Why Is It Impossible To Reach The Speed Of Light?

[johnnya]

This is one thing that I cannot understand no matter how much
I search about it.

I read that it has something to do with mass becoming infinite
or something like that. How does that happen?

Can anyone give me a detailed explanation on why we
can't reach it?

What will happen if we did. Will time stop? Will it have a
negative effect on the universe?

Looking forward for your answers.

 

[novop]

Special relativity predicts that as the speed of a massive object increases, it's momentum also increases. Thus, at very high speeds v >> c, more and more force is required to keep the object at a constant acceleration.

If you apply a constant force to a massive object, the acceleration impressed on the object eventually reaches zero once it's speed is large enough (again, because the momentum of the object increases with increasing speeds when v >> c)

 

[diazona]

Not exponentially, but it does increase more and more for a given increase in speed, in such a way that the momentum approaches infinity as you approach the speed of light. Since force is equal to the rate of change of momentum, a given force will have a smaller and smaller effect on the speed as the object gets faster.

The question of what would happen if we reached the speed of light is meaningless, because we can't.

 

[Timo]

Or perhaps as an alternative to momentum and forces: For any (finite) kinetic energy, the velocity of a massive (free) object is smaller than c:
$$E_{\rm kin} = \left( \frac{1}{\sqrt{1-v^2/c^2}} -1 \right) mc^2 \Rightarrow v = c \sqrt{ 1- \left( \frac{1}{E_{\rm kin}/mc^2 +1} \right)^2 } < c$$

 

[pallidin]

Inertia is the key concept regarding accelerative mass, and is not well understood. Even the fundamental processes behind it are not known.
Many speculations, none proven thus far. Will be undoubtedly one of the most important breakthroughs in physics when done.

 

[pallidin]

What we do know is this: A change in a mass objects velocity RESISTS that change.
The cause is currently unknown.

 

[RocketSurgery]

Although this is only a qualitative answer I think the concept will more clearly motivate the math.

If we let the x-axis be space and the y-axis be time then velocity is a straight line in the spacetime plane.

For a given object let the line have a fixed length and to make it easier to visualize you can think of the line as being a ladder and space as being the floor and time as being a wall.

If the ladder is snug against the wall then the space component is small but the time component is large. if the ladder is nearly horizontal it has a large space component and small time component.

Now if we think of the speed of light as the length of the ladder then it explains why it puts a limit on speed. T

he fastest one could move through space is if the ladder is completely flat on the ground (no time component, only space component) however since the ladder has a specific length, c , this is the "speed which an object can move through space". it cannot go faster than c because the ladder has a fixed length.

Now to explain why a massive particle cannot actually go at the speed of light you can think of mass as being a spring snug against the wall (the time axis). The ladder can slide down the wall (the particle moves faster through space) but the farther down it goes the more difficult it is for it to overcome the repulsion by the spring. Ultimately the spring prevents the ladder from ever falling perfectly flat. This spring is a conceptual (though flawed) analogy for rest mass.

I hope this analogy wasn't too convoluted. This is the best mental model I could think of being as we don't exactly understand inertia at a fundamental level yet.

 

[Mentz114]

This is one thing that I cannot understand no matter how much
I search about it.

I read that it has something to do with mass becoming infinite
or something like that. How does that happen?

Can anyone give me a detailed explanation on why we
can't reach it?

What will happen if we did. Will time stop? Will it have a
negative effect on the universe?

Looking forward for your answers.

It bothers me that you seem to think there is an absolute velocity, and a rocket ( say) could somehow tell when it reached 'light speed'. The only velocity we can ever measure is relative to some other thing. It is perfectly possible that from some rock somewhere in the cosmos you are traveling at .99999999c right now.

