The Elegance and Complexity of Theories Beyond Special and General Relativity

[RiddlerA]

Star moves faster than light!

Cosnider a universe with only two objects, a planet and a star...
From the rest FoR of the Star, the planet appears to be revolving around the star.. And also the planet rotates in its own axis.. (just like our Earth and sun)...
The planet takes 1 second to complete a full rotation in its own axis (like 24 hours for earth)..

Please view the attached image files for better understanding...

Now from the rest FoR of the planet, the star appears to be revolving around the planet for every 1 second..(just like our sun appears to be revolving around the Earth in 24 hours)..

So from the rest frame of the planet, the star is actually moving at a speed six times faster than light(approx.)

How can this happen? what is wrong with this system?

 

[Dale]

Nothing is wrong with it. The rest frame of the planet is non inertial. It is only inertial frames where the speed of light is required to be c.

 

[RiddlerA]

Nothing is wrong with it. The rest frame of the planet is non inertial. It is only inertial frames where the speed of light is required to be c.

So what is the speed of light in non inertial frames?

 

[PAllen]

There is no limit. It can be arbitrarily slower or faster than c (speed of light in vacuum in local inertial frame). Especially, coordinates built from a rotating frame can have many bizarre features in GR.

 

[xlines]

So what is the speed of light in non inertial frames?

Try spinning around yourself and as you revolve, look at Andromeda galaxy up, in the skys: does it speed with warp 10? The two situation are not equivalent: inputing energy to rotate a human body is not the same as putting entire universe into a frenzy spin.

 

[RiddlerA]

Lets consider an inertial frame containing two objects A and B...
Now A and B are both moving anti parallel to each other(closing in on each other) at a velocity 99% of 'c'..

This situation can be described in three ways..

1. A is moving towards B and B is moving towards A... their speeds are below 'c'.

2. From A's rest frame, B is moving towards A at a speed greater than 'c'(double the 99% of 'c')...

3. From B's rest frame, A is moving towards B at a speed greater than 'c'(double the 99% of 'c')...


Im aware that the above case 2 and case 3 are wrong...
Lets just assume that the objects can reach speed of light hypothetically...
And also assume both A and B is accelerating(ya its not an inertial frame anymore)...
From A's rest frame, B appears to be approaching the speed of light 'c'..
Taking time dilation and length contraction into account,
we can say that from A's rest frame, B's time is slowed and space is contracted...
Hypothetically if B reaches speed of light, then its time freezes relative to A...
which means A perceives B to be standing still...
So if B freezes at the speed of light, then its speed should be greatly reduced before the moment it reaches 'c'..
I mean though B is accelerating, from A's rest frame B appears to be decelerating and finally freezed...

THe above story don't make sense at all... Where i went wrong? Please explain...

Thank you all for the replies...
RA...

 

[Dale]

2. From A's rest frame, B is moving towards A at a speed greater than 'c'(double the 99% of 'c')...

3. From B's rest frame, A is moving towards B at a speed greater than 'c'(double the 99% of 'c')...


Im aware that the above case 2 and case 3 are wrong...

Where i went wrong?

The parts in bold are where you went wrong. Velocities don't add that way.

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/einvel2.html#c1

 

[RiddlerA]

The parts in bold are where you went wrong. Velocities don't add that way.

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/einvel2.html#c1

I myself stated those were wrong... The real question is the paragraph below it...
Thanks...

 

[Nugatory]

Hypothetically if B reaches speed of light, then its time freezes relative to A...
which means A perceives B to be standing still...

The bolded part is wrong.

You have to be really careful with these "hypothetically, if something impossible..." arguments, because when you start with a false hypothesis, it's possible to come to very absurd results that tell you only that the hypothesis is false. But in this case, we can avoid the impossibility and the contradictions that come with it by considering a similar but possible case: as the speed of B relative to A approaches the speed of light, and gets closer and closer, B's time slows more... So we won't waste time on the impossible situation in which B's time "freezes"... We'll settle for the case in which B's clock ticks only once in a billion of A's years... Not frozen, but pretty thoroughly glacial, and A would need some fairly sophisticated equipment to tell this situation from a totally stopped clock.

And in that case, when A measures B's speed, A is using A's clocks and rulers, using time and distance as A sees it. The slowing down of B's clock doesn't affect B's speed relative to A - what A sees is an (almost) stopped clock moving past real fast.

Of course, in the absence of acceleration the situation is completely symmetrical and B sees the same thing looking over at A. Yes, they both see the other clock as (nearly) stopped, and using the Lorentz transformations to make sense of this situation is a good way of figuring out what SR is really about.

 

[Dale]

I myself stated those were wrong...

