Simple SR

[vuquta]

Special Relativity.

Assume z=0.

 

Two frames are in relative motion and when their origins are co-located, a light pulse is emitted from their origins.

 

Stationary frame, observer O located is at (vr/c, r/γ, 0).

Moving frame, observer O'1 located is at (0, r/γ, 0) and observer O'2 located is at ( r/γ, 0, 0).

 

When light travels a distance r in O, observers O and O'1 are co-located.

 

Proof

1) Set x = vr/c and t = r/c in the stationary frame since this is the O observer. LT calculates x'=0 and t'= r/(γc).

Hence, ct' = r/γ = √( x'² + y² + z²) = √( 0² + y² + 0²) = y.

So, the coordinate in the O' frame is (0, r/γ, 0).

 

2) Set x' = 0 and t' = r/(γc) in the moving frame since this is the O'1 observer. LT calculates x = vr/c and t = γ/c.

Thus, (ct)² = r² = x² + y² + z² = (vr/c)² + y² + 0² = (vr/c)² + y². So, (vr/c)² + y² = r² and y = r/γ.

So, the coordinate in the O frame is (vr/c, r/γ, 0).

 

Thus, the O and O'1 observers are co-located when light acquires a distance r in the O frame.

 

Next

 

By applying the light postulate in O, light is no more than the coordinate (r,0,0).

 

By applying the light postulate in O', if light is at O'1, then it must also be at the O'2 observer since they are equidistant to the light emission point in O'.

 

Calculating O'2 using LT, x'=r/γ and t' = r/(γc), x = r( 1 + v/c ) and light is therefore located at (r( 1 + v/c ), 0, 0).

 

Thus, two co-located observers conclude two contradictory light positions even though they are located at the same place, light is located at (r,0,0) and also at (r( 1 + v/c ), 0, 0).

 

Contradiction

Edited June 9, 2010 by vuquta

[swansont]

Contradiction

 

Which is your own doing, by putting inconsistent constraints on the problem. A valid contradiction in a thought experiment is impossible; it's just math. But you've been told this before.

 

How is this exercise any different from the flawed examples you've presented before? You've shown you can't validly apply relativity in a 1-D problem, and you're probably making the same exact mistakes in this 2-D problem.

 

I'm not going to go looking for them.

 

edit: though I do wonder how the observer can be at both r/γ and r; the y-direction sees no length contraction. If I'm understanding your setup correctly, ct' = r/γ is not generally true. A circle in the rest frame will not be a circle in the moving frame (and doesn't that sound familiar)

Edited June 10, 2010 by swansont

[vuquta]

Which is your own doing, by putting inconsistent constraints on the problem. A valid contradiction in a thought experiment is impossible; it's just math. But you've been told this before.

 

How is this exercise any different from the flawed examples you've presented before? You've shown you can't validly apply relativity in a 1-D problem, and you're probably making the same exact mistakes in this 2-D problem.

 

I'm not going to go looking for them.

 

edit: though I do wonder how the observer can be at both r/γ and r; the y-direction sees no length contraction. If I'm understanding your setup correctly, ct' = r/γ is not generally true. A circle in the rest frame will not be a circle in the moving frame (and doesn't that sound familiar)

 

 

1) I hope you can see the two observers in the moving frame are equidistant to the origin in the moving frame. There is nothing illegal about this.

2) There is another observer in the stationary frame. There is nothing illegal about that.

3) When the two origins are co-located, a light pulse is emitted. This is standard SR.

4) Based on the above no opionion are used. Strictly LT, the light postulate and the relativity postulate are used for the conclusions.

 

 

 

edit: though I do wonder how the observer can be at both r/γ and r; the y-direction sees no length contraction. If I'm understanding your setup correctly, ct' = r/γ is not generally true. A circle in the rest frame will not be a circle in the moving frame (and doesn't that sound familiar)

 

An observer cannot be at both places and I did not say they could. But two different observers in relative motion can co-locate.

 

The two observers that co-locate are simply based on the Pythagorean theorem derivation of time dilation. If that is true, then my example is true.

 

So, I am on firm footing there. Indeed, the two observers O and O'1 are co-located when light reaches either.

 

ct' = r/γ is not generally true.

This is true for all points in the moving frame that are a distance r/γ from the origin or the light postulate is false in the moving frame.

 

This example requires 2-D.

[swansont]

You are calculating what O'2 sees, but this observer is not co-located with the observer O.

[vuquta]

You are calculating what O'2 sees, but this observer is not co-located with the observer O.

 

Agreed.

But, based on the co-location of O and O'1, O'1 and O'2 are equidistant to the light emission point in the moving frame at all times.

