Question about Special Relativity

[Fanghur]

There is something that I have been wondering for quite a while now about Einstein's theory of special relativity; basically it is this. According to Einstein it is impossible for anything to travel faster (or even at) the speed of light because as an object approached the SOL (speed of light) the energy required to move it even faster increases exponentially until it reaches infinity at the SOL.

 

But what I don't understand is this; lets just for arguments sake say that Einstein was wrong about the whole "mass increases approaching the speed of light" thing. If I'm not mistaken it would still require an infinite amount of energy to make something move faster than the SOL because the only practical way we have of accelerating particles to even close to the SOL is by using electromagnetic fields; which of course propagate at the speed of light.

 

So of course it would be impossible to accelerate something to faster than the speed of light because you can't use a force to accelerate a particle to a greater speed than the force itself can propagate; and I'm pretty sure that if you tried to work out that situation mathematically there would still be an infinity in there somewhere.

 

Correct me if I'm wrong; and please keep in mind that I'm no PhD; I only read a lot of physics books.

[Sisyphus]

In a world that obeys classical physics - even one in which light has a finite velocity - there's no upper limit to velocity. If you can throw your fuel out the back of your rocket at 0.99C, and your mass is mostly fuel, then you end up moving faster than C.

[Fanghur]

But that doesn't make any sense; how could something that has a maximum speed be used to accelerate something else to an even greater speed? If the two things were traveling at the exact same speed then the force would no longer be able to keep pushing against the object in question in order to accelerate it further.

 

For example look at two cars that are both moving at 50 km/h; one in front of the other. As long as the car in front and the car behind continue to move at identical speeds they will never come in contact with each other unless they are literally bumper to bumper; in which case even though the car in the back is in contact with the car in the front; since they are moving at identical speeds the car in the back is not applying any force to the car in the front because the car in front is moving away from the car behind at the same speed as the car in behind is moving towards the car in the front.

 

So why wouldn't the same hold true for my original scenario? Because I would think that if the carrier photons were not able to interact with the object they are supposed to accelerate because it was already moving at their maximum speed; then the electromagnetic force could no longer to used for the job. Common sense 101.

[Sisyphus]

Common sense 101.

 

...failed! You're right that you couldn't get something to move faster than C away from you by, say, shooting a laser at it. You could, however, move yourself at faster than C by using good old Newton's third law of motion. You weigh 100kg. You have 1000kg worth of additional mass, all of which you accelerate away from you to C/2. How fast are you now going?

[J.C.MacSwell]

There is something that I have been wondering for quite a while now about Einstein's theory of special relativity; basically it is this. According to Einstein it is impossible for anything to travel faster (or even at) the speed of light because as an object approached the SOL (speed of light) the energy required to move it even faster increases exponentially until it reaches infinity at the SOL.

 

But what I don't understand is this; lets just for arguments sake say that Einstein was wrong about the whole "mass increases approaching the speed of light" thing. If I'm not mistaken it would still require an infinite amount of energy to make something move faster than the SOL because the only practical way we have of accelerating particles to even close to the SOL is by using electromagnetic fields; which of course propagate at the speed of light.

 

So of course it would be impossible to accelerate something to faster than the speed of light because you can't use a force to accelerate a particle to a greater speed than the force itself can propagate; and I'm pretty sure that if you tried to work out that situation mathematically there would still be an infinity in there somewhere.

 

Correct me if I'm wrong; and please keep in mind that I'm no PhD; I only read a lot of physics books.

 

You would need something faster than the speed of light to mediate the force. So if your hypothetical "classical universe" did not have that, then nothing could get to that speed, but merely approach it.

 

So I think your logic is reasonable given the right assumptions. It is somewhat analogous to SR where an infinite amount of energy would be required.

[Janus]

But that doesn't make any sense; how could something that has a maximum speed be used to accelerate something else to an even greater speed?

By throwing it backwards opposite of the direction you want to go. In other words, a rocket.

 

The ideal rocket equation is:

 

[math]V_f = Ve \ln \left ( \frac{M_r + M_f}{M-r} \right )[/math]

 

where Ve is the velocity of your exhaust, Mr is the mass of your rocket and Mf is the mass of your fuel. Note that the only thing that limits your final velocity is the amount of fuel you can carry.

