# Speeds greater than the speed of light

Hi Jake ( and welcome to PF )

I am using coordinate notation such that (x,t) is (distance,time)

Alice's frame (S)

Event A: Alice sends a superluminal signal at (0,0)
Event B: Bob receives the superluminal signal at (10,5)

The signal travels 10 lightseconds in 5 seconds (2c) in Alice's frame.

Bob's frame (S')

Event A' = (0,0)
Event B' = (x',t')

Using the Lorentz transformation

$$x^{\prime} = \gamma (x - vt)$$

$$t^{\prime} = \gamma \left( t - \frac{vx}{c^2} \right)$$

where $$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$

Assume Bob is moving at 0.8c relative to Alice and units such that c=1.

$$\gamma = \frac{1}{\sqrt{1 - \frac{0.8^2}{1^2}}} = 1.666$$

$$x^{\prime} = 1.666 ( 10 - 0.8*5) = 10$$

$$t^{\prime} = 1.666 \left( 5 - \frac{0.8*10}{1^2} \right)= -5$$

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Hi I apprecciated the clear cut presentation of the maths but couldnt help note a certain paradoxical self negation in your final derivation.
It appears to me that if the signal actually went back in time it couldn't possibly arrive at x=10 and conversely if it did arrive at x=10, it by definition was traveling forward in time no matter what the maths say.

There is the negative time interval: minus 5 seconds.

DrGreg
Gold Member
RandallB, re post #50

I'm trying to work out if there is any common ground between us, anything we can both agree on. That was the purpose of my post #48 and your dismissal of my entire post as "obviously flawed" doesn't help me in that objective. (By the way, what is "obvious" to you need not be obvious to anyone else, otherwise we wouldn't be having this discussion.) Let me spell this out in pedantic detail.

Here are some problems and solutions. I'd like to to consider each problem in isolation and tell me if you accept each solution, or, if not, why not. (All times are in seconds, all distance in light-seconds.)

Question 1: Relative to inertial observer Carol, H and K are the events

$$(T_H, X_H) = (0, 0)$$
$$(T_K, X_K) = (12, 48)$$​

Regardless of whether you think this is possible in the real Universe, if information were to travel from H to K, what would its speed be relative to Carol?

Answer 1: 4 (in the + direction).

Question 2: Relative to inertial observer David, L and M are the events

$$(T'_L, X'_L) = (-44, 64)$$
$$(T'_M, X'_M) = (-35, 28)$$​

Regardless of whether you think this is possible in the real Universe, if information were to travel from L to M, what would its speed be relative to David?

Answer 2: -4 (i.e. 4 in the - direction).

Question 3: Relative to inertial observer Elizabeth, P and Q are the events

$$(T''_P, X''_P) = (0, 0)$$
$$(T''_Q, X''_Q) = (-21, 0)$$​

Do you think all inertial observers agree that Q occurs before P? If not, give an example of an observer who disagrees.

Answer 3: Yes, all observers agree, because $T'''_Q = \gamma(T''_Q - vX''_Q) = -21\gamma < 0$, whatever the value of v and $\gamma$.

Question 4: Relative to inertial observer Alice, E, F and G are the events

$$(t_E, x_E) = (0, 0)$$
$$(t_F, x_F) = (12, 48)$$
$$(t_G, x_G) = (-21, 0)$$​

Bob moves at speed 4/5 relative to Alice (in the + direction) with clocks & distances synced such that in Bob's coordinates

$$(t'_E, x'_E) = (0, 0)$$​

What are the coordinates of F and G relative to Bob?

$$(t'_F, x'_F) = (-44, 64)$$
$$(t'_G, x'_G) = (-35, 28)$$​

I know we are going to disagree at a later stage but can you please restrict yourself to commenting on the above 4 independent problems and let me know if you disagree with any of the solutions?

I've asked these questions so we can avoid wasting time arguing over things we agree on and concentrate on where we disagree.

I'm a noob here and have a related question, or maybe it's the same one: Can the relative speed of two objects exceed the speed of light? I would assume so, for example the diametrically opposing objects on the edges of the observable universe that each move away from earth at the speed of light. Grateful for you expert advice.

There is a loophole, however...Lookup "Alcubierre Drive" on Wikipedia for more info.
Your loophole requires the existance of exotic matter.

HallsofIvy
Homework Helper
I'm a noob here and have a related question, or maybe it's the same one: Can the relative speed of two objects exceed the speed of light? I would assume so, for example the diametrically opposing objects on the edges of the observable universe that each move away from earth at the speed of light. Grateful for you expert advice.
No. In relativity speeds don't add like that. If object A is moving away from the earth with speed .99c relative to the earth and object B is moving in the opposite direction with speed .99c relative to the eath then the speed of each relative to the other would be
$$\frac{.99c+ .99c}{1+ \frac{(.99c)(.99c)}{c^2}}= \frac{1.98c}{1+.9801}$$
= .99994904975c still slightly less than c.

