What happens to speed and velocity in a rotating frame of reference?

[somega]

Let's say I can see a star in a distance of 1ly in front of me.
I'm sitting on an office chair and begin to rotate with a speed of 1/s (1 rotation per second).
Then I consider me as not rotating and everything else as rotating.
Now I will see the star traveling a circular path with the incredible length of 2*pi*1ly in 1 second.

What is wrong about saying the star is traveling with a speed of 2*pi*ly/s ≈ 2*10^8*c?

 

[hutchphd]

Because you are obviously and self-evidently rotating. That's it.

 

[A.T.]

What is wrong about saying the star is traveling with a speed of 2*pi*ly/s ≈ 2*10^8*c?

Nothing is wrong about that.

 

[somega]

Because you are obviously and self-evidently rotating. That's it.

Isn't it in the eye of the beholder?

 

[somega]

Nothing is wrong about that.

Does it mean superluminal is possible in a rotating frame?

 

[A.T.]

Does it mean superluminal is possible in a rotating frame?

Yes.

 

[A.T.]

Isn't it in the eye of the beholder?

No.

 

[Ibix]

Does it mean superluminal is possible in a rotating frame

It's impossible to overtake a light pulse. That fact doesn't change just because you use a rotating frame. But the speed a light pulse travels at is only 3×10⁸m/s in an inertial frame. If doesn't even have a single defined value, as measured in a rotating frame.

 

[PeterDonis]

Because you are obviously and self-evidently rotating.

You are conflating two different senses of the word "rotating".

While spinning in his chair, the OP will feel forces that he does not feel when not spinning (here "not spinning" means "the rest of the universe doesn't appear to be rotating around him"). In that sense "rotation" is an invariant (but explaining exactly what is invariant requires quite a bit more work than you might expect).

However, the OP was asking about whether or not it is valid for him to use a reference frame in which he, spinning in his chair, is at rest and not rotating, and the rest of the universe is rotating around him. The answer to that is yes, it is valid. But such a reference frame will not be inertial, so things that are true in inertial frames (such as that the speed of light, $3 \times 10^8$ meters/second, is the maximum possible coordinate speed for any object in the frame) will not be true in such a frame.

 

[hutchphd]

Your point is well taken but I am not convinced as to the intent of the OP question.
In your context is there an invalid reference frame? It seems to me only a question of utility.

 

[somega]

Now that I am rotating I fire a bullet that will hit the star.
Let's ignore the speed that the bullet is going away from me.
The bullet will arrive at the circular path of the star.
The star will hit the bullet from the side.
From my point of view, they will hit each other with a speed of 2*10^8*c.
Is this possible?

 

[FactChecker]

Now that I am rotating I fire a bullet that will hit the star.
Let's ignore the speed that the bullet is going away from me.
The bullet will arrive at the circular path of the star.
The star will hit the bullet from the side.
From my point of view, they will hit each other with a speed of 2*10^8*c.
Is this possible?

You are going astray here. Both the bullet and the star will be circling at essentially the same rate.

PS. Even without the complication of the non-inertial frame, there is a great deal of SR that does not simply apply on the large, galactic scale.

 

[A.T.]

From my point of view, they will hit each other with a speed of 2*10^8*c.

No.

 

[somega]

If in my reference frame I am not rotating there should be no centrifugal force and the bullet will take a straight line?
So the bullet is not rotating and the star is rotating so they will hit at 2*10^8*c?

 

[FactChecker]

Suppose you fire the bullet from the center of rotation so there is no lateral velocity. The bullet will go straight in the direction it was pointing when it was fired. You are rotating away from that direction. So you can not really say that your rotating reference frame is as good as any other.

 

[A.T.]

If in my reference frame I am not rotating there should be no centrifugal force and the bullet will take a straight line?

If the reference frame is rotating there are centrifugal and Coriolis forces.

 

[Nugatory]

What is wrong about saying the star is traveling with a speed of 2*pi*ly/s ≈ 2*10^8*c?

