Question for you, Wiccan_Child

[Resha Caner]

We really do consider length to contract...

A digression from the other relativity thread, if you don't mind. Others could probably answer this as well, but it was you who said it, so it's you I'm asking to clarify.

OK. We have two ships (A and B) moving at constant velocity, but with a relative velocity. So, they are both in their own inertial frame. Doesn't A observe the length of B to contract and B observes the length of A to contract? It's not that A (or B) observes his own length to contract, correct? Nor is it that A is in some objective frame so that the length of B "really" contracts, but the length of A doesn't. Correct?

I admit I'm good at confusing myself with this. When I was in highschool, I thought I created a paradox about length contraction. Then I though I had resolved the paradox. Now I'm not sure again.

 

[[serious]]

A digression from the other relativity thread, if you don't mind. Others could probably answer this as well, but it was you who said it, so it's you I'm asking to clarify.

OK. We have two ships (A and B) moving at constant velocity, but with a relative velocity. So, they are both in their own inertial frame. Doesn't A observe the length of B to contract and B observes the length of A to contract? It's not that A (or B) observes his own length to contract, correct? Nor is it that A is in some objective frame so that the length of B "really" contracts, but the length of A doesn't. Correct?

I admit I'm good at confusing myself with this. When I was in highschool, I thought I created a paradox about length contraction. Then I though I had resolved the paradox. Now I'm not sure again.

Both observe the other to be contracted. Both have equally "real" observations.If you can give me the specific paradox, I can probably resolve it for you.

 

[Resha Caner]

Both observe the other to be contracted. Both have equally "real" observations.

But they don't observe themselves contracting. As far as their perception of their own length goes, nothing has changed. Correct? That will be an important point to what follows.

That same issue is important to the Twin Paradox. There has always been the problem that relativity eliminates the fixed reference frame, so how does one know which twin will age more?

The proposed solution is that if both start in the same inertial frame, it is the one who accelerates into a different inertial frame (and then decelerates back) who will age less. And, IIRC this was verified experimentally with clocks.

Despite that similarity, I believe my (apparent) paradox differs from the Twin Paradox ... as well as the Ladder Paradox, etc. But, if it does turn out that this is just a different version of one of those, this will be easy.

So, let's do a little back-and-forth to work into this.

First, there is a possible 'picture' of why 'c' at least appears to be the maximum velocity. The picture goes something like this. Suppose the pilot of spaceship B is seated in a chair that can slingshot him forward inside his ship. A observes the length of B shortening such that, when B accelerates to a relative speed of c the observed length is zero (B no longer appears as a 3D object, but as a 2D object).

Suppose that ship B is now traveling at a speed of c. Even if the pilot of B activates his slingshot chair, A will not observe an increase in the pilot's speed. Since A observes B as a 2D object, activating the slingshot chair doesn't appear to move him at all. He would travel a length of zero.

Hmm. But the picture does have a problem. Switch to the inertial frame of B. He also observes a relative speed of c, and A appears to him to be 2D. But, he observes himself as being 3D. So, if he activates his sling shot chair, the relative speed between himself and A will be the relative speed of the ships plus the relative speed of his chair to ship B. So, doesn't he observe a relative speed larger than c?

I must be wrong with that picture somehow. So how is it that the pilot in B will activate his slingshot chair, and yet still observe only a relative velocity of c between himself and A?

Before moving on to the actual paradox I posed in highschool, we need to resolve this issue first. An answer to this issue might resolve my highschool paradox as well.

 

[[serious]]

But they don't observe themselves contracting. As far as their perception of their own length goes, nothing has changed. Correct? That will be an important point to what follows.

That same issue is important to the Twin Paradox. There has always been the problem that relativity eliminates the fixed reference frame, so how does one know which twin will age more?

The proposed solution is that if both start in the same inertial frame, it is the one who accelerates into a different inertial frame (and then decelerates back) who will age less. And, IIRC this was verified experimentally with clocks.

Yes, the acceleration changes the reference frame. That provides the difference between the two

Despite that similarity, I believe my (apparent) paradox differs from the Twin Paradox ... as well as the Ladder Paradox, etc. But, if it does turn out that this is just a different version of one of those, this will be easy.

So, let's do a little back-and-forth to work into this.

First, there is a possible 'picture' of why 'c' at least appears to be the maximum velocity. The picture goes something like this. Suppose the pilot of spaceship B is seated in a chair that can slingshot him forward inside his ship. A observes the length of B shortening such that, when B accelerates to a relative speed of c the observed length is zero (B no longer appears as a 3D object, but as a 2D object).

Let's not do that as it's impossible. Objects with mass cannot reach the speed of light. Let's talk about specific measurements.

Suppose that ship B is now traveling at a speed of c. Even if the pilot of B activates his slingshot chair, A will not observe an increase in the pilot's speed. Since A observes B as a 2D object, activating the slingshot chair doesn't appear to move him at all. He would travel a length of zero.

Hmm. But the picture does have a problem. Switch to the inertial frame of B. He also observes a relative speed of c, and A appears to him to be 2D. But, he observes himself as being 3D. So, if he activates his sling shot chair, the relative speed between himself and A will be the relative speed of the ships plus the relative speed of his chair to ship B. So, doesn't he observe a relative speed larger than c?

I must be wrong with that picture somehow. So how is it that the pilot in B will activate his slingshot chair, and yet still observe only a relative velocity of c between himself and A?

Before moving on to the actual paradox I posed in highschool, we need to resolve this issue first. An answer to this issue might resolve my highschool paradox as well.

Keep in mind that you don't just have length changes, you also have time dilation.