 

Hence, by the light postulate, if O'1 is struck by the light then O'2 is struck by the light since they are equidistant to the light emission point in the frame.

 

So, the light postulate forces light to be at O'2.

 

This in fact is what O'1 sees at co-location.

 

To confirm this, you may plug into LT that location in the stationary system of coords which is t = r/c( 1 + v/c) and x = r( 1 + v/c ) and

 

x'= r/γ and t' = r/(cγ) just as predicted. Hence, this location O'2 is a place of simultaneity with O'1.

 

So, LT has no disagreement here.

[swansont]

You are erring, once again, by requiring absolute simultaneity. O will not see the light simultaneously strike O'1 and O'2; that's an observation that will only be made in O'. Simultaneity is relative. It is not a contradiction to confirm this.

 

AFAICT, this is the same mistake you have made in basically every thread you have started on relativity.

[vuquta]

You are erring, once again, by requiring absolute simultaneity. O will not see the light simultaneously strike O'1 and O'2; that's an observation that will only be made in O'. Simultaneity is relative. It is not a contradiction to confirm this.

 

AFAICT, this is the same mistake you have made in basically every thread you have started on relativity.

 

Nope, this is not correct.

 

I never said O would claim they are simultaneous. O does not.

I said O'1 and O'2 would claim simultaneity and I proved that with LT.

 

1) Do you agree O and O'1 are co-located when light strikes them.

2) Do you agree O'1 and O'2 must see the light as simultaneous since LT calculates that and since they are equidistant to the origin, this is a simple application of the light postulate. Light cannot be at O'1 and not at O'2.

[swansont]

Light cannot be at O'1 and not at O'2.

 

As observed in frame O', that's true. But only in O'. If you state it for any other frame, it is the same as saying the strikes will be seen as simultaneous. As soon as you invoke the frame O in your calculation, that's what you are doing.

[vuquta]

As observed in frame O', that's true. But only in O'. If you state it for any other frame, it is the same as saying the strikes will be seen as simultaneous. As soon as you invoke the frame O in your calculation, that's what you are doing.

 

Yes, the simultaneity is in the moving frame.

 

O'1 and O'2 see the light as simultaneous and O will not agree. This was proven with LT and is consistent with R of S.

 

Now, I did not invoke anything about O concerning the O' frame except that the light will be at O'2 when O and O'1 are co-located.

 

Do you agree or disagree.

[swansont]

Yes, the simultaneity is in the moving frame.

 

O'1 and O'2 see the light as simultaneous and O will not agree. This was proven with LT and is consistent with R of S.

 

Now, I did not invoke anything about O concerning the O' frame except that the light will be at O'2 when O and O'1 are co-located.

 

Do you agree or disagree.

 

I disagree. You cannot state that a priori, and your calculation shows that it is not true. O does not see that, and there is no expectation that it should hold. O does not think that O'1 and O'2 are equidistant from the origin.

 

"the light will be at O'2 when O and O'1 are co-located" is an observer-dependent statement. It is only true in one frame. You cannot state these things without mentioning which observer is involved; if you fail to do so, the implication is that it is true for all observers.

[vuquta]

I disagree. You cannot state that a priori, and your calculation shows that it is not true. O does not see that, and there is no expectation that it should hold. O does not think that O'1 and O'2 are equidistant from the origin.

I am not considering O's views so this does not apply.

 

"the light will be at O'2 when O and O'1 are co-located" is an observer-dependent statement. It is only true in one frame. You cannot state these things without mentioning which observer is involved; if you fail to do so, the implication is that it is true for all observers.

 

 

OK, I only meant from the view of O'1.

 

When O and O'1 are co-located, O'1 concludes light is located at O'2.

 

How about this statement above, do you agree?

 

 

But, we need to be careful about observer views when it comes to light.

 

 

Given an event E, the light cone classifies all events in spacetime into 5 distinct categories:

 

Events on the future light cone of E.

Events on the past light cone of E.

Events inside the future light cone of E are those affected by a material particle emitted at E.

Events inside the past light cone of E are those that can emit a material particle and affect what is happening at E.

All other events are in the (absolute) elsewhere of E and are those that cannot affect or be affected by E.

 

"The above classifications hold true in any frame of reference; that is, an event judged to be in the light cone by one observer, will also be judged to be in the same light cone by all other observers, no matter their frame of reference. This is why the concept is so powerful."

 

http://en.wikipedia.org/wiki/Light_cone

[swansont]

I am not considering O's views so this does not apply.

 

You used a Lorentz transform, which means you moved from one reference frame to another, so you are doing that. If you only consider a single frame, there is no need to do this. You just have light travel and clock synchronization.

