Help resolving absolute motion paradox please.

[rede96]

Hi, I’ve been teaching myself SR, currently reading Relativity for the Questioning Mind by Daniel F Styer.

My math is not great, so I have been avoiding the heavy calculations for now. I will learn those too, but I wanted to understand the concepts first.

For the most part I think I have got a good grasp on the basic concepts but then every now and then something comes up that stumps me.

Like this thought experiment below.

Looking at it from your FoR, imagine 3 rockets in a vertical line (A,B and C), separated by some distance. The rockets are at rest wrt each other and are traveling from left to right with some small velocity. (To approximate Euclidian space for now.)

You see another rocket ship (D) traveling at an angle to the first 3 ships, which creates 3 events as shown below.

Event 1, D passes in front of A
http://img51.imageshack.us/img51/2075/56169348.jpg

Event 2, D passes in front of B
http://img821.imageshack.us/img821/2290/38031069.jpg

Event 3, D passes a distance in front of C as it accelerated between events 2 and 3.
http://img713.imageshack.us/img713/4276/84602899.jpg

The problem I am having is when I am looking at this from A’s FoR. D looks to pass A and B in straight line, but then suddenly veers off to the left and does not pass C in straight line path.

A must think that D has turned off its original trajectory, as A, B and C have not accelerated so he knows that they are still at rest wrt each other.

So he radios D and asks what happened.

D says that he is following a laser and has not veered off his original trajectory, although he did accelerate after passing Ship B.

To cut to the chase, A would know that if D has not moved from his original trajectory, the only way that D could have appeared to move from its initial straight line path is if A,B,C and D were all in motion.

Or in other words, this scenario could not have happened if any FoR was 'at rest'

All that seems fairly simple, but I thought it was not possible for any frame to say that they have absolute motion.

So what am I missing please?

 

[Dale]

You are not missing anything. This is correct (except for the "at rest" bit which is irrelevant). The trajectory that something is heading on is frame-variant.

In general, because of the way that the Lorentz transform mixes up time and space you can immediately know that if an object's worldline is curved in time in one frame that it will also be curved in space in other frames.

 

[rede96]

You are not missing anything. This is correct (except for the "at rest" bit which is irrelevant). The trajectory that something is heading on is frame-variant.

In general, because of the way that the Lorentz transform mixes up time and space you can immediately know that if an object's worldline is curved in time in one frame that it will also be curved in space in other frames.

Thanks DaleSpam.

I've been watching the Standford Special Relativity lectures on YouTube, but I am way off doing Lorentz transforms for now. (They are next on my list to conquer when I get some time!)

Just so I understand this then, are you saying that it is perfectly ok for the ABC frame to deduce that their FoR must be in motion in order for the 3 events to happen?

 

[Dale]

No, I am saying that you cannot use any experiment to deduce that a frame is in motion any more than you can use an experiment to deduce that it is in rest. It is completely irrelevant.

The rocket D's path is curved in time in the frame of the laser source, so it will be curved in space in other frames, regardless of which frame is "in motion".

Note that the laser beam is not a worldline, it is a worldsheet. D's worldline is entirely contained inside the beam's worldsheet. In some frames, such as ABC's frame, the beam is moving, so the fact that D's trajectory curves does not imply that it ever leaves the beam nor does it imply that the beam itself is curved.

 

[rede96]

The rocket D's path is curved in time in the frame of the laser, so it will be curved in space in other frames, regardless of which frame is "in motion".

Ok, I think I get it now. Because D accelerates, D's path becomes curved in time in its frame. (Ignoring the laser for now.) Then when it is transformed, it becomes curved in space in the ABC frame.

So the captain of Ship A could deduce through the amount of curvature, D's acceleration.

Is that right?

So jumping ahead slightly, would it be right to say that due to the equivalence principle, as gravity causes the same curved time path in one frame, it shows as a curvature in space in all other frames? (I’ve not really studied any GR yet.)

 

[ZikZak]

There is very little inherently Einsteinean in your question. Exactly the same "Paradox" arises in Newtonian physics. Acceleration can manifest as change in speed (when it's parallel with velocity) or change in direction (when it's perpendicular) or some combination. Velocity is entirely relative, so you're free to choose a reference frame in which it's parallel to the acceleration, as you've done in the figures, or one in which it's perpendicular, or some other angle.

For instance, you might toss a tennis ball in the air and claim that its trajectory was one-dimensional and never veered from the path of a vertical laser beam. Someone riding by you in a car will disagree and claim that the trajectory was a curved path: a parabola, in fact, while still never departing the laser beam.

In each frame the laws of physics are the same, even if they disagree on the trajectory.

As far as SR goes, pilot D knows he is the one who accelerated because he measures the acceleration with an accelerometer. Pilot A measures no such acceleration and can consider himself inertial, or at rest. Both pilots deduce it must have been pilot D who did the accelerating and nothing can be deduced about any absolute motion.

 

[rede96]

There is very little inherently Einsteinean in your question. Exactly the same "Paradox" arises in Newtonian physics. Acceleration can manifest as change in speed (when it's parallel with velocity) or change in direction (when it's perpendicular) or some combination. Velocity is entirely relative, so you're free to choose a reference frame in which it's parallel to the acceleration, as you've done in the figures, or one in which it's perpendicular, or some other angle.