Assume there is a circular device on the moon. it consists of lazer activated propulsion plates that move a ball upon activation. now imagine this to be very big. then you shine a lazer from earth powerful enough to activate the plates. you then rotate it faster and faster until such time the beam that reaches the moon's surface is travelling faster than c across it's surface triggering the plates at greater than c speeds. would the ball travel faster than c??

A.T.
Assume there is a circular device on the moon. it consists of lazer activated propulsion plates that move a ball upon activation. now imagine this to be very big. then you shine a lazer from earth powerful enough to activate the plates. you then rotate it faster and faster until such time the beam that reaches the moon's surface is travelling faster than c across it's surface triggering the plates at greater than c speeds. would the ball travel faster than c??
No. You could just as well ask if the ball would travel at infinite speed, if you trigger the platforms simultaneously.

I think this thread is done.

In nature, no phenomena has been observed to actually propagate faster then EM waves. Special relativity/general relativity, and the post-parameterized Newtonian models model near light speed behaviors quite well. Until superluminal phenomena are observed, this is a debate worthy of a coffee break. Mind you, even the most cooky theorists would not try to cook up things that refute observable nature--that is for mathematicians to explore.

IF there is such a thing, nature will eventually show us. Until then, c is c.

No. You could just as well ask if the ball would travel at infinite speed, if you trigger the platforms simultaneously.

just out of curiosity, what speed would the ball achieve if such an experiment could be conducted?

And when approaching c, how much mass is gained as a ratio of itself of the actual ball?

Sorry to be a pain.

wow ..i have gone through a bag of popcorn reading this thread ..
first the org question concerns speed of light ..
it is my understanding that C is verabale .. ie in gravaty field water gas's diamond ect..
thus in "space" which is not empty it will change..
the vacuum speed ref.. should we not be considering the speed of a gamma partial ?
any way my real question of the moment is
a super lumen signal is energy of some kind [[which as we know is that e= mC thingy]]
\IF we assume that supper lumen singling is possible then is not time travel it self possible again due to that mater and energy are two sides of a sea saw?

an another question .. in my small mind as we approach C and mass increases would we not collapses to a micro black hole?

Alll right, I sent e-mails to the previous gent/lady "azzkika".

First, let's use what we know about the physical universe.
1) No signal can be sent at any rate exceeding that of light traversing a vacuum.
2) As you build "speed" (I hate that term), MASS is NOT generated.
3) Gravitational field curve space, they do not slow light trajectories.
4) Mass distributions (i.e. glass, dust, water) transparent to light slow light BECAUSE their electrons interact witht the light. Remember you lessons on polarization and complex susceptibility.

SO
1) Even if you could say get in front of an electromagnetic signal, what use is it since the you have not intercepted the signal. Causality still holds. Superluminal signals have nothing to do with "time travel" unless you are speaking of going forwards.

2) E=mc^2 does not mean that MASS increases when kinetic energy increases. The equation is fully written with a relativistic gamma multiplying the rest mass, m_0. It is the momentum that you are fighting to go faster. The change in momentum is what skyrockets. This is why it takes so much energy to go from 0.99c to 0.999c. Therefore, even if you could get a Ferrari traveling near c, you would not get a black hole. See Schutz's book or Misner Thorne and Wheeler's book. This interpretation took a little longer to understand.

3) Light always travels along null trajectories. This means that electromagnetic waves irrespective of wavelength (power line through hard gamma) travel the SAME route in a vacuum. For those of you who would ask about refractive effects, remember refraction requires charges to be present.

4) This is suitable for another thread or is nicely explained in "Modern Optics" by Fowles.
It is also in Jackson, for those daring enough.

first:

No. In relativity speeds don't add like that. If object A is moving away from the earth with speed .99c relative to the earth and object B is moving in the opposite direction with speed .99c relative to the eath then the speed of each relative to the other would be
$$\frac{.99c+ .99c}{1+ \frac{(.99c)(.99c)}{c^2}}= \frac{1.98c}{1+.9801}$$
= .99994904975c still slightly less than c.
wow .. you know i kinda thought the question was a good one.
i have never understood that before..
should have gone to collage i guess..