[vuquta]

You used a Lorentz transform, which means you moved from one reference frame to another, so you are doing that. If you only consider a single frame, there is no need to do this. You just have light travel and clock synchronization.

 

OK, that is simple.

 

The location of O'2 is vr/c + r/γ² because of length contraction when O and O'1 are co-located in the frame of O.

 

This is because the elapsed time in O is r/c. The origin of O' is located at vt = vr/c, plus the length contracted distance of r/γ².

 

Therefore when O and O'1 are co-located, O'2 is at the coordinate (vr/c + r/γ²,0,0) in the coordinates of O.

 

Since the postulates of SR applies to all frames, then O cannot refute the conclusion of O'1 that light is at O'2 or the relativity postulate and/or the light postulate is false for O'.

 

Yet, light is only at (r,0,0) based on the truth of the light postulate in O.

 

So, light is at (r,0,0) based on the light postulate for O and light is at (vr/c + r/γ²,0,0) based on the conjunction of the two light postulate in O'.

 

Both cannot be true.

[swansont]

I appears you are requiring O and O'1 to be both co-located and have the light pulse arrive when they do. Is that true in both frames of reference?

[vuquta]

It appears you are requiring O and O'1 to be both co-located and have the light pulse arrive when they do. Is that true in both frames of reference?

 

Yes it is true in both frames, that was my intent. Here was part of my original post.

 

1) Set x = vr/c and t = r/c in the stationary frame since this is the O observer. LT calculates x'=0 and t'= r/(γc).

Hence, ct' = r/γ = √( x'² + y² + z²) = √( 0² + y² + 0²) = y.

So, the coordinate in the O' frame is (0, r/γ, 0).

 

2) Set x' = 0 and t' = r/(γc) in the moving frame since this is the O'1 observer. LT calculates x = vr/c and t = γ/c.

Thus, (ct)² = r² = x² + y² + z² = (vr/c)² + y² + 0² = (vr/c)² + y². So, (vr/c)² + y² = r² and y = r/γ.

So, the coordinate in the O frame is (vr/c, r/γ, 0).

 

Thus, the O and O'1 observers are co-located when light acquires a distance r in the O frame.

[swansont]

Therefore when O and O'1 are co-located, O'2 is at the coordinate (vr/c + r/γ²,0,0) in the coordinates of O.

 

Since the postulates of SR applies to all frames, then O cannot refute the conclusion of O'1 that light is at O'2 or the relativity postulate and/or the light postulate is false for O'.

 

O cannot refute O'1's conclusion since it applies to the O' frame (which is the basic concept of relativity), but the conclusion of O is that the light strikes at O'1 and O'2 are not simultaneous.

[vuquta]

O cannot refute O'1's conclusion since it applies to the O' frame (which is the basic concept of relativity), but the conclusion of O is that the light strikes at O'1 and O'2 are not simultaneous.

 

1) but the conclusion of O is that the light strikes at O'1 and O'2 are not simultaneous

Agreed. I depend on this.

 

2. O cannot refute O'1's conclusion

 

Since O cannot refute the conclusion of O'1, then where is O'2 when O and O'1 are co-located? It is located at the position (vr/c + r/γ²,0,0) in the stationary system of coords.

 

So, when O and O'1 are at the same place, light is at (r,0,0) according to O and light is at (vr/c + r/γ²,0,0) according to O'1. Neither cionclusion can be refuted and the conclusion of O cannot carry more weight than the conclusion of O'1.

 

Hence, when two observers are at the same place, one light sphere is located at two different places.

[swansont]

Hence, when two observers are at the same place, one light sphere is located at two different places.

 

Two different places, but also two different times. A light sphere is not "located" as if there is a universal set of coordinates. It applies only to the rest frame of an observer.

 

Light hits O'2 at different times and places, depending on the observer. According to O, light hits O'2 after O'1 has passed by, but that's understandable because O'2 is moving away from the beam.

[vuquta]

Two different places, but also two different times. A light sphere is not "located" as if there is a universal set of coordinates. It applies only to the rest frame of an observer.

 

Light hits O'2 at different times and places, depending on the observer. According to O, light hits O'2 after O'1 has passed by, but that's understandable because O'2 is moving away from the beam.

 

This is all correct but only if we demarcate the two frames in logic and ignore some factual results.

 

When O'1 and O are at the same place, light is at two different places in one coordinate system.

 

We are not talking dynamics here as required by SR. This is a static logic.

 

When the O and O' are at the same place, each conclude light is at a different position in the same coordinate system of O. Clocks have no impact on the location of light. A clock is just a bystander and yes, each will have a different time on their respective clocks.

