Understanding Relativity: The Sophomore's Question and Einstein's Answer

[OneEye]

Sohomore = Gk. Soph- (wise) + moros (fool).

I don't doubt that this question has been posited in this forum any number of times, and so I apologize if I am being a chore to anyone, but...

Dr. Einstein, Relativity, 37
As a consequence of its motion the clock goes more slowly than when at rest. Here also the velocity c plays the part of an unattainable limiting velocity.

In response, the sophomore cracks: "Relative to what?"

How does the professor answer the sophomore?

 

[Doc Al]

Sohomore = Gk. Soph- (wise) + moros (fool).

Check out the origin of the word "sophomore" here: http://www.straightdope.com/mailbag/msophomore.html

In response, the sophomore cracks: "Relative to what?"

What relative to what?

 

[OneEye]

Check out the origin of the word "sophomore" here: http://www.straightdope.com/mailbag/msophomore.html

Not going to directly gainsay your reference, but both Webster's and AHD support the etymology I gave.

What relative to what?

In SR, velocity is a relation, not an inherent quality. In my own inertial frame, my velocity is always 0. If I may not exceed the speed of light, the question is, in relation to what may I not exceed the speed of light?

 

[jdavel]

If I may not exceed the speed of light, the question is, in relation to what may I not exceed the speed of light?

In relation to anyone who measures your speed.

 

[Doc Al]

In SR, velocity is a relation, not an inherent quality. In my own inertial frame, my velocity is always 0. If I may not exceed the speed of light, the question is, in relation to what may I not exceed the speed of light?

I would think that velocity is always a relation, even pre-SR. In any case, the statement would be that no material object (including you) can exceed the speed of light with respect to any inertial frame.

 

[Dissident Dan]

I think that the idea is that your time slows in comparison to another observer's. If you are only going 100 mph relative to me, your clock won't appear to have slowed down very much (probably not even measurable). However, relative to someone else, you may be going 0.3c, in which case you would appear to slow down appreciably. However, relative velocities should be symmetric, right? So, the observer should be going the same 0.3c compared to you, in which case he would have slowed down relative to you. How can you both slow down relative to each other?

 

[jdavel]

Dissident Dan asked: "How can you both slow down relative to each other?"

Here's an analogy. Suppose you look at somebody through a big magnifying glass. He looks bigger to you. But you look bigger to him. How can that be?

 

[OneEye]

I would think that velocity is always a relation, even pre-SR.

Assuming I have this right: In a Newtonian context, there is a K₀ coordinate frame with respect to which every other inertial frame is in motion, and a v₀ "really" at rest. SR expressly denies any such concepts. This is precisely the innovation of Relativity: That no object can, in and of itself, be said to be "in motion" or "at rest."

In any case, the statement would be that no material object (including you) can exceed the speed of light with respect to any inertial frame.

So, then, if we could apply a simultaenous acceleration to every object in the universe such that every object was accelerated by .7c, then we have "upped the ante", and are now at liberty to accelerate any one object in the universe by (say) .4c relative to the rest of the universe, and this would be a legitimate velocity, even thought it is 1.1c WRT our original frame of reference? Or, perhaps, we could accelerate every object in the universe by .5c three times in a row. Now the entire universe is moving at 1.5c WRT the original condition, but there has been no contemporaneous change in relative velocity for any object in the universe. If such a process took place over a few billion years, who would be the wiser?

And another question comes to mind: Are we quite sure that no two objects in the entire universe have an aggregate relative velocity greater than c?

A third consideration: Imagine an empty, closed universe in which exists only a spacecraft with an impulse engine (http://chaos.fullerton.edu/~jimw/staif2000.pdf). The impulse engine is running, and has been running for an immeasurable period of time. The relative velocity of the craft is always 0, and yet the craft is experiencing a continuous acceleration from the impulse engine. (Or is it? In any case, energy is being expended, force is being applied, so one would expect that work was being performed). This is the conclusion that we reach if we say that the speed-of-light limitation applies only to an object's velocity relative to any other inertial frame. BUT, as soon as the spacecraft jettisons one pop can, the rules change, and the days of acceleration are numbered. Right?

