and, presumably, when acted upon by the same set of forces. One clock takes one path through space-time; and the other clock takes a different path through space-time. One path may be a geodesic; the other path may be some arbitrary accelerated path. One then brings the clocks back together and compares the readings of the two clocks. Generally speaking, in the presence of a gravitational force, the two clocks will record a different interval of time. If one wants to compare time intervals as measured by the two clocks without bringing them together again, then (once more) you have to elaborate what procedure you are using for making that measurement - and why such a procedure may yield physically relevant results.

Of course, it may be possible to do all of this in the case of the accelerating spaceship, but the point is that one can’t simply manipulate the symbols of the Lorentz transformation (or the space-time metric of the general theory for that matter) and hope that by so doing it will yield something meaningful. One actually needs to be clear and explicit about the foundations of your theory.

#### Linguofreak

##### Well-known member
In the realm of special relativity, one needs to consider the problematic issue of clock synchronisation. In the specific theory, Einstein proposed an explicit synchronisation procedure for clocks in different inertial reference frames, but the special theory does not describe a more elaborate procedure for clock synchronisation between, say, an inertial reference frame and an accelerating reference frame. I would suggest that anyone with an interest in the subject of incorporating accelerations in the special theory of relativity pay particularly close attention to what they mean by clock synchronisation in this particular instance.

jedidia's questions don't really have much to do with clock synchronization.

He's basically asking about elapsed proper time for a relativistic spacecraft traveling between two stars. He isn't really asking about the elapsed time in the rest frame, only what the spacecraft worldline looks like in the rest frame during the acceleration/deceleration phase, which is easy enough to answer: a constant acceleration trajectory is a hyperbola on a Minkowski diagram. And the elapsed proper time on an arbitrary worldline in flat spacetime can be calculated without invoking general relativity.

Even if he were asking about elapsed time in the rest frame, the spread in stellar velocities in a given area of a given galaxy is typically non-relativistic, so the source and destination stars can synchronize their clocks under the assumption that they're more or less in the same reference frame, and it thus makes sense to ask what the elapsed time at Star A is when the spacecraft arrives at Star B (because A and B are in the same reference frame, so the two stars agree on what events are simultaneous, and thus the moment "when the spacecraft reaches star B" can be precisely defined at A, which wouldn't be the case if the relative velocities of the two stars were relativistic).

There is nothing here that requires GR to handle (unless jedidia wants to go full Tau Zero and deal with large intergalactic distances, in which case Hubble expansion becomes a factor).

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• jedidia

#### MontBlanc2012

##### Member
jedidia's questions don't really have much to do with clock synchronization.

He's basically asking about elapsed proper time for a relativistic spacecraft traveling between two stars. He isn't really asking about the elapsed time in the rest frame, only what the spacecraft worldline looks like in the rest frame during the acceleration/deceleration phase, which is easy enough to answer: a constant acceleration trajectory is a hyperbola on a Minkowski diagram. And the elapsed proper time on an arbitrary worldline in flat spacetime can be calculated without invoking general relativity.

Even if he were asking about elapsed time in the rest frame, the spread in stellar velocities in a given area of a given galaxy is typically non-relativistic, so the source and destination stars can synchronize their clocks under the assumption that they're more or less in the same reference frame, and it thus makes sense to ask what the elapsed time at Star A is when the spacecraft arrives at Star B (because A and B are in the same reference frame, so the two stars agree on what events are simultaneous, and thus the moment "when the spacecraft reaches star B" can be precisely defined at A, which wouldn't be the case if the relative velocities of the two stars were relativistic).

There is nothing here that requires GR to handle (unless jedidia wants to go full Tau Zero and deal with large intergalactic distances, in which case Hubble expansion becomes a factor).

The use of special relativity to address question relating to accelerating reference frames relies on the Clock Hypothesis - namely that “when a clock is accelerated the effects of the motion on the rate of the clock is no more that associated with its instantaneous velocity - the acceleration adds nothing.”

This question of the validity of the Clock Hypothesis remains an open issue. If the Clock Hypothesis holds, then it is entirely appropriate to use special relativity to address jedidia’s questions. If it does not, then you might need to think again.

To quote the paper, I referenced earlier: “This brings to mind Einstein’s equivalence principle introduced in the analysis of accelerated frames of reference in general relativity. The simplest formulation of this principle states that on a local scale, the physical effects of a gravitational field are indistinguishable from the effects of an accelerated frame of reference.” ( - and specifically, in this case, with respect to the real, measurable time dilations that result from a gravitational field.) The equivalence principle, of course, lies at the heart of general relativity - and no doubt Einstein developed the general theory because he recognised that the special theory was not sufficient to house the concept.

Again, I would suggest that you be cautious when you attempt to combine accelerations with special relativity.