[J.C.MacSwell]

By throwing it backwards opposite of the direction you want to go. In other words, a rocket.

 

The ideal rocket equation is:

 

[math]V_f = Ve \ln \left ( \frac{M_r + M_f}{M-r} \right )[/math]

 

where Ve is the velocity of your exhaust, Mr is the mass of your rocket and Mf is the mass of your fuel. Note that the only thing that limits your final velocity is the amount of fuel you can carry.

 

Isn't his point that you cannot "throw it back" from anything approaching light speed without invoking some "extra classical" force mediator?

 

Similarly you can't claim to be able to break light speed in the real world by "throwing back" an equal weight at, say, 2C.

[Sisyphus]

Isn't his point that you cannot "throw it back" from anything approaching light speed without invoking some "extra classical" force mediator?

 

Similarly you can't claim to be able to break light speed in the real world by "throwing back" an equal weight at, say, 2C.

 

I don't get what you mean. You don't need to throw it back at more than C. Throw an equal weight back at C to move at C. Throw half the remainder back at C (meaning it's now at zero velocity) and you've got 2C.

[J.C.MacSwell]

I don't get what you mean. You don't need to throw it back at more than C. Throw an equal weight back at C to move at C. Throw half the remainder back at C (meaning it's now at zero velocity) and you've got 2C.

 

That was referring to the real world or SR where you obviously cannot accelerate anything to C, or to 2C which I picked arbitrarily.

 

The point was that in Fanghur's classical space, you lose the ability to push back against anything as you approach C as well, from the perspective of an absolute and preferred reference. The math is different, but it is conceptually similar.

 

Keep in mind we are not discussing the actual physics, just a "what if" based on the pre-Einstein expectations or Newtonian physics approaching the light speed limit.

 

I don't see anything wrong with his conclusion based on the right set of assumptions. The question is more to the history- did scientists expect mass could be accelerated beyond C, and if so how, or by what means?

[Sisyphus]

Why would you lose the ability to push back?

[swansont]

Do you require any interaction to exceed c for this to work? Since Galilean transformations, constant c in all frames and no preferred frame are incompatible conditions, what does a contradiction actually tell us?

 

If I'm in a rocket going 0.9c going relative to some observer, I should still get to say I'm at rest in my frame. If I eject a mass equal to my own going 0.25c relative to me, I will recoil at .25c, thus my final speed is 1.35c with a Galilean transformation. There was no interaction that required exceeding c in my frame.

[D H]

The point was that in Fanghur's classical space, you lose the ability to push back against anything as you approach C as well, from the perspective of an absolute and preferred reference. The math is different, but it is conceptually similar.

The math is the non-relativistic rocket equation, which has no theoretical upper limit on velocity.

 

The very best we can get out of chemical propulsion systems is about 4.5 kilometers/second exhaust velocity. Orbital velocity in low Earth orbit is about 6.7 kilometers/second. In other words, every time we put a vehicle in orbit we exceed the exhaust velocity. The exhaust velocity is not a limit to the velocity that can be achieved.

[J.C.MacSwell]

The math is the non-relativistic rocket equation, which has no theoretical upper limit on velocity.

The very best we can get out of chemical propulsion systems is about 4.5 kilometers/second exhaust velocity. Orbital velocity in low Earth orbit is about 6.7 kilometers/second. In other words, every time we put a vehicle in orbit we exceed the exhaust velocity. The exhaust velocity is not a limit to the velocity that can be achieved.

 

... and was that equation believed to hold up, back in 1904, or was it believed to break down?

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Do you require any interaction to exceed c for this to work? Since Galilean transformations, constant c in all frames and no preferred frame are incompatible conditions, what does a contradiction actually tell us?

 

If I'm in a rocket going 0.9c going relative to some observer, I should still get to say I'm at rest in my frame. If I eject a mass equal to my own going 0.25c relative to me, I will recoil at .25c, thus my final speed is 1.35c with a Galilean transformation. There was no interaction that required exceeding c in my frame.

 

Relative to an absolute frame, yes, and I believe it was generally thought that there was one

[swansont]

... and was that equation believed to hold up, back in 1904, or was it believed to break down?