So then, I don't think this is the right answer. But maybe someone can clear all this up for me.

Still puzzled.

 

[russ_watters]

In relation to anyone who measures your speed.

...and in relation to whatever object you choose to measure your speed from.

So, then, if we could apply a simultaenous acceleration to every object in the universe such that every object was accelerated by .7c...

What follows is Newtonian physics applied to a non-Newtonian situation. Sorry, OneEye, no, it just doesn't work that way.

And another question comes to mind: Are we quite sure that no two objects in the entire universe have an aggregate relative velocity greater than c?

Of course not - nor need we be. Thats why its important to understand what a "theory" is.

A third consideration: Imagine an empty, closed universe in which exists only a spacecraft with an impulse engine

Again, you're applying Newtonian physics to a non-Newtonian situation. Sorry, OneEye, no, it just doesn't work that way.

So then, I don't think this is the right answer. But maybe someone can clear all this up for me.

Nothing at all changes for those inside the spacecraft from before to after the soda can is jettisoned. They still feel the same acceleration force.

 

[OneEye]

What follows is Newtonian physics applied to a non-Newtonian situation.

I would like a little more elaboration on this, please. How am I applying Newtonian physics to a non-Newtonian situation? Is it because the velocities are very near light?

I am especially perplexed about the second model - the spaceship in the void. I cannot see the Newtonian aspects of this model - or rather, how the situation can be said to be either "Newtonian" or "Relativistic" - or why the consideration I posed can be said to invoke a Newtonian perspective.

A little help?

(And anyway, thanks for your time!)

 

[Doc Al]

So, then, if we could apply a simultaenous acceleration to every object in the universe such that every object was accelerated by .7c, then we have "upped the ante", and are now at liberty to accelerate any one object in the universe by (say) .4c relative to the rest of the universe, and this would be a legitimate velocity, even thought it is 1.1c WRT our original frame of reference? Or, perhaps, we could accelerate every object in the universe by .5c three times in a row. Now the entire universe is moving at 1.5c WRT the original condition, but there has been no contemporaneous change in relative velocity for any object in the universe.

I have no idea how one would accelerate every object in the universe (never mind simultaneously). But I think your point is: Can't one view relative velocities as additive? If ship B moves at 0.7c relative to ship A, and ship C moves at 0.7c relative to ship B, then doesn't ship C move at speed 1.4c relative to A?

The answer is no. Under SR, velocities do not simply add like they do under Newtonian physics.

Check out this thread to see examples of how velocities add under SR: https://www.physicsforums.com/showthread.php?t=16948

 

[OneEye]

I think your point is: Can't one view relative velocities as additive? If ship B moves at 0.7c relative to ship A, and ship C moves at 0.7c relative to ship B, then doesn't ship C move at speed 1.4c relative to A?

Thanks, Doc. No, I am not asking a question about adding velocities under SR. Didn't check your link, but Dr. Einstein has already acquainted me with the fact that:

[tex]W={ { v+w } \over { 1 + { vw \over c^2 } } } [/tex]

The question is, if c is a limiting velocity, then what is that limiting velocity relative to?

The reason that I proposed those peculiar cases was to eliminate from the model every consideration of relational velocity.

It seems to me that we have three basic ways of intepreting the idea of c as a limiting velocity:

1) c is the limiting relative velocity in the universe - i.e., no two bodies in the universe can have a relative velocity greater than or equal to c.

2) no individual body in the universe can exceed c in and of itself. This second idea is, I think, incompatible with a foundational tenet of SR to the effect that all velocity is relative velocity.

3) Some other possibility which has not yet been mentioned. ()

If feel caught in no-man's land between relative and absolute velocity. Can anyone help me out of here?