Nothing. That “speed” is a coordinate velocity, the change in the position coordinates with the time coordinate. Clearly you can make a coordinate velocity come out to be whatever you want just by choosing coordinates that make it so, and you have chosen coordinates in which the star is moving at that absurd superliminal speed. Note, however, that the coordinate speed of a flash of light next to the star and moving in the same direction isn’t $c$, it’s something even greater than the “speed” of the star so the star still isn’t moving faster than light.

When people say that the speed of light is $c$, there is unstated qualification - we are using coordinates in which Newton’s first law works so that there is no coordinate acceleration (that is, change in the coordinate velocity with time) when there are no forces. These frames are often referred to as “inertial”. Your rotating frame is not inertial; there is no force between the Earth and the star, yet the star is not moving in a straight line at a constant speed and even locally on Earth you will see odd things like the Coriolis effect.

From my point of view, they will hit each other with a speed of 2*10^8*c.

You are mistaken here. If you fire the bullet straight up (and at escape velocity, so that it doesn’t fall back to earth) its coordinate velocity will steadily increase as it moves away from the earth, for the same reason that the star has an enormous coordinate velocity just because it is far away.

 

[somega]

What about the kinetic energy of the star?

Let's start from the beginning.
I am sitting on my office chair (not rotating) and seeing the star in front of me.
The star is not moving.
Now I begin to rotate on my office chair.
The star is now moving with a very high speed on a circular path around me.
Does the star now have kinetic energy?
Did I create a huge amount of energy just be rotating on my office chair?

 

[jbriggs444]

Does it mean superluminal is possible in a rotating frame?

It means that speeds greater than c are possible, but not speeds faster than light. Light moving in the same direction as the apparent motion of the star will still pass it.

 

[PeterDonis]

In your context is there an invalid reference frame?

Sure, it's easy to construct invalid reference frames: just make the mapping between coordinates and points in spacetime not be one-to-one, for example. But a reference frame is not invalid just because it happens to be non-inertial and allow coordinate speeds greater than $c$.

 

[PeterDonis]

If in my reference frame I am not rotating there should be no centrifugal force

Wrong. The centrifugal force is there because the frame is non-inertial. It has nothing to do with your motion or lack thereof relative to the frame.

Let's start from the beginning.

Yes, let's, by you answering this question: What textbooks or other references have you studied to learn about how non-inertial, rotating frames work? This isn't even an issue specific to relativity: rotating frames have to be handled properly (differently from inertial frames) even in Newtonian physics.

Does the star now have kinetic energy?

Yes. Kinetic energy is frame-dependent.

The question that has an invariant, frame-independent meaning is: did you do any work on the star by starting to spin in your chair? And the answer to that is obviously no.

 

[PeroK]

What about the kinetic energy of the star?

Let's start from the beginning.
I am sitting on my office chair (not rotating) and seeing the star in front of me.
The star is not moving.
Now I begin to rotate on my office chair.
The star is now moving with a very high speed on a circular path around me.
Does the star now have kinetic energy?
Did I create a huge amount of energy just be rotating on my office chair?

You are falling into your own trap in a way. You start with the implicit assumption that "sitting in your chair" is somehow a fixed reference frame and that things that are not moving in that frame are "really" not moving.

Then, you start spinning in your chair and things that are "really" not moving are now moving and have somehow assumed a kinetic energy. And this seems odd to you.

But, if we do start from the beginning, then sitting in your chair, you are on the surface of a rotating Earth, which is itself orbiting the Sun, which is orbiting the Galactic centre ...

While you are sitting in your chair, therefore, a distant star is already moving about in a daily and yearly pattern. You do not need to spin in your chair to create relative motion between you and the star. Or, to give the star an arbitrary kinetic energy, based on the motion of the Earth within the solar system.