 

The NY Times certainly thought it broke down.

http://kottke.org/09/07/best-correction-ever

 

Relative to an absolute frame, yes, and I believe it was generally thought that there was one

 

But is that part of the current thought experiment? I thought we were using the assumptions of relativity but with Galilean transformations.

[J.C.MacSwell]

The NY Times certainly thought it broke down.

http://kottke.org/09/07/best-correction-ever

 

Funny!

 

 

But is that part of the current thought experiment? I thought we were using the assumptions of relativity but with Galilean transformations.

 

 

 

I was thinking otherwise. At least I thought Fanghur was thinking otherwise.

[D H]

You can't have it both ways. If throw out c as an upper limit to velocity you inherently throw out the first postulate of special relativity. Some alternative derivations of special relativity start with by postulating a causal, flat universe. This postulate that the relative velocity of any two objects must have some finite upper limit and that this upper limit is the same to all observers. Reconciling this philosophical reasoning with Maxwell's equations says that c is that upper limit.

 

While Galilean relativity appears to be correct at velocities that are small compared to c, it is not a universal law. If you want to pretend that it is a universal law (which is what I thought was the point of this thread), you need to throw out things like the speed of light being constant -- and you need to throw out causality.

[timo]

Typo correction:

The ideal rocket equation is:

[math]V_f = Ve \ln \left ( \frac{M_r + M_f}{M-r} \right )[/math]

where Ve is the velocity of your exhaust, Mr is the mass of your rocket and Mf is the mass of your fuel. Note that the only thing that limits your final velocity is the amount of fuel you can carry.

Should read

[math] V_f = V_e \ln \left[ \frac{ M_r + M_f } { M_r } \right] = V_e \ln \left[ 1+\frac{ M_f } { M_r } \right] [/math]

where the rightmost term added by me is a trivial step from Janus version but shows the critical value, the mass-to-fuel ratio, more prominently. Note: Mr is the mass of the rocket without the fuel.

 

 

Not sure if that was a typo or if I missed something:

While Galilean relativity appears to be correct at velocities that are small compared to c, it is not a universal law. If you want to pretend that it is a universal law (which is what I thought was the point of this thread), you need to throw out things like the speed of light being constant -- and you need to throw out causality.

I think I disagree with the "need to throw out causality": When velocities are not restricted then the full set of events with times >t lie in the future of any event at t. Order of cause and effect seem to be conserved in nonrelativistic mechanics. That pretty much meets my requirements for causality, as far as I see right now.

[D H]

Typo correction:

 

Should read

[math] V_f = V_e \ln \left[ \frac{ M_r + M_f } { M_r } \right] = V_e \ln \left[ 1+\frac{ M_f } { M_r } \right] [/math]

Concur.

 

 

I think I disagree with the "need to throw out causality": When velocities are not restricted then the full set of events with times >t lie in the future of any event at t. Order of cause and effect seem to be conserved in nonrelativistic mechanics. That pretty much meets my requirements for causality, as far as I see right now.

Multiple physicists have derived special relativity by showing that causality coupled with unbounded velocities leads to a contradiction. One of the two has to go. Unfortunately, I don't have time to chase the papers down today. I have a big deadline tomorrow. I'll try to find some references later this week.

[J.C.MacSwell]

Concur.

 

 

 

Multiple physicists have derived special relativity by showing that causality coupled with unbounded velocities leads to a contradiction. One of the two has to go. Unfortunately, I don't have time to chase the papers down today. I have a big deadline tomorrow. I'll try to find some references later this week.

 

That doesn't hold though, in non-relativistic mechanics, where everyone agrees on simultaneity and time proceeds for everyone at the same rate.