This problem has been analysed by Rindler, who worked out the motion for a rocket undergoing constant proper acceleration. This has been well presented in an article by Greg Egan, which you can find here
http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html

 

[I_am_learning]

Don't get upset that these weired scientist have found the ultimate limit for our velocity.
Read this thread:
https://www.physicsforums.com/showthread.php?t=332737

 

[stevmg]

It bothers me that you seem to think there is an absolute velocity, and a rocket ( say) could somehow tell when it reached 'light speed'. The only velocity we can ever measure is relative to some other thing. It is perfectly possible that from some rock somewhere in the cosmos you are traveling at .99999999c right now.

This problem has been analysed by Rindler, who worked out the motion for a rocket undergoing constant proper acceleration. This has been well presented in an article by Greg Egan, which you can find here
http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html

I read that paper to which you referred, and there is no way that a person with my training in mathematics or physics (Have a BA and MA in math - but emphasis on statistics in the Masters program) could ever understand it and it is NOT intuitive.

I prefer to think of the asymptotic concept - infinitely approaching but never reaching (in this case, c.) Sort of like the grasshopper jumping half the distance to the wall with each jump... always getting closer but never reaching. This is equivalent to an open set of real numbers. No matter how hard or how long you push on an object, it will always have a finite speed less than c. c is NOT included in the set of attainable speeds, but everything shy of it, no matter how shy, is.

My rhetorical question is can the speed of light, c, vary (to wit, light does slow down in a medium.) How do we know that in our locale, our "empty" universe is really that "empty." I know, Michelson-Morley and the non-existent ether, but you know what I mean... is c constant everywhere?

 

[stevmg]

By the way, it should not be possible to go faster than c unless space itself expanded giving velocities an artificial "kick." By the relative velocity addition formulas no matter how many different times you leap-frogged from one frame of reference to another, the ultimate difference between the final FR and the initial FR would still be less than c (except for space expansion which shoves distance between two points.)

 

[stevmg]

The increase in mass has been verified by the various particle accelerators in the past century, so that is theoretically (by conservation of energy-momentum) and experimentally proven.

 

[nonequilibrium]

Special relativity predicts that as the speed of a massive object increases, it's momentum also increases. Thus, at very high speeds v >> c, more and more force is required to keep the object at a constant acceleration.

If you apply a constant force to a massive object, the acceleration impressed on the object eventually reaches zero once it's speed is large enough (again, because the momentum of the object increases with increasing speeds when v >> c)

What are you talking about? First of all you said "v >> c" twice, and second of all: Newtonian Mechanics also predicts that momentum p increases when v increases because p = mv, yet it (in Newtonian Mechanics) it doesn't require more force to accelerate a moving mass with the same amount as a non-moving mass. And third of all, I don't agree with "If you apply a constant force to a massive object, the acceleration impressed on the object eventually reaches zero once it's speed is large enough", because you're delivering work, and if the object would acquire a constant speed (i.e. not gain kinetic energy by your work), energy will be disappearing.

As for the OP's question: I don't know if there's an intuitive answer (in the way that by simple reasoning you will conclude this). In the end it's a result of the non-intuitive belief (with experimental verifcation) that c is constant.

 

[Mentz114]

I read that paper to which you referred, and there is no way that a person with my training in mathematics or physics (Have a BA and MA in math - but emphasis on statistics in the Masters program) could ever understand it and it is NOT intuitive.
I prefer to think of the asymptotic concept - infinitely approaching but never reaching (in this case, c.) Sort of like the grasshopper jumping half the distance to the wall with each jump... always getting closer but never reaching. This is equivalent to an open set of real numbers. No matter how hard or how long you push on an object, it will always have a finite speed less than c. c is NOT included in the set of attainable speeds, but everything shy of it, no matter how shy, is.

OK. In the section 'Free Fall' he describes what happens if the rocket leaves something behind from the point it starts accelerating ( drops 'Adam'). As the rocket moves way the occupants see Adam falling away from them with ever increasing velocity. But from the rocket's frame they never see him reach the speed of light, because the local time coordinate axes never touch the horizon. So they see Adam forever dwindling and red-shifting but never totally disappering. From Adam's point of view, there is no horizon and nothing at all unusual happens. One could interpret this by saying that when Adam crosses the horizon ( which only the rocket is aware of) that the rocket has reached c wrt Adam and vice-versa. But the observer on the rocket never see this ( I've already said that ).

