Exactly Why C is impossible for Massive Objects?

[AndromedaRXJ]

not being able to reach c comes first, what follows is the measurements/calculations. For example, does it even makes sense to say "it would require infinite energy to reach c"? The question is; why mass cannot "go" exactly c?

That did cross my mind, but I guess I thought it was acceptable to ask it that way.

Rephrasing the question; why does any finite amount of energy(or force) applied to the particle always result in a speed less than c? And is it energy or force? And does this only apply to a change in speed but not necessarily a change in velocity?

 

[mfb]

So then inertia only increases with a change in speed but not a change in velocity in Special Relativity, right?

A change in speed is always a change in velocity.

I'm trying to understand where the need for infinite energy(or force) comes from to reach c.

$$E=\gamma m c^2 = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}$$ As v approaches c, the energy grows without bounds. In other words, every finite energy corresponds to a velocity below c.

You can derive similar equations like $F=m \gamma^3 a$ or, equivalently, $a=\frac{F}{m \gamma^3}$. As the velocity increases, γ increases so quickly that you never reach c, independent of the force and acceleration track you have.

 

[nitsuj]

That did cross my mind, but I guess I thought it was acceptable to ask it that way.

Rephrasing the question; why does any finite amount of energy(or force) applied to the particle always result in a speed less than c? And is it energy or force? And does this only apply to a change in speed but not necessarily a change in velocity?

It totally is "acceptable" to ask it the way you did. But it is important to "see" the perspective of the inquiry.

I wish I could answer the question from the perspective you are asking, I only "understand" SR geometrically. I don't know the energy / mass equivalence well enough yet to explain c with it the same way I can with length/time. Saying mass "stays still-length" and energy "moves-time" doesn't clarify anything lol,

 

[Samshorn]

The very definition of a "frame" (by which you must mean a standard inertial frame, since otherwise the "speeds" defined in terms of it need not satisfy that composition formula) is based on the dynamics and inertia of physical entities.

Whose definition would that be?

Yours, I would think. You're not claiming you can define an inertial frame without reference to inertia (dynamics), are you?

I don't see any self contradiction in saying that one can talk about positions and their rate of change of position with respect to time (velocities) without reference to the masses or forces acting on the bodies (dynamics).

Again, the velocities you're talking about are defined only in terms of an inertial frame, which is defined based on the inertial behavior (dynamics) of physical objects and phenomena. Without that connection, there is no physical meaning to equations such as the velocity composition formula. (Obviously that formula doesn't apply to any arbitrary kinematically defined system of coordinates.)

Early relativity courses talk about the kinematics first (the description of position and velocities, and how they transform). Later on in the course dynamics will often be introduced.

Those courses are just witlessly mirroring a well-known malapropism in Einstein's 1905 presentation, where he begins with a section labeled "Kinematical Part", and yet the very first sentence is "Let us take a system of coordinates in which the equations of Newtonian mechanics hold good", and the entire discussion that follows consists of statements that are true only on the basis of those dynamically defined inertial coordinates. Indeed that operational dynamical basis is the only thing that gives physical meaning to those statements. Thus the entire discussion is based squarely on mechanics (dynamics). What he meant by "kinematical" was really just that in this section he establishes the systems of measure for space and time, not that these were to be defined without reference to physical phenomena (any more than they are in Newtonian physics), which would be absurd.

Kinematically we can say the Sun goes around the Earth; it is only in terms of inertial coordinates that we can say unequivocally the Earth goes around the Sun (more or less). Kinematically the two twins are on a symmetrical footing - they separate and come back together; it is only by distinguishing between inertial and non-inertial paths that we can say one twin ages more than the other. The speed of light can obviously take arbitrary values in terms of arbitrary kinematic coordinate systems; it is only in terms of an inertial coordinate system that we can say the speed of light is c.

You expressed your "kinematical" statements in terms of speeds defined in terms of inertial frames, so I suggest you think about what you really mean by "speed" and "inertial frame", and if it's possible to give them physical meaning without invoking dynamics and inertia. (It isn't.)

 

[AndromedaRXJ]

A change in speed is always a change in velocity.

I know, but a change in velocity doesn't have to be a change in speed.

That's why I brought up the particle going in a circle at a constant speed of 0.99c. Is it's inertia growing due to it's circular path?

$$E=\gamma m c^2 = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}$$ As v approaches c, the energy grows without bounds. In other words, every finite energy corresponds to a velocity below c.

You can derive similar equations like $F=m \gamma^3 a$ or, equivalently, $a=\frac{F}{m \gamma^3}$. As the velocity increases, γ increases so quickly that you never reach c, independent of the force and acceleration track you have.

So would γ increase to a change in direction?

 

[PAllen]

and yet the very first sentence is "Let us take a system of coordinates in which the equations of Newtonian mechanics hold good",

Oh, then we can say there is no such thing as an inertial frame, because there is no frame where Newtonian mechanics is exactly true (and no, I am not bringing in gravity)? Taking this statement as definitional is itself circular because the point is to derive how Newton's laws of motion need modification in SR. You have to arrive at a new definition of momentum in an inertial frame that is conserved, and the conservation is preserved under the already derived Lorentz transform.