 

[selfAdjoint]

I think your option 1 is the prediction of relativity. Yes all velocity is relative and no relative velocity can be greater than c.

 

[russ_watters]

I would like a little more elaboration on this, please. How am I applying Newtonian physics to a non-Newtonian situation? Is it because the velocities are very near light?

Yes, that's it exactly.

I am especially perplexed about the second model - the spaceship in the void. I cannot see the Newtonian aspects of this model - or rather, how the situation can be said to be either "Newtonian" or "Relativistic" - or why the consideration I posed can be said to invoke a Newtonian perspective.

Unitil the can is released, its neither and either: it has no velocity, so while either can be used, neither tell us anything of value regarding velocity. You can certainly measure some forces of acceleration, but they don't tell you anything about your velocity.

Once you toss the can over the side, which situation you have depends on your velocity relative to the can. I don't know what would be considered the cutoff point (if any) where Einstein's is preferable.

The reason that I proposed those peculiar cases was to eliminate from the model every consideration of relational velocity.

...which we already told you is meaningless. All velocities are "relational" (relative).

So #1 is correct, #2 isn't so much wrong as it is meaningless.

If feel caught in no-man's land between relative and absolute velocity. Can anyone help me out of here?

Its simple but for some reason difficult to accept: "absolute velocity" quite simply does not exist.

Try this: how exactly would you measure "absolute velocity?" Distance over time? Where do you get the distance...?

 

[OneEye]

So #1 is correct, #2 isn't so much wrong as it is meaningless.

Okay, but it occurs to me that if we want to avoid "spooky action at a distance," we are going to have to say that "c is the unattainable upper limit of relative velocity within any event cone". Right?

Its simple but for some reason difficult to accept: "absolute velocity" quite simply does not exist.

Try this: how exactly would you measure "absolute velocity?" Distance over time? Where do you get the distance...?

I didn't really intend to get into that particular question. It is obvious that the concept of absolute velocity is a stumblingblock over which those who study SR routinely trip. As I indicated, I am not suggesting that absolute velocity exists - especially without some proposal as to how to measure it.

I had thought of such a method based on Dr. Einstein's book: A given particle ought, theoretically, to have a given mass. Particle mass can be determined by the traces produced in a particle accelerator. The difference between the detected mass and the theoretical mass should tell us the "true" velocity of the particle directly, and will probably have a derivable component which tells us the velocity of the observer's inertial frame. However, someone on the board stated that the concept of "relativistic mass" had been jettisoned some time ago (even during Dr. Einstein's lifetime), so apparently this method is not effective.

Other than that, I can think of no obvious way to measure absolute velocity. So, I must conclude that, whether absolute velocity exists or not, it appears to be a concept of no value - unless someone can think of a way to measure it.

 

[turin]

You do not need to toss a can out the window to realize that you are moving relative to something. Whince have you acquired the acceleration? Acceleration is not possible in the complete absense of any other bodies. I did not read that link about the impulse engine, but I did see extensive reference to Mach's principle, which hinges on the existence of some extraneous mass distribution (no other masses in the universe = trivialized Mach's principle).

In order to supply an impulse to the ship, something else in the universe must take on the negative of that impulse (conservation of momentum). Then, naturally, wrt this "something else" one can determine a velocity. This negative impulse could have been delivered to a soda can, to exhaust vapors, or even to photons. In any case, there is always something that provides a reference.

 

[OneEye]

Whince have you acquired the acceleration? Acceleration is not possible in the complete absense of any other bodies.

The link I provided was not for the impulse engine design I had in mind (and saw some years ago on the web). The design I was considering had a power unit, a tower proceeding from the unit, and a rotating arm assembly at the top of the tower. Power flowed to the rotating arm assembly, where it energized a coil at one exact position in the rotation of the arm (say, at the 0^o position). The energy in the coil produced a transient effective mass, but only when the coil was in the specific orientation - thus causing an acceleration at the energizing angle.