This is similar to what happens to an observer who accelerates in free-fall from infinity ( as far as one can go) towards a black hole. From the point of view of the distant observer the infalling particle never reaches the horizon, although the coordinate velocity would be c if they did. To the infalling particle there is no horizon and it can cross the horizon in a finite time by its clock.

This seems to indicate that if something does achieve light speed wrt to some observer, this will always be censored from that observer.

My rhetorical question is can the speed of light, c, vary (to wit, light does slow down in a medium.) How do we know that in our locale, our "empty" universe is really that "empty." I know, Michelson-Morley and the non-existent ether, but you know what I mean... is c constant everywhere?

I think this is another topic. There are other threads on the constancy of the speed of light.

 

[bcrowell]

FAQ: Why can't anything go faster than the speed of light?

In flat spacetime, velocities greater than c lead to violations of causality: observer 1 says that event A caused event B, but observer 2, in a different state of motion, says that B caused A. Since violation of causality can produce paradoxes, we suspect that cause and effect can't be propagated at velocities greater than c in flat spacetime. Special relativity is one of the most precisely and extensively verified theories in physics, and in particular no violation of this speed limit for cause and effect has ever been detected -- not by radiation, material particles, or any other method of transmitting information, such as quantum entanglement. Particle accelerators routinely accelerate protons to energies of 1 TeV, where their velocity is 0.9999996c, and the results are exactly as predicted by general relativity: as the velocity approaches c, a given force produces less and less acceleration, so that the protons never exceed c.

The corresponding speed limit in curved spacetime is far from being established. The argument from causality is not watertight. General relativity has spacetimes, such as the Godel solution, that are valid solutions of the field equations, and that violate causality. Hawking's chronology protection conjecture says that this kind of causality violation can't arise from realistic conditions in our universe -- but that's all it is, a conjecture. Nobody has proved it. In fact, there is a major current research program that consists of nothing more than trying to *define* rigorously what the chronology protection conjecture means.

There are certain things we *can* say about FTL, based on the fundamental structure of general relativity. It would definitely be equivalent to time travel, so any science fiction that has routine FTL without routine time travel is just plain wrong. It would probably require the existence of exotic matter, which probably doesn't exist. If it were possible to produce FTL artificially, it would certainly require the manipulation of godlike amounts of matter and energy -- so great that it is unlikely that beings able to carry it out would have anything like ordinary human concerns.

There are many ways that velocities greater than c can appear in relativity without violating any of the above considerations. For example, one can point a laser at the moon and sweep it across, so that the spot moves at a speed greater than c, but that doesn't mean that cause and effect are being propagated at greater than c. Other examples of this kind include a pair of cosmic-sized scissors cutting through a gigantic piece of paper at greater than c; phase velocities greater than c; and distant, observable galaxies receding from us at greater than c, which is interpreted as an effect in which space itself is expanding in the space in between.

 

[stevmg]

FAQ: Why can't anything go faster than the speed of light?

1) In flat spacetime, velocities greater than c lead to violations of causality: observer 1 says that event A caused event B, but observer 2, in a different state of motion, says that B caused A. Since violation of causality can produce paradoxes, we suspect that cause and effect can't be propagated at velocities greater than c in flat spacetime. Special relativity is one of the most precisely and extensively verified theories in physics, and in particular no violation of this speed limit for cause and effect has ever been detected -- not by radiation, material particles, or any other method of transmitting information, such as quantum entanglement. Particle accelerators routinely accelerate protons to energies of 1 TeV, where their velocity is 0.9999996c, and the results are exactly as predicted by general relativity: as the velocity approaches c, a given force produces less and less acceleration, so that the protons never exceed c.

2) The corresponding speed limit in curved spacetime is far from being established. The argument from causality is not watertight. General relativity has spacetimes, such as the Godel solution, that are valid solutions of the field equations, and that violate causality. Hawking's chronology protection conjecture says that this kind of causality violation can't arise from realistic conditions in our universe -- but that's all it is, a conjecture. Nobody has proved it. In fact, there is a major current research program that consists of nothing more than trying to *define* rigorously what the chronology protection conjecture means.

