Causality and the Lorentz transformation

[Joao]

Hi everyone! Sorry for my bad English!

Please, suppose you have a subject A that opens his arms like a "T", in each hand he holds a laser and shoots the light at the same time. There are 2 targets at the same distance and, to A, the light hits both targets simultaneously. I Know that in some referencials light would hit one or other targets first, but:

Is there any possibility that, in some referential, the light hits the target BEFORE the subject shoots the laser? Thanks! =)

 

[PeroK]

Hi everyone! Sorry for my bad English!

Please, suppose you have a subject A that opens his arms like a "T", in each hand he holds a laser and shoots the light at the same time. There are 2 targets at the same distance and, to A, the light hits both targets simultaneously. I Know that in some referencials light would hit one or other targets first, but:

Is there any possibility that, in some referential, the light hits the target BEFORE the subject shoots the laser? Thanks! =)

No. In all reference frames the light travels at $c$.

 

[Joao]

No. In all reference frames the light travels at $c$.

Thanks a lot! And if instead of shooting lasers, he was shooting bullets? Than it would be possible that, in some referential, the bullet hits the target BEFORE the shooter shoots?

 

[Ibix]

shoots the light at the same time.

It doesn't matter here, but it is a good habit when learning relativity to say which frame regards the events are simultaneous. They don't all agree.

Is there any possibility that, in some referential, the light hits the target BEFORE the subject shoots the laser? Thanks! =)

Under some circumstances it's possible to find a frame where one laser hits its target before the other one fires. But there is no frame where a laser hits its target before it has fired.

Thanks a lot! And if instead of shooting lasers, he was shooting bullets? Than it would be possible that, in some referential, the bullet hits the target BEFORE the shooter shoots?

Same answer.

 

[PeroK]

Thanks a lot! And if instead of shooting lasers, he was shooting bullets? Than it would be possible that, in some referential, the bullet hits the target BEFORE the shooter shoots?

No. The bullets travel less than the speed of light. The basic concept of a particle trajectory is unchanged by relativity. All observers will agree that a particle traveled from point A to point B.

 

[Nugatory]

All observers will agree that a particle traveled from point A to point B.

... and that the particle left A before it arrived at B, and that a flash of light leaving A along with the particle will reach B before the particle does.
(PeroK already knows this, of course)

 

[Joao]

Thanks a lot! You are awesome! =)

One last scenario, just to see if I got it right...

And if there were some magic referential that travel faster then the speed of light, than, for he, would it be possible mathematically that the bullet hits the target before it was fired?

I mean, is the speed of light barrier what garanties causality?

Thanks again! =)

 

[PeroK]

Thanks a lot! You are awesome! =)

One last scenario, just to see if I got it right...

And if there were some magic referential that travel faster then the speed of light, than, for he, would it be possible mathematically that the bullet hits the target before it was fired?

I mean, is the speed of light barrier what garanties causality?

Thanks again! =)

There is no way within the theory of relativity to assign a reference frame to an object traveling faster than light (or, indeed, at the speed of light).

 

[DrStupid]

And if there were some magic referential that travel faster then the speed of light, than, for he, would it be possible mathematically that the bullet hits the target before it was fired?

As PeroK already mentioned there is no such frame of reference. But you can replace the laser beam with a chase light and let the patter running faster than c. In that case the direction of the "propagation" becomes frame-dependent.

 

[pixel]

I mean, is the speed of light barrier what garanties causality?

It can be shown using the Lorentz Transformation that if a signal traveled faster than light, a reference frame can be found in which the signal is received before it is sent, thus violating causality.

 

[Joao]

Thanks a lot everyone! Really really Thanks! All my doubts were clarified! Now I have a depper understanding of the world we live in! Now I will be thinking about what the faster then light communication and quantum eraser phenomena means about the structure of the universe... I guess time flow and causality aren't the most fundamental things after all! Hehehehehe! So, thanks again! =)

 

[Ibix]

Thanks! All my doubts were clarified! Now I have a depper understanding of the world we live in! Now I will be thinking about what the faster then light communication and quantum eraser phenomena means about the structure of the universe... I

A word of warning: if you haven't learned the mathematical structure underlying relativity and quantum theory then you do not understand them, whatever you may think. That means any thinking you do about them is like criticising Shakespeare's play "Macbeth" based on nithing but the summary "Macbeth plots to become king, then there's some fighting".

By all means do read non-mathematical treatments of subjects that interest you. It'sa great way to notivate yourself. But if you want to understand them enough to think about what they mean, you need the maths. Taylor and Wheeler's Spacetime Physics is a good source for relativity.

 

[PeroK]

It can be shown using the Lorentz Transformation that if a signal traveled faster than light, a reference frame can be found in which the signal is received before it is sent, thus violating causality.

You can use the Lorentz Transformation to show this. But, if you analyse the problem another way, you get a different answer. I would say that this is more than a causality violation, it is outside the scope of the SR equations. For example:

Consider a spaceship moving away from Earth at $c/2$. The spaceship sends a message back to Earth at a velocity of $-2c$ in the ship's frame. The velocity of this signal in the Earth's frame would have to be:

$\frac{-2c + \frac{c}{2}}{1 - 1} = \frac{-3c/2}{0}$

Hence undefined. Any further analysis using SR and the Lorentz Transformation becomes absurd.

In fact, if you have the message travel at $-4c$, then in the Earth's frame this becomes $+\frac{9c}{2}$. I.e. the message is traveling away from the spaceship and away from the Earth.

This, in part, explains the causality problem. If the message starts from the ship and is sent away from the Earth, then the message never reaches the Earth. However, a kinematic solution in the Earth's frame is, of course, that the message started from the Earth at an earlier time, reached the ship at a later time and continued away from the Earth.

Suffice it to say that if the ship and the Earth cannot agree on the direction that the message is travelling, then all bets are off!

 

[DrStupid]

Consider a spaceship moving away from Earth at $c/2$. The spaceship sends a message back to Earth at a velocity of $-2c$ in the ship's frame. The velocity of this signal in the Earth's frame would have to be:

$\frac{-2c + \frac{c}{2}}{1 - 1} = \frac{-3c/2}{0}$

Hence undefined. Any further analysis using SR and the Lorentz Transformation becomes absurd.

The result is correct but your conclusion is wrong. It is no problem to transform the position of the message from the ship frame into the Earth frame. If you do that you will see why you get the strange result for the velocity. It is a special case of the relativity of simultaneity.

 

[georgir]

A pair of events can be objectively (independently of reference frame selection) categorized to have one of the following relationship:
- space-like separated - they are simultaneous in some reference frame - their order can be either option in other frames, but they always require FTL travel to connect;
- time-like separated - they occur at the same location in some reference frame - their order is always the same in all other reference frames, their locations differ but are always close enough to connect with sub-light travel;
- light-like separated - the border case, they are on each other's light-cone - their order is always the same in all reference frames, they always require exactly light-speed travel to connect; you can find a reference frame to get them arbitrarily close to simultaneous and co-located, but never completely so.

As to your second post and question, generally yes - arbitrary FTL travel would be identical with "time travel" or causality violation. But there are possible models with FTL restricted in some ways that still preserve causality.

 

[Joao]

Thanks a lot everyone! Really! =)