I know you stated that they were wrong, but you also asked why they were wrong. That is why, you used the wrong formula for velocity addition.

Regarding your "travelling at c" question. It is a bad question, at least for this forum. SR says nothing massive can go as fast as light, so the question itself cannot be answered by SR. If you want a non-SR answer you need to take the question somewhere else.

 

[RiddlerA]

I know you stated that they were wrong, but you also asked why they were wrong. That is why, you used the wrong formula for velocity addition.

Regarding your "travelling at c" question. It is a bad question, at least for this forum. SR says nothing massive can go as fast as light, so the question itself cannot be answered by SR. If you want a non-SR answer you need to take the question somewhere else.

Sorry and that formula helped a lot.. so thank you...
And my question was related to GR and this is the GR forum, that's why i posted here...
Sorry if the topic deviated from your scope of the forum...

 

[RiddlerA]

The bolded part is wrong.

You have to be really careful with these "hypothetically, if something impossible..." arguments, because when you start with a false hypothesis, it's possible to come to very absurd results that tell you only that the hypothesis is false. But in this case, we can avoid the impossibility and the contradictions that come with it by considering a similar but possible case: as the speed of B relative to A approaches the speed of light, and gets closer and closer, B's time slows more... So we won't waste time on the impossible situation in which B's time "freezes"... We'll settle for the case in which B's clock ticks only once in a billion of A's years... Not frozen, but pretty thoroughly glacial, and A would need some fairly sophisticated equipment to tell this situation from a totally stopped clock.

And in that case, when A measures B's speed, A is using A's clocks and rulers, using time and distance as A sees it. The slowing down of B's clock doesn't affect B's speed relative to A - what A sees is an (almost) stopped clock moving past real fast.

Of course, in the absence of acceleration the situation is completely symmetrical and B sees the same thing looking over at A. Yes, they both see the other clock as (nearly) stopped, and using the Lorentz transformations to make sense of this situation is a good way of figuring out what SR is really about.

That makes a lot of sense now, thank you for the answer...
I have one more doubt,
Assume an object which is moving relative to everything else in the universe...
The object was never at rest in any frame... And its speed is lower than 'c'...
So now can i say that the object has no rest mass?
And i also need to know why speed of light is assumed to be the cosmological speed limit?

 

[Dale]

And my question was related to GR and this is the GR forum, that's why i posted here...

It didn't seem to be a GR-related question, but a massive object cannot travel as fast as light locally in GR either. In GR massive objects have timelike worldlines and massless "objects" have null worldlines. The distinction is very strong.

 

[D H]

And i also need to know why speed of light is assumed to be the cosmological speed limit?

Maxwell's equations from the theoretical side, the Michelson-Morley experiment from the experimental side.

 

[Nugatory]

Assume an object which is moving relative to everything else in the universe...
The object was never at rest in any frame... And its speed is lower than 'c'...
So now can i say that the object has no rest mass?

No. The object is always at rest in a frame that is moving along with the object, which is to say the frame of an observer who happens to be riding along with that object. (It would only be very slightly sloppy to call that frame "the object's frame"). Whatever mass the object has in that frame is its rest mass. So we can define the rest mass of an object as the mass it would have if measured by some observer not moving relative to that object.

And I also need to know why speed of light is assumed to be the cosmological speed limit?

Not so - the speed of light is not ASSUMED to be the cosmic speed limit. "Nothing can travel faster than light" is not an assumption. It is a (somewhat oversimplified) summary of results derived by mathematics and logic from a much simpler and more natural assumption that also has substantial experimental evidence behind it.

The starting assumption is the Principle of Relativity, which basically says that the laws of physics won't change out from underneath you just because you're moving. If you do an experiment in a sealed, climate-controlled, vibration-isolated, not affected by anything happening outside the walls, no windows laboratory in December when the Earth and your lab is moving in one direction at about ten miles a second... And then do the same experiment in June, six months later when the Earth is on the other side of the sun and moving in the opposite direction at ten miles a second... You'll get the same results. Likewise, we don't expect experiments to give different results at different times of the day, even though the Earth's rotation means that we're constantly moving in different directions.

That's a very common-sense assumption, and it is supported by a huge amount of experimental evidence, including the famous Michelson-Morley experiment.

Now comes the twist... The speed of light can be CALCULATED from the laws of physics (specifically, Maxwell's equations of electricity and magnetism). Therefore, if the laws of physics don't change when the observer happens to moving, then all observers, regardless of their motion, must use the same Maxwell's equations and hence must calculate and see the same value for the speed of light.

