Place to discuss the Theory of Relativity

[Doc Al]

I scanned this thread quickly and didn't see this point made: you can't just accelerate something indefinitely at a constant acceleration because the relativistic mass increases as speed increases. Since F= ma, in order to have constant acceleration with increasing mass, you would need to increase the force. As the speed nears c, the mass, and therefore force required to accelerate, goes to infinity.

Careful here. At relativistic speeds, F no longer equals ma, even if you replace m by the "relativistic mass". Assuming that the force and acceleration are in the same direction as the object's velocity, and that the acceleration a is that measured by an inertial observer who measures the object to be moving at speed v, the relationship would be:
$$F = m \gamma^3 a$$
where m is the usual invariant ("rest") mass. ($m \gamma^3$ is sometimes called the longitudinal mass.)

So, yes, as the speed increases it requires (greatly) increasing force to maintain a constant acceleration with respect to the inertial observer.

 

[learningphysics]

Wrong, the fact that it takes infinite amount of energy doesn't prevent you from assuming you have this infinite amount of energy. Prove that there is a finite amount of energy in the universe. Or, prove that there is no way possible to take energy from another universe and that there are not an infinite number of universes.

Well then why is exceeding the speed of light a non-physical situation? If one can possesses infinite amounts of energy, what prevents it.

I'm troubled by where this is going... infinity is not even a real number.

I did not ask for any such thing, I asked you to prove that relativistic mass is in anyway related to this. Please check your definition of inertial mass

Relativistic mass = inertial mass

 

[EnumaElish]

I've never brought up a Newtonian universe as that would be completely absurd when dealing with velocities that approach anything near c. Relativistic mass IS NOT relativitity in a Newtonian universe - perhaps you need to read up more on the basics of special relativity.

I am working on my reading list. In the meantime, just bear with me and assume a universe that is Newtonian in every aspect but relativistic mass. We know the universe isn't Newtonian (even I know that!), and what I am asking you to assume is absurd, non-physical, maybe even stupid. But I think I have a point to make. I will give you the benefit of an infinite supply of energy available instantly on demand. In such a universe, it would still be impossible to exceed the speed of light because of the relativistic mass effect. Every incremental amount of energy that the ship consumes would add to the ship's relativistic mass far more than it would add to the ship's velocity.

 

[Aer]

Relativistic mass = inertial mass

How wrong you are. Well, perhaps my knowledge of what relativistic mass is lacking. So I will challenge you to prove yourself correct in the statement Relativistic mass = inertial mass

 

[Aer]

I am working on my reading list. In the meantime, just bear with me and assume a universe that is Newtonian in every aspect but relativistic mass. We know the universe isn't Newtonian (even I know that!), and what I am asking you to assume is absurd, non-physical, maybe even stupid. But I think I have a point to make. I will give you the benefit of an infinite supply of energy available instantly on demand. In such a universe, it would still be impossible to exceed the speed of light because of the relativistic mass effect. Every incremental amount of energy that the ship consumes would add to the ship's relativistic mass far more than it would add to the ship's velocity.

I don't wish to subject myself to what I consider nonsense, prove to me that it is not nonsense and I'll reconsider.

 

[Janus]

Well then why is exceeding the speed of light a non-physical situation? If one can possesses infinite amounts of energy, what prevents it.

It takes an infinite amount of energy to reach the speed of light, exceeding it is another animal completely.

The equation for the total energy for an object(including the energy equivalnet of its rest mass) is:

$$E = \frac{Mc^2}{ \sqrt{1- \frac{v^2}{c^2}}}$$

Notice that if v>c, the bottom half of the equation becomes a squareroot of a negative number.

Meaning that the total energy of the object becomes non-physical.

 

[EnumaElish]

I am working on my reading list. In the meantime, just bear with me and assume a universe that is Newtonian in every aspect but relativistic mass. We know the universe isn't Newtonian (even I know that!), and what I am asking you to assume is absurd, non-physical, maybe even stupid. But I think I have a point to make. I will give you the benefit of an infinite supply of energy available instantly on demand. In such a universe, it would still be impossible to exceed the speed of light because of the relativistic mass effect. Every incremental amount of energy that the ship consumes would add to the ship's relativistic mass far more than it would add to the ship's velocity.