I have no idea whether this would work or not. And that's not really the point. I was simply trying to provide a mental model which would allow investigation of the statement that "the velocity c plays the part of an unattainable limiting velocity."

HTH.

 

[turin]

I was simply trying to provide a mental model which would allow investigation of the statement that "the velocity c plays the part of an unattainable limiting velocity."

Just think about what you have to do in order to accelerate. You have to push on (off of) something. Whatever that something is cannot be pushed to a speed faster than c relative to you. Therefore, relative to it, you cannot push yourself to a speed faster than c. Then, for succesive pushes, the SR velocity addition formula kicks in.

 

[OneEye]

Just think about what you have to do in order to accelerate. You have to push on (off of) something. Whatever that something is cannot be pushed to a speed faster than c relative to you. Therefore, relative to it, you cannot push yourself to a speed faster than c. Then, for succesive pushes, the SR velocity addition formula kicks in.

All right, this gives us another window into the question:

I and my twin are flying through space together with atomic bombs strapped to our backs. We pass near a MACHO, against which we (my twin and I) have a relative velocity of .99c. I touch off my bomb, and it gives me a boost of .1c relative to my twin. Or does it? Does the presence of the MACHO in my event cone prevent me from gaining the velocity boost which I would otherwise have gained should the MACHO have not been nearby?

 

[Doc Al]

I and my twin are flying through space together with atomic bombs strapped to our backs. We pass near a MACHO, against which we (my twin and I) have a relative velocity of .99c. I touch off my bomb, and it gives me a boost of .1c relative to my twin. Or does it?

Why wouldn't it? Your speed with respect to some other observer (the MACHO, or whatever) is irrelevant.

 

[turin]

OneEye,
It is probably better to consider it in terms of energy. That way, you avoid the problem of posing something like: "I have specified X. Is X true?"

You and your twin are at rest wrt each other (I'm assuming). Then, as y'all are sitting there chatting about how you are going to set off an atomic bomb on your back and survive, a MACHO flies by at 0.99c. You know this because you are equiped with a MACHOFSD (MACHO Fly-by Speed Detector). That startles you and makes you accidently press the red buttom. This gives you an increment of energy of:

ΔE ~ 0.005 m_youc²

which corresponds to a β of 0.1

How can the reading on the MACHOFSD effect this amount of energy? If you had pressed the red button out of resolve instead of accidentally, would the energy be less than 0.005 m_youc²?

 

[OneEye]

Why wouldn't it? Your speed with respect to some other observer (the MACHO, or whatever) is irrelevant.

Because...

...the statement would be that no material object (including you) can exceed the speed of light with respect to any inertial frame.

 

[Doc Al]

Because...

...the statement would be that no material object (including you) can exceed the speed of light with respect to any inertial frame.

Ah... but you won't exceed the speed of light with respect to the MACHO. Your puny 0.1c change in speed, when added relativistically to the 0.99c you were traveling--pre explosion--relative to the MACHO, makes your new speed relative to the MACHO just 0.9918c. No problem!

 

[OneEye]

Ah... but you won't exceed the speed of light with respect to the MACHO. Your puny 0.1c change in speed, when added relativistically to the 0.99c you were traveling--pre explosion--relative to the MACHO, makes your new speed relative to the MACHO just 0.9918c. No problem!

...hmmm... using the formula for the addition of velocity...

[tex]W={ { v+w } \over { 1+{vw \over c^2}}}[/tex]

...it is, in fact, impossible to produce a velocity [itex]W[/itex] greater than that of light.

However, I had thought that this formula was only to be applied when an observer K was watching a body B accelerate in frame K' when B had erstwhile shared the reference frame K'. The example given in Relativity is someone observing a passenger on a train who is walking down the aisle - which, under Newtonian mechanics, would be calculated as [itex]W=v+w[/itex], but in SR must be calculated under the formula shown above. Neither I nor my twin have ever been in contact with the MACHO, and so I am not sure that this formula for the addition of velocity applies here.