3) There are certain things we *can* say about FTL, based on the fundamental structure of general relativity. It would definitely be equivalent to time travel, so any science fiction that has routine FTL without routine time travel is just plain wrong. It would probably require the existence of exotic matter, which probably doesn't exist. If it were possible to produce FTL artificially, it would certainly require the manipulation of godlike amounts of matter and energy -- so great that it is unlikely that beings able to carry it out would have anything like ordinary human concerns.

4) There are many ways that velocities greater than c can appear in relativity without violating any of the above considerations. For example, one can point a laser at the moon and sweep it across, so that the spot moves at a speed greater than c, but that doesn't mean that cause and effect are being propagated at greater than c. Other examples of this kind include a pair of cosmic-sized scissors cutting through a gigantic piece of paper at greater than c; phase velocities greater than c; and distant, observable galaxies receding from us at greater than c, which is interpreted as an effect in which space itself is expanding in the space in between.

1) That's what I thought. I have the flat brain, so that made sense.

2) I thought we lived in a universe of curved spacetime because I thought that is what causes the spontaneous generation of gravity ("the parable of the two travelers.")

3) What is "FTL?"

4) The laser moving across the moon is is moving nothing. Just a bunch of little separate dots being generated. No different than looking at a ruler of any length at both ends at the same time which is "spacelike.". That's the old searchlight paradigm. Any cosmic scissors that were so large that the ends would be forced to move at equal to or greater than c would not do so. The scissors would close at an angular velocity commensurate with the tips of the scissors moving at just below c (you can't push any mass to c or greater) and the tips, no matter how small, would still have some mass and therefore couldn't be pushed that fast. The business about expanding space makes sense because that allows distance to be built up without doing anything to a moving photon and yet kick up its "apparent" speed.

5) (This is not to bcrowell) - Lighten up on the other guy (novop) who wrote v >> c. He got his ">" and "<" turned around. Happened to me all the time and I majored in math. Always had to rush through tests at the end to fix that.

 

[nonequilibrium]

5) (This is not to bcrowell) - Lighten up on the other guy (novop) who wrote v >> c. He got his ">" and "<" turned around. Happened to me all the time and I majored in math. Always had to rush through tests at the end to fix that.

In all honesty, I think my reply was fair. I considered the reversal of "<<" but even that didn't make sense, cause the context pointed to the fact he was talking about speeds close to the speed of lights.

On another matter, and more on topic: interesting posts so far, very insightful

 

[bcrowell]

Thanks, stevemg, for your comments on my FAQ :-)

2) I thought we lived in a universe of curved spacetime because I thought that is what causes the spontaneous generation of gravity ("the parable of the two travelers.")

Yes.

3) What is "FTL?"

Faster than light. Thanks, I'll clarify that.

4) The laser moving across the moon[...]

I think we're in agreement here, aren't we?

 

[stevmg]

In all honesty, I think my reply was fair. I considered the reversal of "<<" but even that didn't make sense, cause the context pointed to the fact he was talking about speeds close to the speed of lights.

On another matter, and more on topic: interesting posts so far, very insightful

Your comments are fair. Anything we write is fair as that is how information is given back and forth so there are no "wrong" answers in that sense. As I look at his (novop's) comments again I believe he means "as v approaches c [from below.]" In both cases, if one uses that interpretation, his statements make sense. Maybe he should have written "as v --> c^-"

To bcrowell - Yes, I have learned a lot, particularly how dumb I am but that's how I have to learn - by making misteaks and getting corrected.