I don't wish to subject myself to what I consider nonsense, prove to me that it is not nonsense and I'll reconsider.

No, I freely admit it is nonsense. But it proves a point.

 

[Aer]

No, I freely admit it is nonsense. But it proves a point.

Nonsense is used to prove something other than nonsense?

 

[pervect]

Relativistic mass = inertial mass

Nooooooooooooooooooo!

 

[pervect]

I am working on my reading list. In the meantime, just bear with me and assume a universe that is Newtonian in every aspect but relativistic mass. We know the universe isn't Newtonian (even I know that!), and what I am asking you to assume is absurd, non-physical, maybe even stupid.

This doesn't give me any clear mental picture of what sort of "universe" you are talking about. Writing down a Lagrangian for this universe would be nice (perhaps it's too much to ask of you?)

By Lagrangian I mean the Lagrangain of a particle in potential V, i.e. for Newtonian mechanics

L(x,xdot) = .5*m*xdot^2 - V(x)

where V(x) is the potential.

Anyway, a little bit of thinking about what you might mean more or less convinces me that such a universe must have some sort of "preferred frame". I assume in your universe velocities add linearly. I think that this is what you mean by Newtonian, you want the Gallilean transform to be the transform between moving and non-moving coordinte systems.

x' = x-vt
t' = t

and not the Lorentz transform

x' = gamma*(x-vt)
t' = gamma*(t-vx/c^2)

Right?

Then this universe would have to be some sort of preferred frame that distinguishes high velocities, where it is difficult to accelerate, from low velocities, where it is not.

The point I've been trying to make is that it is the Lorentz transform itself that makes it impossible to exceed 'c' in relativity, and that "relativistic mass" is not needed to explain why it is impossible to exceed 'c'.

The Lorentz transform gives us a velocity addition formula by means of which one can add any fininte number of velocities less than 'c' and arrive at a velocity less than 'c'. Thus it becomes obvious that one is not going to exceed 'c' by means of acceleration.

[add]
It may be belaboring the obvious, but I want to point out that acceleration is just the process of accumulating velocity via addition

i.e. v(t+dt) = v(t) + a*dt

(using the Galiean transform, which makes velocities add linearly)

the relativistic version using the Lorentz transforms is

v(t+dt) = v(t) + (1-(v/c)^2)*a*dt

thus, the process of acceleration consists of adding incrementally to one's existing velocity.
[end add]

Thus it is obvious from the Lorentz transform and the velocity addition formula alone that one cannot accelerate to the velocity of 'c', the concept of relativistic mass is not needed.

 

[learningphysics]

It takes an infinite amount of energy to reach the speed of light, exceeding it is another animal completely.

The equation for the total energy for an object(including the energy equivalnet of its rest mass) is:

$$E = \frac{Mc^2}{ \sqrt{1- \frac{v^2}{c^2}}}$$

Notice that if v>c, the bottom half of the equation becomes a squareroot of a negative number.

Meaning that the total energy of the object becomes non-physical.

E is also undefined when v=c and the denominator is 0.

 

[learningphysics]

How wrong you are. Well, perhaps my knowledge of what relativistic mass is lacking. So I will challenge you to prove yourself correct in the statement Relativistic mass = inertial mass

It is not really proving... it's a matter of definition. I was using inertial mass as defined as p/v, where p is the relativistic momentum and v is velocity. This happens to be the definition of relativistic mass. In Rindler's "Introduction to Special Relativity" he calls p/v "relativistic inertial mass".

Anyway, I was able to find some lecture notes here and there using this definition of inertial mass.

http://physics.tamuk.edu/~hewett/ModPhy1/Unit1/SpecialRelativity/RelativeView/Mass/Mass.html

The comment about the spinning flywheel is one of the reasons I personally find relativistic mass helpful.

