# Speed of light in relative frames and acceleration

I have a fog in my brain as I am trying to wrap my head around a problem and I am not sure how to word it so I will do my best.

If object A is moving near the speed of light but without acceleration then it could be said to be at rest. Measuring a beam of light that passes its position would come up with the expected answer of C. So object A decides to accelerate to achieve the speed of C (or near). We know this will take infinite mass-energy. So firing a rocket in this frame will accelerate the object, it will take energy, and it will increase speed. But I am trying to wrap my head around the image since the object A is already near the speed of light. (from the perspective of Z)

It seems to me that this could all be taken as slices. That each frame of reference is taken individually and almost like its own little universe. A is near the speed of light, but only relative to another frame. (Z) When A measures a beam of light, the frame of reference switches to its frame, therefore all measurements are now changed (or recalibrated, or?) At this point its calculations of mass, energy etc are relative to itself so even though it is taking more energy the calculations work for it.....

Hmmm seems to make more sense as i write it... almost like all frames of reference are warped bubbles and as you move from one frame to another the warping affect keeps all equations in line. The acceleration, from A's point, is moving from rest, firing a rocket and picking up speed. They are using, at first, zero energy and then burning energy to accelerate to the speed of light in their frame

From Z's perspective A is already moving near the speed of light. No acceleration (energy already used to reach that speed) and then they expend more energy to accelerate. While A thinks they are moving at much greater velocity. Z's reference point might only see a small percentage change is speed and acceleration, BUT still below C (speed of light)

My original thought got me confused because it occurred to me that the earth could be moving near the speed of light right now and we would not even know it. Then we fire a rocket to accelerate and it tries to reach the speed of light. Then it cuts its engines but continues without acceleration. It launches a ship and it accelerates to near the speed of light and so on...

It was just an interesting thought to think the earth could already be moving near C from some point of view, but all our measurements are from our frame. I am starting to see the relationship between mass and energy now (even as I write) It will be relative to the frame you are in. We would measure a lot more energy in our current frame than an outside observer of our frame would. If that outside observer viewed us at a faster speed.

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Simon Bridge
Homework Helper
Observer A will always measure itself to have speed 0.
So how does it know it is travelling close to c?

You imagine it is travelling from some place, Z, and that place is receding at close to c ... along with lots of other things that are slow moving in Z ... so the common, and confusion-generating, deduction is that A is the one moving. Especially if A had to do some work to get into this situation.

When A decides to do some work - in order to make objects in Zs reference frame move faster. Doing this changes reference frame - A still measures itself to be stationary but objects in Z are now a bit faster, the distances between them are getting smaller, and their clocks get slower and so on. Light is still travelling at c. Z and A agree about the relative speed but disagree about who is doing the moving.

The relationship between energy and rest-mass will be the same in all frames. Masses in frames moving with respect to us will have a higher total energy - the extra energy is kinetic energy.

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more of a clarification of what is happening. I agree that A can state they are stationary, (as long as there is no acceleration or deceleration) but I would imagine that we could also state that we are traveling at 99% of C in relation to some object.

My question initially was how does object A (already at 99% of C) accelerate from there to another 99% of C? Its actually more confusing than that and hard to state in writing, but as I was saying in my first post; As I write it becomes more clear. "A" can do this by changing its frame of reference, so we must consider A stationary at all times in order to accelerate to C(except during acceleration or deceleration)

Partly I was thinking that as A accelerated to C the energy it uses is infinite and is harder to do. I used to think this was like resistance and we would be able to measure the difficulty of the task. Like pushing against a rubber barrier that gets harder and harder and we can feel the difference. I am now beginning to think ( or understand) that there is no resistance. it does not get measurable harder. It is just that our frame constantly changes so we do not notice any discernible difference.

Simon Bridge
Homework Helper
more of a clarification of what is happening. I agree that A can state they are stationary, (as long as there is no acceleration or deceleration) but I would imagine that we could also state that we are traveling at 99% of C in relation to some object.
We cannot properly do that - however we can deduce that some object is moving at 0.99c in our reference frame and so deduce that we are also moving at 0.99c in the objects reference frame. When you are getting used to relativity it is a good idea to get pedantic about exactly what you mean.
My question initially was how does object A (already at 99% of C) accelerate from there to another 99% of C?
It does not need to accelerate to 0.99c - as it is already there. If it does some work it may accelerate to some higher percentage of the speed of light... which is to say, it changes reference frame to one in which the previous object is moving at a higher relative speed.

Its actually more confusing than that and hard to state in writing, but as I was saying in my first post; As I write it becomes more clear. "A" can do this by changing its frame of reference, so we must consider A stationary at all times in order to accelerate to C(except during acceleration or deceleration)

Partly I was thinking that as A accelerated to C the energy it uses is infinite and is harder to do. I used to think this was like resistance and we would be able to measure the difficulty of the task. Like pushing against a rubber barrier that gets harder and harder and we can feel the difference. I am now beginning to think ( or understand) that there is no resistance. it does not get measurable harder. It is just that our frame constantly changes so we do not notice any discernible difference.
If each incremental acceleration got harder as you approached the speed of light, you'd be able to use this "resistance" to work out your absolute speed. Since absolute speed cannot be determined (there is no such thing) you won't expect to experience anything special about your acceleration.

Ger
Speed of light (information) is constant in every frame. Every frame can be defined as an inertial frame, even if one is accelerating away/towards another frame. Small objects, in reference to the observer frame are moving in observers frame space, not necessarly along a geodesic (corrections needed) and not with the same speed. Going from one frame (observer) to object frame needs transformation which should not change any physical process in neither frames, but can give rise to different measurement basic parameters. Things look shorter, things seem to live longer. Things do seem to have more energy. Frames are not changing, it is the point of view changing.

pervect
Staff Emeritus
I have a fog in my brain as I am trying to wrap my head around a problem and I am not sure how to word it so I will do my best.

If object A is moving near the speed of light but without acceleration then it could be said to be at rest. Measuring a beam of light that passes its position would come up with the expected answer of C. So object A decides to accelerate to achieve the speed of C
This can't be done. Your brain will remain foggy as long as you insist that it can be done. IT seems simple enough to me to just say "it can't be done", but until your brain accepts that, it'll remain foggy. Historically from similar questions, people seem to have some foggy reasoning for thinking it can be done and therfore refuse to accept that it can't be done. (I can't say for sure if you'll follow this historical pattern). But they can't really explain why they think it can be done either, so we reach an impasse.

Looking at the mathematical proofs that it can't be done might be one way to get rid of the fog. Look at the Lorentz transforms, or the relativistic velocity additon formulas, and satisfy yourself that no matter how many times you do a Lorentz boost, you'll always come up with a velocity less than 'c'.

Have you studied the Lorentz transform at all? If not, that's probably the place to start.

Every frame can be defined as an inertial frame, even if one is accelerating away/towards another frame.
An inertial frame is one which is not undergoing acceleration.

Ger
An inertial frame is one which is not undergoing acceleration.
Frame A and B are accellerating away from frame C. Both with same accelleration along same vector. Perpendiculair both frames drift away with a constant speed. You can not say that there is no acceleration, there is. It is the relative constant speed between two (mass) frames which counts. for constant speeds, the Lorentz transformations do work in euclidian space. For acellerating frames, one needs to 'curve' the space to have things Lorentz invariant again.