 

[outandbeyond2004]

Why not think about measuring distances especially those that change with time? To say that something is 5 miles long is meaningless without specifying what that thing is. Always for distance measurement you have to specify two points. It would be meaningless to say that Paris is five miles. The distance between Paris's center and London's center is about 300 km. If speed is always the rate at which the distance btw two given points is changing, then a speed specification always implies a particular relationship btw two points. It's always relative, after all, by definition! Besides, a pair of points may have the same relative velocity as many other pair of points. Just saying that the speed is blah blah is meaningless without an common understanding of which two points are in question.

A tenuous exception: it may be possible to "construct" a frame of reference in which the laws of physics attain their simplest form: a preferred frame of reference. When the CRB was measured, a dipole was found -- one half appeared redshifted and the other half blueshifted. It's as though the solar system is in motion wrt the CRB with a particular velocity. An observer at rest wrt the CRB would not "see" any such dipole. Whether one could treat the CRB rf as the "absolute rest" rf, I don't know. If it is, though, then all velocities measured wrt the CRB rf could be considered to be absolute velocities, because no two points are really specified. Yet, what does it mean to move blah blah wrt the CRB?

 

[HallsofIvy]

"1) c is the limiting relative velocity in the universe - i.e., no two bodies in the universe can have a relative velocity greater than or equal to c."

Yes, exactly. And that's all there is- all motion is relative to something. The point is the the relative velocity of anybody relative to any other body must be less than the speed of light.

"2) no individual body in the universe can exceed c in and of itself. This second idea is, I think, incompatible with a foundational tenet of SR to the effect that all velocity is relative velocity."

Yes- although as pointed out before, that's not just a "foundational tenet of SR". It is also fundamental to classical physics (so-called "Gallilean relativity"). What initiated special relativity was the discovery during the 19th century that magnetic force IS proportional to velocity which seemed to blow that out of the water.

 

[Doc Al]

However, I had thought that this formula was only to be applied when an observer K was watching a body B accelerate in frame K' when B had erstwhile shared the reference frame K'. The example given in Relativity is someone observing a passenger on a train who is walking down the aisle - which, under Newtonian mechanics, would be calculated as [itex]W=v+w[/itex], but in SR must be calculated under the formula shown above. Neither I nor my twin have ever been in contact with the MACHO, and so I am not sure that this formula for the addition of velocity applies here.

How is your example with the MACHO any different than Einstein's with the train? In the train example, the observer is someone on the ground. There is no requirement that this observer has to have "shared" the reference frame with either the moving train or the passenger.

In your example with the MACHO, I take the MACHO as the frame comparable to that of the ground observer, your twin is comparable to the moving train, and you are the passenger. So from the MACHO's inertial frame, your speed relative to the MACHO is calculated just like the ground observer calculates the speed of the walking passenger: using the relativistic addition of velocities formula.

 

[OneEye]

Why not think about measuring distances especially those that change with time? To say that something is 5 miles long is meaningless without specifying what that thing is. Always for distance measurement you have to specify two points. It would be meaningless to say that Paris is five miles. The distance between Paris's center and London's center is about 300 km. If speed is always the rate at which the distance btw two given points is changing, then a speed specification always implies a particular relationship btw two points. It's always relative, after all, by definition! Besides, a pair of points may have the same relative velocity as many other pair of points. Just saying that the speed is blah blah is meaningless without an common understanding of which two points are in question.

A tenuous exception: it may be possible to "construct" a frame of reference in which the laws of physics attain their simplest form: a preferred frame of reference. When the CRB was measured, a dipole was found -- one half appeared redshifted and the other half blueshifted. It's as though the solar system is in motion wrt the CRB with a particular velocity. An observer at rest wrt the CRB would not "see" any such dipole. Whether one could treat the CRB rf as the "absolute rest" rf, I don't know. If it is, though, then all velocities measured wrt the CRB rf could be considered to be absolute velocities, because no two points are really specified. Yet, what does it mean to move blah blah wrt the CRB?

outandbeyond2004: What is a CRB, and do you have a reference for this dipole measurement? It sounds very interesting!