This prof says we can interpret the relativistic mass as inertial mass:
http://www.physics.hku.hk/~phys1303/Chap4.doc

The beginning of Section 7.1 here talks about the principle of equivalence and relativistic mass-energy
http://ankh-morpork.maths.qmul.ac.uk/~saha/teach/relativity/notes.pdf

 

[learningphysics]

Nooooooooooooooooooo!

Can you elaborate?

 

[EnumaElish]

(perhaps it's too much to ask of you?)

It is. (I'd hate to disappoint.)

Anyway, a little bit of thinking about what you might mean more or less convinces me that such a universe must have some sort of "preferred frame". I assume in your universe velocities add linearly.

I guess I am okay with both of these. Velocities add linearly so in principle one could exceed the speed of light but cannot because their "relativistic" mass keeps increasing with their velocity. They just cannot go on speeding up indefinitely.

acceleration is just the process of accumulating velocity via addition

i.e. v(t+dt) = v(t) + a*dt

Alternatively, a = x''(t), (I am guessing that these two definitions would be equivalent in this nonsense universe.)

Thus it is obvious from the Lorentz transform and the velocity addition formula alone that one cannot accelerate to the velocity of 'c', the concept of relativistic mass is not needed.

I agree that it is not necessary, but sufficient.

 

[EnumaElish]

Nonsense is used to prove something other than nonsense?

Yes, as in reductio ad absurdum.

 

[Aer]

Velocities add linearly so in principle one could exceed the speed of light but cannot because their "relativistic" mass keeps increasing with their velocity. They just cannot go on speeding up indefinitely.

Where were you educated? Lazy teachers would be better off not teaching their students anything rather than teaching them wrongly. SIGH

Alternatively, a = x''(t),

"Alternatively"?! assume v(t) = x'(t), then a(t)=x''(t) follows from the same equation. Maybe that is what you meant, but it sounds like you are saying that is an unrelated defintion.

 

[EnumaElish]

Where were you educated? Lazy teachers would be better off not teaching their students anything rather than teaching them wrongly. SIGH

I never had physics beyond HS, where relativity was not covered. Everything was Newtonian. I might add that I find it quite unbecoming (even a little unsettling) to disparage educators you don't even know.

"Alternatively"?! assume v(t) = x'(t), then a(t)=x''(t) follows from the same equation. Maybe that is what you meant, but it sounds like you are saying that is an unrelated defintion.

You should be pickier about the words you use. I'd say they are two independent definitions. I would never say they are unrelated. (Well, almost never.) See pervect's post #31 above, where he defines x''(t) as coordinate acceleration vs. v'(t) as proper acceleration. I am not a physicist, so I would only guess that there can be a difference between the two. To my uneducated understanding, that does not mean that the two are always different, only that they can be different.

 

[Aer]

I never had physics beyond HS, where relativity was not covered. Everything was Newtonian. I might add that I find it quite unbecoming (even a little unsettling) to disparage educators you don't even know.

To each his own... I recall having a few ignorant teachers in HS whom I still think would be better off spending their days out of the classroom and my HS was rated within the top 100 in the US when I attended.

You should be pickier about the words you use. I'd say they are two independent definitions. I would never say they are unrelated. (Well, almost never.) See pervect's post #31 above, where he defines x''(t) as coordinate acceleration vs. v'(t) as proper acceleration. I am not a physicist, so I would only guess that there can be a difference between the two. To my uneducated understanding, that does not mean that the two are always different, only that they can be different.

As I said, it seemed like you meant the other thing as I don't see the point of your response if that is what you initially meant.

 

[EnumaElish]

To each his own... I recall having a few ignorant teachers in HS whom I still think would be better off spending their days out of the classroom

Hell if I care. Just leave mine out of your comments. (Yeah, that's a hot button for me.)

My advice to you is to try to take responsibility for what you have become, or where you have ended up in, especially if you think you could have done better.

