High School Deep Space Speed Limit: What Prevents Exceeding Light Speed?

[cmw]

TL;DR

Deep space speed limit

This is probably a dumb question. I'm not a physicist and took basic physics a very long time ago.

If an object was in deep space, a long way away from gravitational fields and was subjected to a constant 1g acceleration in a straight line what prevents it from eventually exceeding light speed (in approximately 356 days)? Velocity is only meaningful if measured relative to something else, but relative to the object, it is still traveling at an ever increasing speed. If an observer on earth watched the object what would they see?
Just curious.

Thanks,

Chris

 

[Drakkith]

Nature itself prevents it. Objects under accelerating simply cannot exceed the speed of light. I can give you all the formulas and experiments supporting this, but at the end of the day it's simply that the way the universe works is such that no object can exceed the speed of light*.

Velocity is only meaningful if measured relative to something else, but relative to the object, it is still traveling at an ever increasing speed.

We can say that the object feels a constant acceleration, but we can't measure its velocity relative to itself.

If an observer on earth watched the object what would they see?

They would see the object continue to accelerate forever but with a decreasing acceleration over time such that the object never exceeds the speed of light. This reconciles with the observer on board the spacecraft who feels a constant, never decreasing acceleration when we include time dilation I believe.

*With the caveat that we are either talking about 'local' spacetime and not dealing with cosmological expansion, where the expansion of space can cause objects to recede from each other at any velocity, even FTL velocities. But these objects are not moving faster than a beam of light would locally.

 

[Dale]

TL;DR Summary: Deep space speed limit

If an object was in deep space, a long way away from gravitational fields and was subjected to a constant 1g acceleration in a straight line what prevents it from eventually exceeding light speed (in approximately 356 days)?

That isn’t how acceleration works. You are assuming that a constant acceleration produces a parabolic time vs position profile. However, it does not. A constant acceleration produces a hyperbolic profile. That hyperbola has an asymptote of c, so you never reach c (let alone exceed it).

 

[Nugatory]

If an object was in deep space, a long way away from gravitational fields and was subjected to a constant 1g acceleration in a straight line what prevents it from eventually exceeding light speed (in approximately 356 days)?

Google for “relativistic velocity addition”.

We have a rocket that is initially at rest relative to the earth. It fires its motors for one second and increases its speed from zero to 9.8 meters/second relative to the earth - that’s what accelerating at 1g means.

We can just as reasonably describe this situation as saying that the rocket is still at rest but the earth is moving backwards at 9.8 meters per second - to be more precise, we should say that the rocket is at rest relative to a hypothetical object that was already moving relative to the earth at 9.8 meters per second.

Now when the rocket fires its motors for another second it increases its speed relative to that hypothetical object from zero to 9.8 meters per second. But because of the way relativistic velocity addition works, its speed relative to the earth does not increase from 9.8 meters per second to 19.6 meters second, but instead to something less. That is, after two seconds the speed of the rocket relative to the earth is less than the sum of the speed of the rocket relative to the hypothetical object and the speed of the hypothetical object relative to the earth. If you work through the math in the relativistic velocity addition formula you will find that the speed of the rocket relative to the earth never exceeds $c$.

The difference between straightforwardly adding the speeds and the correct relativistic formula for adding speeds is almost unnoticeable for speeds less than many thousands of kilometers per second - which is why we usually don’t notice it - but sufficiently sensitive measurements have been done and seen this effect. In fact, it was first detected by Fizeau a half-century before the discovery of relativity.

 

[cmw]

Google for “relativistic velocity addition”.

We have a rocket that is initially at rest relative to the earth. It fires its motors for one second and increases its speed from zero to 9.8 meters/second relative to the earth - that’s what accelerating at 1g means.

We can just as reasonably describe this situation as saying that the rocket is still at rest but the earth is moving backwards at 9.8 meters per second - to be more precise, we should say that the rocket is at rest relative to a hypothetical object that was already moving relative to the earth at 9.8 meters per second.

Now when the rocket fires its motors for another second it increases its speed relative to that hypothetical object from zero to 9.8 meters per second. But because of the way relativistic velocity addition works, its speed relative to the earth does not increase from 9.8 meters per second to 19.6 meters second, but instead to something less. That is, after two seconds the speed of the rocket relative to the earth is less than the sum of the speed of the rocket relative to the hypothetical object and the speed of the hypothetical object relative to the earth. If you work through the math in the relativistic velocity addition formula you will find that the speed of the rocket relative to the earth never exceeds $c$.

The difference between straightforwardly adding the speeds and the correct relativistic formula for adding speeds is almost unnoticeable for speeds less than many thousands of kilometers per second - which is why we usually don’t notice it - but sufficiently sensitive measurements have been done and seen this effect. In fact, it was first detected by Fizeau a half-century before the discovery of relativity.

Thanks I'll look into relativistic velocity addition.

 

[PeroK]

Thanks I'll look into relativistic velocity addition.

You can also calculate the relationship between the "proper" acceleration of the object in its own rest frame (in this case a constant $a = g$) and the acceleration measured in the original rest frame ( e.g. as measured on Earth). If we denote that acceleration as $a'$, then we find that:$$a = \gamma^3 a'$$where$$\gamma = \frac{1}{\sqrt{1- v^2/c^2}}$$And you see that as the relative velocity $v$ increases towards $c$ the gamma factor ($\gamma$) increases without limit and the measured acceleration of the object reduces towards zero.

You don't need to go to deep space to test this equation. CERN and other particle accelerators routinely accelerate charged particles with huge proper acceleration and to very high energy and yet never to or beyond $c$.

In fact, the total energy of a particle of mass $m$ is given by:$$E = \gamma mc^2$$And again you can see that as $v$ approaches $c$ the energy of the particle increases without limit. In other words, no matter how much energy you give a particle its speed never actually reaches $c$.