By the way - my brain runs a lot, lot slower than even sound (STS,) much less light.

stevmg

 

[stevmg]

1) OK. In the section 'Free Fall' he describes what happens if the rocket leaves something behind from the point it starts accelerating ( drops 'Adam'). As the rocket moves way the occupants see Adam falling away from them with ever increasing velocity. But from the rocket's frame they never see him reach the speed of light, because the local time coordinate axes never touch the horizon. So they see Adam forever dwindling and red-shifting but never totally disappering. From Adam's point of view, there is no horizon and nothing at all unusual happens. One could interpret this by saying that when Adam crosses the horizon ( which only the rocket is aware of) that the rocket has reached c wrt Adam and vice-versa. But the observer on the rocket never see this ( I've already said that ).

This is similar to what happens to an observer who accelerates in free-fall from infinity ( as far as one can go) towards a black hole. From the point of view of the distant observer the infalling particle never reaches the horizon, although the coordinate velocity would be c if they did. To the infalling particle there is no horizon and it can cross the horizon in a finite time by its clock.

This seems to indicate that if something does achieve light speed wrt to some observer, this will always be censored from that observer.

2) [Speed of light constancy]

I think this is another topic. There are other threads on the constancy of the speed of light.

1) I am going to have to download the article and go through it slowly. I think you mean horizon a little differently than I am used to.

2) We'll defer that to another post or posts...

 

[stevmg]

To Mentz114

I take it back - there's got to be a more intuitive explanation than Gregan's article...

I still like "the harder you push... the harder the object is to move (accelerate)..." explanation.

I think by integrating v = at and a = a₀/SQRT[1 - v²/c²] (pugging that in for the a and doing that impossible integration) you will arrive at an answer in mathematical terms in which v is always < c no matter how long you push...
Of course a = dv/dt. a₀ is where you start...

When I get the energy up, I will try to do just that... but I bet you it has already been done.

Someone out there , give us a hand with this if you know this has been done.

 

[diazona]

Well actually acceleration is not quite as useful a concept to work with in relativity as it is in non-relativistic mechanics. If you want a mathematical argument, I would suggest thinking in terms of force and momentum. The relevant equations are
$$\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t}$$
and
$$\vec{p} = \frac{m\vec{v}}{\sqrt{1 - v^2/c^2}}$$
You can integrate both sides of the first equation to get
$$\int\vec{F}\mathrm{d}t = \Delta\vec{p}$$
which tells you that you can produce a change in momentum which is as large as you like just by pushing hard enough for a long enough time. However, if you solve the second equation for v, you get
$$\vec{v} = \frac{c\vec{p}}{\sqrt{m^2c^2 + p^2}}$$
This is basically c times some factor, which is roughly (momentum)/(momentum + something positive). That factor will always be smaller than 1, no matter how large the momentum is, and so the velocity will always be smaller than c.

 

[Integral]

Another path to answer the OPs question lies in Einstein's other postulate. That is the laws of physics are the same in all inertial frames. No matter how fast you are moving wrt to some reference point you will still measure the speed of light to be c. In other words no matter how fast you are moving the lights in your spaceship will still work the same. In addition if you have been observing a light signal from your reference point you will never cease to receive it.

 

[stevmg]

Well actually acceleration is not quite as useful a concept to work with in relativity as it is in non-relativistic mechanics. If you want a mathematical argument, I would suggest thinking in terms of force and momentum. The relevant equations are
$$\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t}$$
and
$$\vec{p} = \frac{m\vec{v}}{\sqrt{1 - v^2/c^2}}$$
You can integrate both sides of the first equation to get
$$\int\vec{F}\mathrm{d}t = \Delta\vec{p}$$
which tells you that you can produce a change in momentum which is as large as you like just by pushing hard enough for a long enough time. However, if you solve the second equation for v, you get
$$\vec{v} = \frac{c\vec{p}}{\sqrt{m^2c^2 + p^2}}$$
This is basically c times some factor, which is roughly (momentum)/(momentum + something positive). That factor will always be smaller than 1, no matter how large the momentum is, and so the velocity will always be smaller than c.

Using plain algebra (no calculus) I was able to "derive" an equation which said essentially what you just stated. I would like to post it but it is too brutal (and/or too painful) to do.