Alll right, I sent e-mails to the previous gent/lady "azzkika".
First, let's use what we know about the physical universe.
1) No signal can be sent at any rate exceeding that of light traversing a vacuum.
oh i agree, in my small mind the only way to seem to excede C would be folding or worm holes

2) As you build "speed" (I hate that term), MASS is NOT generated.
3) Gravitational field curve space, they do not slow light trajectories.
ugh here i agree- i referring to appearance time laps of that light from egality two diff sources one will arrive later due to the apparent effects of the gravity fields between us and sorce and the other source has no gravity wells between us..
ie if space it self is warped then ther is an appearent streching
4) Mass distributions (i.e. glass, dust, water) transparent to light slow light BECAUSE their electrons interact witht the light. Remember you lessons on polarization and complex susceptibility.

SO
1) Even if you could say get in front of an electromagnetic signal, what use is it since the you have not intercepted the signal. Causality still holds. Superluminal signals have nothing to do with "time travel" unless you are speaking of going forwards.

2) E=mc^2 does not mean that MASS increases when kinetic energy increases. The equation is fully written with a relativistic gamma multiplying the rest mass, m_0. It is the momentum that you are fighting to go faster. The change in momentum is what skyrockets. This is why it takes so much energy to go from 0.99c to 0.999c. Therefore, even if you could get a Ferrari traveling near c, you would not get a black hole. See Schutz's book or Misner Thorne and Wheeler's book. This interpretation took a little longer to understand.
humm seems i must have skimped that day.. as i thought that as you neared C yur mass increased.. will have to attempt correcting this miss understanding..
thanks for bringing it to my attention..
3) Light always travels along null trajectories. This means that electromagnetic waves irrespective of wavelength (power line through hard gamma) travel the SAME route in a vacuum. For those of you who would ask about refractive effects, remember refraction requires charges to be present.

4) This is suitable for another thread or is nicely explained in "Modern Optics" by Fowles.
It is also in Jackson, for those daring enough.

No. In relativity speeds don't add like that. If object A is moving away from the earth with speed .99c relative to the earth and object B is moving in the opposite direction with speed .99c relative to the eath then the speed of each relative to the other would be
$$\frac{.99c+ .99c}{1+ \frac{(.99c)(.99c)}{c^2}}= \frac{1.98c}{1+.9801}$$
= .99994904975c still slightly less than c.

Thanks for your clear answer, which aligns with other explanations I have seen. I have great difficulties in accepting it though, as it obviously defies logic. I would think that since object A and object B are not interrelated in any way they would simply be invisible to each other once their relative velocities exceed c. One interesting implication of your answer also seems to be that everything in the universe is visible to us, because nothing will ever escape the event horizon due to relative velocities exceeding the speed of light.

But these are obviously just my amateurish speculations, and that's usually put to rest by empirical evidence. It's hard to imagine empirical evidence for something we can't see, but is the inverse may have been proven (that we never loose sight of anything)? grateful for the direction to some such experiment in that case.

A.T.
I have great difficulties in accepting it though, as it obviously defies logic.
It doesn't defy logic, just intuition and presumptions. Pure logic alone doesn't tell you how nature behaves.

JesseM
Thanks for your clear answer, which aligns with other explanations I have seen. I have great difficulties in accepting it though, as it obviously defies logic.
It may seem less illogical if you understand that each observer defines "speed" in terms of distance/time on rulers and clocks at rest relative to themselves, and that each observer also measures the rulers and clocks of other observers to be distorted (rulers shrunk, clocks slowed down). So, the fact that a third observer sees A and B separating at faster than the speed of light does not imply that A and B measure each other to be moving away faster than light.
carstenk said:
I would think that since object A and object B are not interrelated in any way they would simply be invisible to each other once their relative velocities exceed c. One interesting implication of your answer also seems to be that everything in the universe is visible to us, because nothing will ever escape the event horizon due to relative velocities exceeding the speed of light.
The formula above is only intended to work in special relativity where spacetime itself doesn't behave in a dynamical way--in general relativity where spacetime is curved by mass, there actually can be an event horizon between sufficiently distant galaxies because the space between them is expanding faster than a light beam can bridge the gap (see http://www.scientificamerican.com/article.cfm?id=misconceptions-about-the-2005-03 [Broken] for some more on this).

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Ich
I would think that since object A and object B are not interrelated in any way they would simply be invisible to each other once their relative velocities exceed c.
Well, the whole point of HallsofIvy's answer is that relative velocities never exceed c. So I don't see where your difficulties come from - except that you maybe hadn't time to read carefully, as you answered hastily.