 

[OneEye]

How is your example with the MACHO any different than Einstein's with the train? In the train example, the observer is someone on the ground. There is no requirement that this observer has to have "shared" the reference frame with either the moving train or the passenger.

In your example with the MACHO, I take the MACHO as the frame comparable to that of the ground observer, your twin is comparable to the moving train, and you are the passenger. So from the MACHO's inertial frame, your speed relative to the MACHO is calculated just like the ground observer calculates the speed of the walking passenger: using the relativistic addition of velocities formula.

Yes, you are quite correct.

The MACHO is the observer on the embankment. My twin and I are in the "train" frame of reference. My atomic thrust is the added velocity (w). v is the relative velocity between the MACHO and my twin and I (prior to the atom bomb thrust).

The professor answers the sophomore thus:

"No matter what force is applied to any object, there is no frame of reference in which it will observed that the accelerated object attains the speed of light, because the relativistic formula for the addition of velocity can never produce a velocity equal to that of light. So, whatever happens to the accelerated object in itself is immaterial, because no one will ever observe an acceleration which produces a velocity greater than that of light."

Have I expressed that clearly and correctly?

(I am aware that, if v=0 and w=c, then the end result is c, but this case is impossible to obtain, since this would require an infinite amount of energy, and is probably not physically possible anyway.)

 

[turin]

Have I expressed that clearly and correctly?

It would seem so. I have just a couple of picky adjustments to your (the person's to whomever the quote belongs) choice of wording:

"No matter what force is applied to¹ any object, there is no frame of reference in which it will observed that the accelerated object attains the speed of light, because the relativistic formula for the addition of velocity can never produce a velocity equal to that of light².

¹ Replace with: "acceleration is experienced by"
The explanation can be entirely kinematical and does not require an appeal to dynamics.

² Replace with: ". The relativistic formula for the addition of velocities shows how this can maintain a consistent set of kinematical laws"
No formula ever requires anything of the physical universe. It is the other way around. The universe has spoken (in its own mysterious language), and the physicists have interpretted that into the relevant kinematical equation. The expressions of a relativistic formula do not influence the universe (as far as I know).

 

[outandbeyond2004]

CRB = Cosmic Radiation Background.

Nice graphic in this webpage showing the dipole:
http://csep10.phys.utk.edu/astr162/lect/cosmology/cbr.html

 

[OneEye]

outandbeyond2004: Fascinating!

So, then, given that the CBR is anisotropic, it might be taken to provide a sort of "grid" to the entire universe against which any other velocity or position might be measured - the dreaded K₀ and v₀?

Need to grind on that for a while!

 

[turin]

outandbeyond,
Do you know of any evidence of objects in the universe tending to slow down under the anisotropy of the impingent momentum of this radiation? Do you know of any direct physical effects attributed to bombardment by this radiation, specific to its wavelength? Do you have a link to an explanation (of can you give one yourself) of "galactic coordinates?"

 

[outandbeyond2004]

Well, a photon colliding with an electron, say, often would give the electron pause <smile>. Compton scattering and Thomson scattering. Rayleigh scattering is proportional to the 4th power of the frequency. Other kinds of scattering possible. Come to think of it, you want Thomson scattering; Compton scattering is for high-energy photons.

http://farside.ph.utexas.edu/teaching/jk1/lectures/node85.html

Let me know if this is too technical.

 

[OneEye]

outandbeyond,
Do you know of any evidence of objects in the universe tending to slow down under the anisotropy of the impingent momentum of this radiation? Do you know of any direct physical effects attributed to bombardment by this radiation, specific to its wavelength? Do you have a link to an explanation (of can you give one yourself) of "galactic coordinates?"

Of course, the fact that we can measure the CBR indicates a rather ordinary interaction with, say, earthbound observers.

The fact that we can map the CBR implicates it as a reference frame.