I also integrated dv/dt = a₀SQRT[(c² - v²)/c²] to get sin^-1(v/c) = a₀t/c which v/c (or $$\beta$$) must always be between -1 and +1 for any t, but I don't know if that is right or not.

 

[stevmg]

Another path to answer the OPs question lies in Einstein's other postulate. That is the laws of physics are the same in all inertial frames. No matter how fast you are moving wrt to some reference point you will still measure the speed of light to be c. In other words no matter how fast you are moving the lights in your spaceship will still work the same. In addition if you have been observing a light signal from your reference point you will never cease to receive it.

Now this IS intuitive!

 

[diazona]

I also integrated dv/dt = a₀SQRT[(c² - v²)/c²] to get sin^-1(v/c) = a₀t/c which v/c (or $$\beta$$) must always be between -1 and +1 for any t, but I don't know if that is right or not.

It seems like a good argument to me, as long as you're careful to use the right definitions of acceleration and velocity. I think your a₀ would be the force applied divided by the mass of the object being pushed, and v would be the observed velocity of the object relative to some fixed (i.e. inertial) reference.

 

[novop]

What are you talking about? First of all you said "v >> c" twice, and second of all: Newtonian Mechanics also predicts that momentum p increases when v increases because p = mv, yet it (in Newtonian Mechanics) it doesn't require more force to accelerate a moving mass with the same amount as a non-moving mass.

Yes, my mistake. I meant as v approaches c. Using a greater than sign to convey a pointed arrow, on a physics board, is not a good idea.

I can understand that I didn't properly communicate my point. The difference is that Newtonian physics predicts that momentum increases linearly with increasing speed, whereas in relativity, this simply doesn't work.

When v is close to c, the classic equation $$p = mv$$ doesn't hold. But $$p=\gamma{m}{v}$$ does in fact hold.

Now, if

$$F=\frac{dp}{dt}$$

then

$$\frac{dv}{dt}=\frac{F}{\gamma{m}}$$

So for a given force F, the acceleration of a massive object does, in fact, approach zero as v approaches c.

 

[stevmg]

It seems like a good argument to me, as long as you're careful to use the right definitions of acceleration and velocity. I think your a₀ would be the force applied divided by the mass of the object being pushed, and v would be the observed velocity of the object relative to some fixed (i.e. inertial) reference.

Thanks for (essentially) the "thumbs up" on my math. I have to work it out in specific examples to see if it makes any sense as the sines are a periodic function (sort of like the modulo system) and it wouldn't make sense for the v/c (or $$\beta$$) to increase for a while with t and then go the other way. Of course arcsines are not periodic, just the angles they represent are periodic, and the $$\beta$$ in this case is, hopefully, a monotonic increasing value as t increases (I hope.) All this would show is that no matter how long one applies a given force onto a given mass m₀ which gives that m₀ an initial acceleration of a₀ for an instant but forever decreses as the mass of the object increases with velocity (or energy,) one will never reach c no matter how long this same force is applied. This, of course, would not be true in classical physics as m (and therefore a) does not change so there does exist a t for which c = at does hold, no matter how small the a is.

Yes, my mistake. I meant as v approaches c. Using a greater than sign to convey a pointed arrow, on a physics board, is not a good idea.

I can understand that I didn't properly communicate my point. The difference is that Newtonian physics predicts that momentum increases linearly with increasing speed, whereas in relativity, this simply doesn't work.

When v is close to c, the classic equation $$p = mv$$ doesn't hold. But $$p=\gamma{m}{v}$$ does in fact hold.

Now, if

$$F=\frac{dp}{dt}$$

then

$$\frac{dv}{dt}=\frac{F}{\gamma{m}}$$

So for a given force F, the acceleration of a massive object does, in fact, approach zero as v approaches c.

Very, very nice mathematical demonstration of the veracity of the statement that v is always < c.

I will post my algebraic (no calculus) derivation of the same when I have time to do so. As I said, it is painful and to write it on this forum will take me thirteen forevers.

 

[DrGreg]

But $$p=\gamma{m}{v}$$ does in fact hold.