Don't say definately that no matter can go faster than the speed of light. There is always the theoretical sub-atomic particle, the tachyon. For a quick overview, http://en.wikipedia.org/wiki/Tachyon" [Broken]

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JesseM
Don't say definately that no matter can go faster than the speed of light. There is always the theoretical sub-atomic particle, the tachyon. For a quick overview, http://en.wikipedia.org/wiki/Tachyon" [Broken]
Tachyons can't be ruled out absolutely, but they'd violate either relativity or causality (meaning you could use them to send messages into the past). See the discussion on this thread for example.

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The formula above is only intended to work in special relativity where spacetime itself doesn't behave in a dynamical way--in general relativity where spacetime is curved by mass, there actually can be an event horizon between sufficiently distant galaxies because the space between them is expanding faster than a light beam can bridge the gap (see http://www.scientificamerican.com/article.cfm?id=misconceptions-about-the-2005-03 for some more on this).

Thanks for your patient explanations, which is highly appreciated. Unfortunately the article on Scientific American that you refer to requires a paid subscription that I currently can't justify. But your answer is interesting, since it in fact seems to (politely) refute HallsofIvy's original explanation, and in fact say that relative speeds (based on the expanding universe at least) above the speed of light is indeed possible in the scenario I originally described, since the farthest observable objects on diametrically opposite sides of the earth are indeed escaping at the speed of light because of the expanding universe. I'm just puzzled that this is not advertised more, because I had known I would obviously not have asked my original question, and a lot of the discussion here could instead concentrate on the implications of the cases where c is actually exceeded.

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JesseM
Thanks for your patient explanations, which is highly appreciated. Unfortunately the article on Scientific American that you refer to requires a paid subscription that I currently can't justify.
Sorry, I didn't notice that, it had been free for a long time...anyway I found a free PDF copy on an MIT page here:

carstenk said:
But your answer is interesting, since it in fact seems to (politely) refute HallsofIvy's original explanation, and in fact say that relative speeds (based on the expanding universe at least) above the speed of light is indeed possible in the scenario I originally described, since the farthest observable objects on diametrically opposite sides of the earth are indeed escaping at the speed of light because of the expanding universe. I'm just puzzled that this is not advertised more, because I had known I would obviously not have asked my original question, and a lot of the discussion here could instead concentrate on the implications of the cases where c is actually exceeded.
The problem is that to deal with cosmological scenarios we have to deal with non-inertial coordinate systems, while the restriction that nothing can travel faster than c is only intended to apply in inertial frames, the way that we can define a non-inertial coordinate system in GR is totally arbitrary (you could define a coordinate system where you were moving faster than c relative to some object in your own room, for example, although presumably light itself would move even faster in such a coordinate system). In general relativity all large-scale coordinate systems are non-inertial, one can only define "local" inertial frames in very small neighborhoods around freefalling observers, a consequence of the "equivalence principle" which is discussed in http://www.aei.mpg.de/einsteinOnline/en/spotlights/equivalence_principle/index.html [Broken].

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Sorry, I didn't notice that, it had been free for a long time...anyway I found a free PDF copy on an MIT page here:

The problem is that to deal with cosmological scenarios we have to deal with non-inertial coordinate systems, while the restriction that nothing can travel faster than c is only intended to apply in inertial frames, the way that we can define a non-inertial coordinate system in GR is totally arbitrary (you could define a coordinate system where you were moving faster than c relative to some object in your own room, for example, although presumably light itself would move even faster in such a coordinate system). In general relativity all large-scale coordinate systems are non-inertial, one can only define "local" inertial frames in very small neighborhoods around freefalling observers, a consequence of the "equivalence principle" which is discussed in http://www.aei.mpg.de/einsteinOnline/en/spotlights/equivalence_principle/index.html [Broken].

I must ask the experimental basis for this conclusion.

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JesseM
I must ask the experimental basis for this conclusion.
A coordinate system isn't defined by experiment, it's just a way we choose to label events in spacetime, we can define it however we want. And purely as a theoretical matter, it's possible to show that if the tensor equations of GR work in one coordinate system in a given spacetime (say, Schwarzschild coordinates in a black hole spacetime), they will be unchanged under a totally arbitrary coordinate system (such an arbitrary transformation is called a diffeomorphism, and the equations of GR are 'diffeomorphism invariant').

A coordinate system isn't defined by experiment, it's just a way we choose to label events in spacetime, we can define it however we want. And purely as a theoretical matter, it's possible to show that if the tensor equations of GR work in one coordinate system in a given spacetime (say, Schwarzschild coordinates in a black hole spacetime), they will be unchanged under a totally arbitrary coordinate system (such an arbitrary transformation is called a diffeomorphism, and the equations of GR are 'diffeomorphism invariant').

Yea, you can do whatever you want.

I would like to see an experiment to verify this assertion.

There must be a way to perform a comparison.

JesseM