 

[pervect]

Let's say I can see a star in a distance of 1ly in front of me.
I'm sitting on an office chair and begin to rotate with a speed of 1/s (1 rotation per second).
Then I consider me as not rotating and everything else as rotating.

This is basically where you're going wrong. You wish to apply the methods that you are familiar with from a non-rotating frame to a rotating frame, but the sort of methods you are familiar with are not applicable to a rotating frame.

Calling the rotating frame "non-rotating" still won't allow you to apply the methods that you appear to be using to a rotating frame.

The easiest solution is to simply not use rotating frames. A frame is something that one is free to chose, and it's generally wise to make the easiest possible choice, rather than to confuse oneself by making it unnecessarily complex.

The next step up is to learn how to deal with rotating frames in the context of Newtonian physics. AT has some helpful posts in this regard.

At the end of this, you should be able to work a problem in a non-rotating frame, or a rotating frame, transform a known solution from a non-rotating frame to a rotating frame and vica-versa, and demonstrate that the solutions are equivalent, so that it doesn't matter which frame you decide to use.

From your attempt to work out the collison speed, I would say you are not at this point yet.

After learning how to deal with rotating frames correctly in Newtonian physics, the next step would probably be to learn how to do special relativity in a non-rotating frame. The math isn't that hard, but some of the concepts may be challenging. A certain amount of un-learning may be necessary, this always seems to be tricky.

Only after you have mastered rotating Newtonian frames and non-rotating relativistic frames would I recommend even trying to learn about relativistically rotating frames.

 

[A.T.]

What about the kinetic energy of the star?

What does this have to do with rotating frames? The kinetic energy of an object can be different between inertial frames too. Try to understand the simple stuff first.

https://en.wikipedia.org/wiki/Kinetic_energy#Frame_of_reference

 

[haushofer]

My 2 cents: what people often state as "nothing can go faster than the speed of light" is actually "a physical object cannot be physically accelerated beyond the speed of light".

You are rotating. You feel inertial forces. Not the star. So: no problem.

 

[1977ub]

It is important to understand that the difference between inertial vs. non-inertial frames is not simply a matter of geometry - it is something physical, which needs to be detected.

 

[Dale]

Let's start from the beginning.
I am sitting on my office chair (not rotating) and seeing the star in front of me.
The star is not moving.
Now I begin to rotate on my office chair.

So, in this reference frame not only are there time varying centrifugal and Coriolis forces, there is also the Euler force. This is a messy reference frame.

Does the star now have kinetic energy?
Did I create a huge amount of energy just be rotating on my office chair?

Yes, however energy is not conserved in this frame, so creating it is not really a big deal. You won’t be able to do anything with this energy.

 

[olgerm]

This wikipedia article is releated to the topic https://en.wikipedia.org/wiki/Absolute_rotation . Main point is that there is a preferred non-rotating (inertial) type of frame of references.

 

[FactChecker]

This wikipedia article is releated to the topic https://en.wikipedia.org/wiki/Absolute_rotation . Main point is that there is a preferred non-rotating (inertial) type of frame of references.

I think this is too strong a statement. It implies that the same result can not be obtained if the object is assumed stationary and the universe orbiting. That is debatable. Although the calculations may be much more complicated, they may be valid. (see The Machian Origin of the Centrifugal Force)

 

[Khashishi]

Speed and velocity depend on your choice of coordinates, and you can easily choose coordinates where speed > c. However, if you create a plot of the worldline of any object, you will find that it stays within the lightcone which expands outward at the speed of light relative to a local inertial observer. You aren't local to the object, so the light cones might not appear to travel at c to you. In special relativity, it's easy to simply only use inertial reference frames with flat coordinates, and then light will travel at c everywhere, but in general relativity, it isn't so simple because spacetime is curved, so you might have light traveling faster than c, such as light falling into a black hole using flat coordinates of an asymptotically distant observer. You are probably making things more difficult by using a rotating frame of reference, but it is not necessarily wrong, as long as you apply your coordinates correctly.