Now, if

$$F=\frac{dp}{dt}$$

then

$$\frac{dv}{dt}=\frac{F}{\gamma{m}}$$

So for a given force F, the acceleration of a massive object does, in fact, approach zero as v approaches c.

This is almost a valid argument apart from a technical error in the maths. You have treated $\gamma$ as a constant, but it isn't. I will leave it as an exercise for the reader that the correct formula turns out to be

$$\frac{dv}{dt}=\frac{F}{\gamma^3\,m}$$​

But the final conclusion still holds true.

 

[Mentz114]

I still like "the harder you push... the harder the object is to move (accelerate)..."

This is true if you're accelerating a particle in an accelerator, where the force is coming from the lab frame and is translated into the particle's frame.

But from a rocket ship's frame, there no change in inertia however long they accelerate.

 

[starthaus]

I liked the proof based on dynamics using the equation of relativistic momentum.
Here is an alternative proof using kinematics only:
Accelerated motion in SR gives:
$$v=\frac{at}{\sqrt{1+(at/c)^2}}$$

When $$t->oo$$ $$v->c$$ while, all along $$v<c$$There is a caveat to all these "proofs": they have all been derived using SR. But, SR assumes $$v<c$$ in its very own derivation of fundamental formulas. So, all we are doing is confirming that SR is consistent.

 

[novop]

This is almost a valid argument apart from a technical error in the maths. You have treated $\gamma$ as a constant, but it isn't. I will leave it as an exercise for the reader that the correct formula turns out to be

$$\frac{dv}{dt}=\frac{F}{\gamma^3\,m}$$​

But the final conclusion still holds true.

Of course, thanks Greg.

 

[stevmg]

I liked the proof based on dynamics using the equation of relativistic momentum.
Here is an alternative proof using kinematics only:
Accelerated motion in SR gives:
$$v=\frac{at}{\sqrt{1+(at/c)^2}}$$

When $$t->oo$$ $$v->c$$ while, all along $$v<c$$


There is a caveat to all these "proofs": they have all been derived using SR. But, SR assumes $$v<c$$ in its very own derivation of fundamental formulas. So, all we are doing is confirming that SR is consistent.

Have we all been using tautological reasoning? Never heard of that before. Sometimes it is hard to differentiate between tautological reasoning for a proof versus starting with a (proposed) conclusion and working backwards with logically equivalent statements until a prior known true statement is reached. Thus an equivalence chain is established (in reverse, but still, none-the-less valid) from the known valid statement and the conclusion.

 

[stevmg]

My algebraic proof is even more weird but does not rely on assuming that v < c. It was very long and painful to enter into the blog.

v = at
v = a₀t(SQRT[1 - v²/c²]) where a₀ = initial acceleration
c²v² = (c² - v²)a₀²t²
c²v² = c²a₀t² - v²a₀²t²
(c² + a₀²t²)v² = c²a₀²t²
v² = c²a₀²t²/(c² + a₀²t²)
invert both sides:
1/v² = (c² + a₀²t²)/[c²a₀²t²]
1/v² = 1/a₀²t² + 1/c²
all terms are positive so 1/a₀²t² is positive and always > 0 for any t > 0
1/v² > 1/c²
inverting back
v² < c², always, regardless of t (no matter how long we supply a force to make the initial acceleration a₀)
thus, because we are dealing with positive numbers here we have v < c, always, always, provided we start at v < c. Thus, if we start at v < c, we can never get to v $$\geq$$ c no matter how long we try.
If we try for v > c, then a is always 0 for any mass and everything falls apart.

Am I making any sense?

 

[stevmg]

This is true if you're accelerating a particle in an accelerator, where the force is coming from the lab frame and is translated into the particle's frame.

That's what I was talking about...

But from a rocket ship's frame, there is no change in inertia however long they accelerate.

I didn't know that. Must read up on SR in an accelerating frame. I have Taylor and Wheeler 1960's edition as well as AP French Spacetime Physics from 1979. I've looked in there. Do you folks know if it is in there?