 

It is the position of light that is at issue.

 

One place of observation cannot produce two different places for one light sphere in one coordinate system.

 

 

 

The next problem comes with the location of O'2 in the coordinates of O.

I am sure you agree it is located at ( vr/c + r/γ² ) in the coordinates of O when O and O'1 are co-located.

 

If you calculate x' with x = ( vr/c + r/γ² ) and t = r/c, you get x' = r/γ as expected, so that is consistent.

 

But, with t' you get t' = r/(cγ)( 1 - v/c ) with x = ( vr/c + r/γ² ) and t = r/c.

 

However, the time on the clock of O'1 and all O' observers is r/(cγ) when O and O'1 are co-located.

 

Therefore, LT contends the place where O'2 is located was struck by the light prior to the co-location of O and O'2 even though O'1 and O'2 are equidistant to the origin/light emission point. So a clock argument under the rules of SR fails in this situation as well.

[swansont]

When the O and O' are at the same place, each conclude light is at a different position in the same coordinate system of O. Clocks have no impact on the location of light. A clock is just a bystander and yes, each will have a different time on their respective clocks.

 

It is the position of light that is at issue.

 

One place of observation cannot produce two different places for one light sphere in one coordinate system.

 

It's different observers. You can't just state that this is impossible. The two observers will not agree when and where O'2 is when light hits that point. Arguing against this has no basis in physics.

 

You do not have one light sphere in one coordinate system. You have two coordinate systems, and you are imposing an artificial constraint.

[vuquta]

It's different observers. You can't just state that this is impossible. The two observers will not agree when and where O'2 is when light hits that point. Arguing against this has no basis in physics.

 

You do not have one light sphere in one coordinate system. You have two coordinate systems, and you are imposing an artificial constraint.

 

You do not have one light sphere in one coordinate system. You have two coordinate systems, and you are imposing an artificial constraint

 

The coordinates from the moving frame were mapped to the stationary system of coordinates. You cannot claim LT is true and then say LT cannot be used to map coords from the moving frame to the stationary frame.

Everything was mapped into the stationary system of coordinates so that one space is used.

 

The co-location O and O'1 is used as a mechanism. This is not artifical since it is all within the domain of logic of SR. So, I am not imposing anything.

 

I merely picked some points at certain places to make sure SR worked correctly.

 

Next, the two observers are at the same place. You can argue they may disagree about time, you can argue they may disagree about measurements. But, you cannot argue they can disagree about the location of one light sphere both using the stationary coordinates. The light event is O'2 being struck by the light.

 

 

The above classifications hold true in any frame of reference; that is, an event judged to be in the light cone by one observer, will also be judged to be in the same light cone by all other observers, no matter their frame of reference. This is why the concept is so powerful.

http://en.wikipedia.org/wiki/Light_cone

[swansont]

The coordinates from the moving frame were mapped to the stationary system of coordinates. You cannot claim LT is true and then say LT cannot be used to map coords from the moving frame to the stationary frame.

Everything was mapped into the stationary system of coordinates so that one space is used.

 

The co-location O and O'1 is used as a mechanism. This is not artifical since it is all within the domain of logic of SR. So, I am not imposing anything.

 

I merely picked some points at certain places to make sure SR worked correctly.

 

It's not the mapping that is the problem, it is the added requirement that events be simultaneous in two frames. What you have shown is that in the O frame, light does not reach O'2 at the same time it hits O'1. O observes it at one place and O'1 observes it somewhere else.

 

Next, the two observers are at the same place. You can argue they may disagree about time, you can argue they may disagree about measurements. But, you cannot argue they can disagree about the location of one light sphere both using the stationary coordinates. The light event is O'2 being struck by the light.

 

You cannot argue about whether the light hits O'2. But you can argue about when the light hits any given point. The circle that describes O' is not a circle when you map it into O. Therefore, according to O, light does not hit all points on that curve simultaneously. You have no basis to demand that it should.

 

The above classifications hold true in any frame of reference; that is, an event judged to be in the light cone by one observer, will also be judged to be in the same light cone by all other observers, no matter their frame of reference. This is why the concept is so powerful.

http://en.wikipedia.org/wiki/Light_cone

 

There is nothing in your argument that questions whether the events are in/on the light cone. The question is where in/on the cone they are located. Since it's light, the events would be on the cone, but located at different points in time.

[vuquta]

It's not the mapping that is the problem, it is the added requirement that events be simultaneous in two frames. What you have shown is that in the O frame, light does not reach O'2 at the same time it hits O'1. O observes it at one place and O'1 observes it somewhere else.

 

Oh, I do not require them to be simultaneous. I do not care.