Having gotten this out of my chest, in what ways were your teachers ignorant? Do you mean technically, or culturally? Isn't blaming teachers (and anyone other than one's self) the easy way out? What about personal responsibility?

Don't your teachers deserve any credit for the fact that you can outsmart almost anybody in this forum?

 

[Aer]

Hell if I care. Just leave mine out of your comments. My advice to you is to try to take responsibility for what you have become, or where you have ended up in, especially if you think you could have done better.

What have I become? Am I evil for pointing out the short-comings in some educators? I never included all educators, in fact many of my HS teachers were a lot better teachers than some of the professors I had in college. However, I cannot say that I had any ignorant college professors - they were all appropriately knowledgeable in their respective fields which we'd all hope to be the case at the university level.

 

[Aer]

Don't your teachers deserve any credit for the fact that you can smart out almost anybody in this forum?

Being able to out smart anyone on these forums is quite a leap considering I've participated in only a handful of discussions.

As for physics in HS, I did have a very good teacher though I don't recall relativity being covered, perhaps it was briefly.

 

[EnumaElish]

And relativistic mass was never included in your curricula? Or was it?

 

[Aer]

And relativistic mass was never included in your curricula? Or was it?

Luckily I still have many of my physics notes. Most of my knowledge of special relativity is self learned though. I never took a class in college devoted solely to special relativity, but my physics classes at least covered the topic. (I have an engineering degree, not a physics degree). One of the best professors I had in college was one of my physics professors and here are his equations:

$$p = \gamma m v$$

$$E = m c^{2} + K = \gamma m c^{2}$$

$$E^{2} = (p c)^{2} + (m c^{2})^{2}$$

As you can see, there is no mention of "relativistic mass" as that equation would be:
$$E = m_r c^{2}$$
where m_r is relativistic mass. Also I checked the index of my physics book which contains no reference to relativistic mass.

Edit: Can someone (I am talking to you management, put a link to latex reference on the main page? OR if it is there, please make it more noticable)

 

[EnumaElish]

One of the best professors I had in college was one of my physics professors and here are his equations:

$$p = \gamma m v$$

$$E = m c^{2} + K = \gamma m c^{2}$$

$$E^{2} = (p c)^{2} + (m c^{2})^{2}$$

As you can see, there is no mention of "relativistic mass" as that equation would be:
$$E = m_r c^{2}$$
where m_r is relativistic mass.

What is K? And what is $\gamma$? What I am wondering is whether m_r is not part of these equations implicitly, or whether it can be derived from them, e.g. m_r = m/b, for a suitably defined b.

 

[Aer]

What is K? And what is $\gamma$? What I am wondering is whether m_r is not part of these equations implicitly, or whether it can be derived from them, e.g. m_r = m/b, for a suitably defined b.

K is kinetic energy and $\gamma$ is the relativity gamma factor (i.e. ${1}/{\sqrt{1-({v}/{c})^{2}}}$)

 

[EnumaElish]

If $m_r$ is defined as $= \gamma m$, then $E = \gamma m c^2= m_r c^2$, so relativistic mass is implicitly in your notes.

 

[Aer]

If $m_r$ is defined as $= \gamma m$, then $E = \gamma m c^2= m_r c^2$, so relativistic mass is implicitly in your notes.

I made $m_r$ up, it is not in my notes. As I said, that is just a definition and a meaningless definition at that. I could also define a "relativistic velocity" as $v_r = \gamma v$, does this velocity have any physical meaning? No it doesn't, athough the two terms $\gamma v$ come up in some equations, it doesn't make any sense to think of it as a "relativisitic velocity" to explain anything.

 

[EnumaElish]

From now on, whenever I use the expression "relativistic mass" (or the notation m_r) I will remember that it was not mentioned in your notes. Or that it is difficult to interpret as a physical entity (because it mixes the two frames, so to speak).

 

[Daminc]

Try the excellent General Relativity from A to B by Robert Geroch, which details different views spacetime from Aristotle to Galileo to Einstein.

Regards,
George

This book has finally arrived. Let's see how I get on