 

In SR faster than light in one frame would mean backwards in time in another. For example, given 3 points in space (with enough distance separation and moving relative to each other) sending messages faster than light, you could get tomorrows lotto winning numbers that you yourself sent off after reading them in the newspaper, with plenty of time to buy a winning ticket. (obviously can't happen and neither can faster than light communication)

[Fanghur]

OK guys; I'm getting the impression that you are missing the point of my question; so let me try and explain it a different way. Let's say that one day we discover some entirely new fundamental force which can be used to accelerate sub-atomic particles just as easily as the electromagnetic force with the only difference being that this force (let's call it the "fifth force") propagates at 50% the speed of light. (I'm just using this to make my point)

 

Now; what I am trying to get at is this. Even though there is no law of physics which forbids an object to travel at 0.5C, and even though the fifth force is just as easy to work with as the electromagnetic force; if I am not mistaken it would be impossible to use this force to accelerate a sub-atomic particle (or any other object for that matter) to 0.5C because then the particle would be moving faster than the force itself could propagate to the particle; and since if the particle was traveling at exactly the propagation speed of the fifth force then the force would no longer be able to accelerate it any further because it would no longer be able to interact with it.

 

Now I'll admit that I'm not a genius in math; but just from looking at this situation logically it seems roughly analogous to taking the tangent of 90 degrees; only the limit in this case is not 90 degrees; it's 0.5C. As the speed of the particle approaches 0.5C; the greater the energy needed to accelerate it even further. And since common sense tells us that something cannot interact with something that it can't make contact with...

 

This hypothetical situation looks remarkably similar to the light barrier; only in that case the limit is the propagation speed of the electromagnetic force, and let's face it; the only practical way we have of accelerating particles is through the use of electromagnetic fields; and since if I'm correct (and I may very well be missing something) then it's only natural that we haven't been able to break the light barrier; because in affect we are using light to try and accelerate something faster than light.

[D H]

That "fifth" force would come in mighty handy for as a propellant.

 

We currently use chemical propulsion for most space vehicles.The best chemical propellant leaves the rocket at about 4.5 km/s, or about 1/67000 light speed. Even though the exhaust leaves the rocket at a tiny fraction of the speed of light, the only theoretical limit to a chemical rocket's speed is the speed of light itself.

 

While there is no theoretical limit to speed in a non-relativistic universe, there is a huge practical limit. Suppose we want to use chemical rockets to get a small 1 kilogram payload to 1/10 the speed of light. Assume the rocket itself is made of some incredibly light material (for example, unobtanium) and ignore relativistic effects. The amount of fuel needed would be, via the rocket equation (see post 17) is the solution to

 

[math]\frac{0.1 c}{4.5\,\text{km}/\text{s}} = \ln\left(1+\frac{M_f}{1\,\text{kg}}\right)[/math]

 

or about 10²⁹⁰⁰ kilograms of fuel, which exceeds the mass of the universe by a mere factor of 10⁵² or so.

 

On the other hand our unobtanium rocket would require only 220 grams of this magical fuel that leaves the vehicle at half light speed to reach 1/10 light speed.

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Now suppose we want to use get to 0.9 c. This magical fuel will come in mighty handy. First let's ignore relativistic effects and use the non-relativistic rocket equation:

 

[math]M_f = \left(\exp\left(\frac{0.9 c}{0.5 c}\right)-1\right)\,M_r[/math]

 

or about 5.05 kg of fuel. One cannot of course relativistic effects in achieving a velocity of 0.9 c. The relativistic equivalent of

 

[math]\Delta v = v_e\ln\left(1+\frac{M_f}{M_r}\right)[/math]

 

is

 

[math]\tanh^{-1}\left(\frac{\Delta v}{c}\right) = \frac {v_e}{c}\,\ln\left(1+\frac{M_f}{M_r}\right)[/math]

 

which yields 18 kilograms of fuel for our magical rocket.

[swaha]

what if the force is impulsive? no pushing case.

[Sisyphus]

Fanghur, you're talking about the same situation as a rocket with a maximum exhaust velocity of 0.5C. The rocket's maximum velocity is not limited to 0.5C. A single projectile with no means of independent propulsion, sure, but not a universal law applicable to all objects.

 

Again, a simple example:

 

You start at rest, and have three projectiles with you, each equal to to your own mass. You fire two at once at 0.5C away from you, in the positive direction. Because of Newton's third law, you (and the third projectile) are now also moving at 0.5C, in the negative direction (as the fired projectiles were half your total mass). Now fire the third, also at 0.5C (relative to you, obviously), in the postive direction. The third projectile now has zero velocity (-0.5C+0.5C=0), and because of Newton's third law, you are now moving at C (-0.5C-0.5C=-C). "Speed limit" exceeded.