Let's walk through this. We have person A, person B, and ship C. Person A is moving at .995c towards person B. This would make person B appear to be 1/10 as thick to person A and vice versa. Now, Person A activates his slingshot chair that accelerates him to .995c relative to the ship. now Person A observes the ship as 1/10 the length and person B would appear to be 1/200 as thick and moving at 0.999987c. Person B sees person A moving at 0.999987c in a ship moving .995c. Now, person B observes the apparent relative speed between Person A and ship C as .995c PROVIDED he remembers to use the other ship's clock (which runs slower due to relativistic effects). So even though person B sees person A accelerate only .005c faster, he can still observe that this translates to .995 c according to the other ship.

To sum up, when you have a third observer trying to determine the relative speed of two objects, he has to remember to use a clock in the inertial reference frame he's speaking about.

 

[Resha Caner]

Let's walk through this. We have person A, person B, and ship C.

D'oh. You relabeled everything! Person A in ship A and person B in ship B would have saved me some mental gymnastics in rearranging the problem in my head.

To sum up, when you have a third observer trying to determine the relative speed of two objects, he has to remember to use a clock in the inertial reference frame he's speaking about.

Ah, yes. That's the piece I was missing. Got it. Now it makes sense.

Let's not do that as it's impossible. Objects with mass cannot reach the speed of light. Let's talk about specific measurements.

Why is it impossible? Impractical to the point that I don't believe it will ever happen, yes. But impossible? Hmm.

You started your example at 0.995c. Why not 0.999995c, or 0.9999999999999995c, or ... Where does it become impossible?

 

[[serious]]

You started your example at 0.995c. Why not 0.999995c, or 0.9999999999999995c, or ... Where does it become impossible?

At 1. You can do calculations for stuff for numbers close to 1c, but once you hit 1c it becomes undefined. I don't like undefined Lorentz transformations.

 

[Tinker Grey]

IIRC, in the denominator of one of those formulas is √(c[sup]2[/sup]-v[sup]2[/sup]). So when you reach the speed of light, you have a divide by zero problem. Hence as [serious] said, the result is undefined.

 

[AV1611VET]

IIRC, in the denominator of one of those formulas is √(c[sup]2[/sup]-v[sup]2[/sup]). So when you reach the speed of light, you have a divide by zero problem. Hence as [serious] said, the result is undefined.

Ya -- I think it's something like: √ 1-(c[sup]2[/sup]/v[sup]2[/sup]).

 

[Tinker Grey]

Ya -- I think it's something like: √ 1-(c[sup]2[/sup]/v[sup]2[/sup]).

Might've been. Thanks. Although, it most likely was v[sup]2[/sup]/c[sup]2[/sup] since otherwise a zero velocity would also result in a divide by zero error.

 

[Resha Caner]

At 1. You can do calculations for stuff for numbers close to 1c, but once you hit 1c it becomes undefined. I don't like undefined Lorentz transformations.

No, not undefined. Technically it's asymptotic, and you can work the math for such situations. That's why we're taught L'Hopital's rule, the convergence of an infinite series, etc. So, with the Lorentz transformation for length, it is asymptotic to zero as v -> c.

Though I will admit this. I was quicker to grasp your example via some finite calculations than I would have been in working through it in the limit.

 

[[serious]]

Ya -- I think it's something like: √ 1-(c[sup]2[/sup]/v[sup]2[/sup]).

This exactly.

Actually, I think I forgot the exponents in one of my calculations. I'm too lazy to redo them and check though.

 

[AV1611VET]

I'm too lazy to redo them and check though.

I know the feeling --

 

[Farinata]

Why is it impossible? Impractical to the point that I don't believe it will ever happen, yes. But impossible? Hmm.

You started your example at 0.995c. Why not 0.999995c, or 0.9999999999999995c, or ... Where does it become impossible?

It's a bit late but I hope this makes sense...

Any massive particle or ship/whatever always has a definable momentarily comoving reference frame where the object is at rest. You can construct a 4-velocity for that particle/ship where all 3 spatial 4-velocity components are zero with the remaining non-zero component being equal to the particle's rest mass (i.e. you can always find a frame where the massive particle/ship is at rest). This means that these particles will always move on time-like worldlines (speed < c). But photons move on null worldlines because the speed of light has to look the same in every reference frame (one of the postulates of special relativity is that there exists an invariant speed which is experimentally linked to the speed of light). Photons don't have a definable 4-velocity and don't have any rest mass. You cannot find a momentarily comoving reference frame of the photon because there is no frame where the photon is at rest.

Massive particles can't travel at the speed of light anymore than photons can travel less than the speed of light. To do otherwise would require introducing a preferred reference frame or breaking special relativity in some way. To say so in the context of SR would be like saying that the massive particle moving at the speed of light has no rest mass which doesn't make sense.

 

[[serious]]

It's a bit late but I hope this makes sense...

Any massive particle or ship/whatever always has a definable momentarily comoving reference frame where the object is at rest. You can construct a 4-velocity for that particle/ship where all 3 spatial 4-velocity components are zero with the remaining non-zero component being equal to the particle's rest mass (i.e. you can always find a frame where the massive particle/ship is at rest). This means that these particles will always move on time-like worldlines (speed < c). But photons move on null worldlines because the speed of light has to look the same in every reference frame (one of the postulates of special relativity is that there exists an invariant speed which is experimentally linked to the speed of light). Photons don't have a definable 4-velocity and don't have any rest mass. You cannot find a momentarily comoving reference frame of the photon because there is no frame where the photon is at rest.

Massive particles can't travel at the speed of light anymore than photons can travel less than the speed of light. To do otherwise would require introducing a preferred reference frame or breaking special relativity in some way.

That's a really great summary. I actually understand why things are as they are much better because of it. Thank you.