 

But, you are not complete in your statement. Two observers at the same place required the light sphere to be in two different places. How is this logical?

 

 

 

 

You cannot argue about whether the light hits O'2. But you can argue about when the light hits any given point. The circle that describes O' is not a circle when you map it into O. Therefore, according to O, light does not hit all points on that curve simultaneously. You have no basis to demand that it should.

 

OK, if I cannot argue about whether the light hits O'2 when light hits O'1, then which do you refute the relativity postulate or the light postulate in the primed frame?

 

 

There is nothing in your argument that questions whether the events are in/on the light cone. The question is where in/on the cone they are located. Since it's light, the events would be on the cone, but located at different points in time.

 

No, you missed the point of the post.

 

No two observes can disagree on whether a position/event is in the light cone. Hey, when observers are at the same place, that is the highest standard.

---

Merged post follows:

Consecutive posts merged

 

 

There is nothing in your argument that questions whether the events are in/on the light cone. The question is where in/on the cone they are located. Since it's light, the events would be on the cone, but located at different points in time.

 

This is false.

 

You cannot argue the two frames agree on the light cone because by design because if you do then you agree with absolute simultanety.

 

So, can you show they agree on the light cone and disagree on the position of the light sphere when O and O'1 are co-located?

 

Can I see the math?

[swansont]

Oh, I do not require them to be simultaneous. I do not care.

 

But, you are not complete in your statement. Two observers at the same place required the light sphere to be in two different places. How is this logical?

 

Depends on your definition of logical. If logical is "what conforms with Lorenz transforms" then it's completely logical. Saying that the light sphere has to be in two places points toward a logic that there is some absolute frame — that light is in a certain place, regardless of frame, i.e. that point is absolute — and that logic fails. One observer says it's in one place, while another observer says it's in a different place. Their observations are relative to their own frame.

 

 

OK, if I cannot argue about whether the light hits O'2 when light hits O'1, then which do you refute the relativity postulate or the light postulate in the primed frame?

 

I refute neither. You are misapplying them.

 

 

No, you missed the point of the post.

 

No two observes can disagree on whether a position/event is in the light cone. Hey, when observers are at the same place, that is the highest standard.

 

No, I think you misunderstand. The observers are not disagreeing on whether the event is in O's light cone. They disagree when the event takes place. The light cone for O and for O' are not identical.

 

 

This is false.

 

You cannot argue the two frames agree on the light cone because by design because if you do then you agree with absolute simultanety.

 

I didn't say they agree on the light cone, I said they agree on whether an event is in or on or outside of the light cone.

 

So, can you show they agree on the light cone and disagree on the position of the light sphere when O and O'1 are co-located?

 

Again, that's not what I claimed.

 

Can I see the math?

 

I don't know. Can you do a web search? It is neither my function nor my desire to teach you relativity. There are plenty of resources out there.

 

Besides, the light cone discussion is just a distraction from the basic error of your scenario. Observations are relative to the frame of reference of the observer. Light does not create some absolute frame.

[vuquta]

Depends on your definition of logical. If logical is "what conforms with Lorenz transforms" then it's completely logical. Saying that the light sphere has to be in two places points toward a logic that there is some absolute frame — that light is in a certain place, regardless of frame, i.e. that point is absolute — and that logic fails. One observer says it's in one place, while another observer says it's in a different place. Their observations are relative to their own frame.

 

Well, LT claims light is located at ( r(1+v/c,0,0) in the O frame when O and O'1 are co-located.

 

This is because O'1 and O'2 are simultaneous and we have been through the math.

 

So, if LT is logical then light is located at at ( r(1+v/c,0,0) when O and O'1 are co-located.

 

But, based on the light postulate, light is only located (r,0,0) in O.

 

How do you resolve this?

 

 

 

 

 

I refute neither. You are misapplying them.

How?

 

 

 

No, I think you misunderstand. The observers are not disagreeing on whether the event is in O's light cone. They disagree when the event takes place. The light cone for O and for O' are not identical.

 

This is wrong. The light cone is identical. The space-time coords are not.

 

 

 

I didn't say they agree on the light cone, I said they agree on whether an event is in or on or outside of the light cone.

 

But, we have been through this.

O'1 and O disagree on the light cone in terms of O'2. Can you show how they agree when they are co-located?

 

 

Besides, the light cone discussion is just a distraction from the basic error of your scenario. Observations are relative to the frame of reference of the observer. Light does not create some absolute frame.

 

No, this does not work. Observations are relative until two observers are at the same place.

 

If they are at the same place and disagree on the light sphere, then there must be more than one.

 

Are you claiming the light sphere is relative?