For instance, you might toss a tennis ball in the air and claim that its trajectory was one-dimensional and never veered from the path of a vertical laser beam. Someone riding by you in a car will disagree and claim that the trajectory was a curved path: a parabola, in fact, while still never departing the laser beam.

In each frame the laws of physics are the same, even if they disagree on the trajectory.

This bit confused me a little. I can understand how accelerating frames will see light bend, but I thought all inertial frames would see a laser as a 'straight line path'.

As far as SR goes, pilot D knows he is the one who accelerated because he measures the acceleration with an accelerometer. Pilot A measures no such acceleration and can consider himself inertial, or at rest. Both pilots deduce it must have been pilot D who did the accelerating and nothing can be deduced about any absolute motion.

That bit is now clear to me now. Although I will need to think a bit about the implications. For example if I watch a spaceship pass over me and head off into space on some arbitrary straight line path, if I see that spaceship deviate from that path it could be because it turned (perpendicular change) or because it accelerated (parallel change).

Observing something that deviates from a straight line trajectory because it 'speed up' is not something I had thought about before!

Thanks.

 

[Dale]

Ok, I think I get it now. Because D accelerates, D's path becomes curved in time in its frame. (Ignoring the laser for now.) Then when it is transformed, it becomes curved in space in the ABC frame.

So the captain of Ship A could deduce through the amount of curvature, D's acceleration.

Is that right?

Yes, that is essentially correct. However there are a few nuances that are not really important for a first-pass, but would become important later.

1) Above I spoke of the laser source's frame rather than D's frame. The reason is that the laser source is (presumably) inertial, whereas D is non-inertial.

2) Because D is non-inertial the very concept of D's frame is ambiguous. There is no standard way of building coordinates around a non-inertial observer. So you would have to define his frame in more detail.

3) Because D is non-inertial any coordinates that you would make would themselves be curved, meaning that you would have to talk about the straightness or curviness of the various worldline lines in terms of tensors rather than in terms of coordinates.

So jumping ahead slightly, would it be right to say that due to the equivalence principle, as gravity causes the same curved time path in one frame, it shows as a curvature in space in all other frames? (I’ve not really studied any GR yet.)

In this example we are talking about the curvature of a worldline, which is a mathematically and conceptually different thing from the curvature of spacetime. Gravity is modeled by a curvature in spacetime, and forces are modeled by the curvature of worldlines. In GR, the curvature of spacetime causes things such as when two initially parallel (at rest) straight worldlines (inertial objects) later converge and intersect.

 

[rede96]

Yes, that is essentially correct. However there are a few nuances that are not really important for a first-pass, but would become important later.

1) Above I spoke of the laser source's frame rather than D's frame. The reason is that the laser source is (presumably) inertial, whereas D is non-inertial.

2) Because D is non-inertial the very concept of D's frame is ambiguous. There is no standard way of building coordinates around a non-inertial observer. So you would have to define his frame in more detail.

3) Because D is non-inertial any coordinates that you would make would themselves be curved, meaning that you would have to talk about the straightness or curviness of the various worldline lines in terms of tensors rather than in terms of coordinates.

Ah ok. As I've said, I've got a lot to learn on the Math side. But this helps me to understand it better thanks.

In this example we are talking about the curvature of a worldline, which is a mathematically and conceptually different thing from the curvature of spacetime. Gravity is modeled by a curvature in spacetime, and forces are modeled by the curvature of worldlines. In GR, the curvature of spacetime causes things such as when two initially parallel (at rest) straight worldlines (inertial objects) later converge and intersect.

Yes I see, one is a curved path through space time, the other is curvature in space time itself.

I guess I just found it interesting that both acceleration and gravity could lead to me seeing the same curved trajectory of an object.

 

[ZikZak]

This bit confused me a little. I can understand how accelerating frames will see light bend, but I thought all inertial frames would see a laser as a 'straight line path'.

I didn't claim that inertial frames wouldn't see a laser as a straight line path, although that's debatable.* I did claim that a tennis ball could follow a parabolic trajectory while remaining in the laser path at all times.

*In both of our examples, the laser beam is a moving line. Is that a "straight line path?"

 

[rede96]

I didn't claim that inertial frames wouldn't see a laser as a straight line path, although that's debatable.*

*In both of our examples, the laser beam is a moving line. Is that a "straight line path?"

Again, sorry for being a bit slow. Do you mean in the same respect as a telegraph pole as viewed from a moving observer would be seen to be moving?

If so, I would have said the a straight line in one inertial frame must be viewed as a straight line in another inertial frame. Which I understood is different to the path an object may be viewed from one frame to another.

I did claim that a tennis ball could follow a parabolic trajectory while remaining in the laser path at all times.

Yes, I see that now. However that is a slightly different situation than the one I described. I guess what I was describing may have been like throwing a ball in a vertical direction, you passing by would see it moving at an angle. But if the ball suddenly accelerated in the line of its trajectory, you would see it move away from the angled path you were observing.

But what I am still struggling to understanding is this.

If I was shining a laser in the direction of the ball, when the ball accelerated, you must see the laser bend too. But if I was just shining a laser vertically and you passed by, you would not see it bend. So how does the ball accelerating affect the path of the laser


EDIT: I think I was too quick there. What you would be observing is the ball's path curve but the laser would still be straight, you would just see the ball at different points along its axis as the ball accelerated.