That's kinda weird, because it says that if I see a beam of light moving at speed c, and I see you moving in the same direction at speed .4c... I'll see the difference between your speed and the beam of light as .6c, but you will see the difference between your speed and the speed of the beam of light as c, not .6c. Because speed is defined as distance traveled divided by time, the only way that can happen is if you and I are measuring distance and time differently.

Special relativity is about how two observers moving relative to one another measure distance and time differently, in such a way that all of the laws of physics, including especially Maxwell's equations and the speed of light, will hold for both observers. The math is actually not all that complicated; Einstein's book (Relativity: The Special and General Theory - pay special attention to appendix 1) manages to get through it with no calculus, just basic algebra. Time dilation, length contraction, and some other stuff follow from this math. And among the the "other stuff" that follows from the math is:

1) No amount of force will accelerate any object of non-zero mass that is moving at a speed below the speed of light relative to you to a speed greater than the speed of light relative to you.
2) If an object can move from point A to point B more quickly than a beam of light could make the same journey, then some observer traveling at some speed somewhere will see the object arrive at point B before it has left point A - and would be able to stop the departure from A even though the arrival at B is known to happen.
3) The equations all include a term that looks like $\sqrt{1-{v^2}/{c^2}}$, and that makes no real-world sense if v is greater than c.

#1 is a pretty convincing argument against any form of traditional transport working at speeds greater than light - a bigger better rocket motor won't ever do the job.
#2 excludes the standard science fiction hyperdrives and wormholes and space-warping mechanisms, unless you are willing to give up any sane notion of cause and effect.
#3 doesn't have to forbid anything. We could argue that it's just saying that the math of special relativity doesn't apply at speeds greater than light... But we'd still have to deal with #1 and #2, and propose some other theory that is consistent with the observed fact that the laws of physics don't change just because you're moving. No one has ever been able to come up with a such a theory, and even if someone could, we'd still have #1 and #2 to deal with.

So, until (and we can bet very long odds against this happening) some seriously compelling experimental evidence comes along to tell us that the Principle of Relativity is wrong... We can conclude that nothing with non-zero rest mass can travel faster than the speed of light.

 

[Chronos]

It appears we have a complex version of a 'division by zero' issue going on here.

 

[RiddlerA]

Special relativity is about how two observers moving relative to one another measure distance and time differently, in such a way that all of the laws of physics, including especially Maxwell's equations and the speed of light, will hold for both observers. The math is actually not all that complicated; Einstein's book (Relativity: The Special and General Theory - pay special attention to appendix 1) manages to get through it with no calculus, just basic algebra. Time dilation, length contraction, and some other stuff follow from this math. And among the the "other stuff" that follows from the math is:

1) No amount of force will accelerate any object of non-zero mass that is moving at a speed below the speed of light relative to you to a speed greater than the speed of light relative to you.
2) If an object can move from point A to point B more quickly than a beam of light could make the same journey, then some observer traveling at some speed somewhere will see the object arrive at point B before it has left point A - and would be able to stop the departure from A even though the arrival at B is known to happen.
3) The equations all include a term that looks like $\sqrt{1-{v^2}/{c^2}}$, and that makes no real-world sense if v is greater than c.

Ahh that makes a lot of sense now... all these days i was unaware of the origin of the 2 postulates of the SR... Thank you so much for taking time to explain all this...

So, until (and we can bet very long odds against this happening) some seriously compelling experimental evidence comes along to tell us that the Principle of Relativity is wrong... We can conclude that nothing with non-zero rest mass can travel faster than the speed of light.

Now that the CERN has detected neutrinos to be traveling faster than light... I hope this will bring a new theory which will be easy and elegant to understand

 

[Dale]

a new theory which will be easy and elegant to understand

Don't count on it.

 

[Nugatory]

Now that the CERN has detected neutrinos to be traveling faster than light...

"CERN has detected neutrinos traveling faster than light" is a huge oversimplification, the sort of thing that the popular press routinely gets wrong when they're trying to write about technical topics for a non-technical audience. Find the CERN paper online, read it, compare with the press coverage, and consider the difference between the statements:
1) I have found the fountain of youth!
2) I have found something that might be the fountain of youth.
3) I have found something that I haven't been able to prove is not the fountain of youth.

Or (before the moderators chase us over there) find the thread on the CERN results and read it.

I hope this will bring a new theory which will be easy and elegant to understand

Elegant, yes, but ease of understanding may be harder to come by. SR is elegant within its domain and can be understood by any motivated person who has understood algebra-based high school classical physics. GR is even more elegant - arguably the most elegant physical theory constructed to date - and applies to a much broader domain, but you have to pay some serious mathematical dues to be able to appreciate it.

I would bet long odds that whatever lies beyond GR is going to require an even higher investment in skull sweat - and it will be worth it to those who choose to pay it.