 

[PAllen]

I know, but a change in velocity doesn't have to be a change in speed.

That's why I brought up the particle going in a circle at a constant speed of 0.99c. Is it's inertia growing due to it's circular path?




So would γ increase to a change in direction?

No and no.

 

[Samshorn]

Oh, then we can say there is no such thing as an inertial frame, because there is no frame where Newtonian mechanics is exactly true (and no, I am not bringing in gravity)?

No, that's why Sommerfeld added the famous clarifying footnote to that statement in Einstein's paper, saying "to the first approximation", by which he meant quasi-statically, which correctly establishes the physical basis, i.e., the coordinate systems Einstein is referring to are those in which mechanical inertia is homogeneous and isotropic.

Taking this statement as definitional is itself circular because the point is to derive how Newton's laws of motion need modification in SR.

That would be true without the footnote. Einstein's statement, as clarified by Sommerfeld's footnote, is correct, and provides the operational basis leading to the need to modify Newton's laws (but not quasi-statically).

You have to arrive at a new definition of momentum in an inertial frame that is conserved, and the conservation is preserved under the already derived Lorentz transform.

The Lorentz transformation is the relationship between relatively moving systems of inertial coordinates, which are defined operationally as coordinate systems in which mechanical inertia is homogeneous and isotropic. Such coordinate systems exist (empirically), and when we impose the requirement that the finite speed c is invariant in all such coordinate systems, it follows that they are related by Lorentz transformations. (We don't define inertial coordinate systems as systems related by Lorentz transformations; that would be false - take any arbitrary coordinate system and apply the Lorentz transformation to get another coordinate system... this doesn't make either of them inertial coordinate systems.)

 

[PAllen]

The Lorentz transformation is the relationship between relatively moving systems of inertial coordinates, which are defined operationally as coordinate systems in which mechanical inertia is homogeneous and isotropic. Such coordinate systems exist (empirically), and when we impose the requirement that the finite speed c is invariant in all such coordinate systems, it follows that they are related by Lorentz transformations. (We don't define inertial coordinate systems as systems related by Lorentz transformations; that would be false - take any arbitrary coordinate system and apply the Lorentz transformation to get another coordinate system... this doesn't make either of them inertial coordinate systems.)

It seems to me, the only aspect of inertial needed is that I can place a configuration of clocks in space, unsupported, and they stay put. At my leisure, I measure (with a reference object) their mutual distances, and they stay the same over time. Then using Einstein synchronization, I synchronize the clocks. I claim I now have an inertial coordinate system without imposing any requirements on motion of bodies in this frame.

It is a non-trivial feature of the universe that such frame can be established at all. It is also non-trivial that infinitely many different such frames can be established.

Having established such frames, I can then verify empirically SR laws of mechanics (in their standard inertial expression) hold in each of them.

This was basically the path to Newtonian mechanics as well (with slow clock transport). You seem to be saying you cannot actually discover laws of mechanics.

 

[Samshorn]

It seems to me, the only aspect of inertial needed is that I can place a configuration of clocks in space, unsupported, and they stay put. At my leisure, I measure (with a reference object) their mutual distances, and they stay the same over time. Then using Einstein synchronization, I synchronize the clocks.

Light is an inertial phenomena, i.e., light has momentum (as does every form of energy), and it's propagation is part of electrodynamics. So you are using the dynamics of an inertial entity to define your coordinate system. Hence this serves just as well to make the point I was making to prevect. Inertial frames are not (and can not) be meaningfully defined without reference to the behavior of some inertial phenomenon.

I claim I now have an inertial coordinate system without imposing any requirements on motion of bodies in this frame.

We can choose to call that an "inertial coordinate system", since it is indeed based on the inertial dynamics of electromagnetism, but of course there was already a pre-existing definition of inertial coordinate systems based on mechanics, namely, the coordinates in terms of which mechanical inertia is homogeneous and (at least quasi-statically) isotropic. The latter are the systems in which (as Einstein put it) "the equations of Newtonian mechanics hold good (at least quasi-statically)". The point is that these two definitions turn out to give precisely the same set of coordinate systems - which is not surprising, since they both represent coordinates in which inertia (mechanical or electromagnetic or anything else) is homogeneous and isotropic.

Having established such frames, I can then verify empirically SR laws of mechanics (in their standard inertial expression) hold in each of them.

Sure, but that's backwards from what actually happened. We already had a definition of inertial coordinate systems based on mechanical dynamics, and then verified empirically (and to the surprise of the experimenters involved) that the speed of light is c in terms of any such coordinate systems.

This was basically the path to Newtonian mechanics as well (with slow clock transport).

Slow clock transport is a red-herring in foundational discussions, since a clock is an ambiguous and high-level construction, and everything that's necessary can be expressed unambiguously in terms of the underlying basic laws that govern inertial behavior, i.e., the usual definitions of inertial coordinate systems based on the homogeneity and isotropy of inertia.

You seem to be saying you cannot actually discover laws of mechanics.

To the contrary, I'm describing how they were discovered (and what they mean).

 

[PAllen]

Slow clock transport is a red-herring in foundational discussions, since a clock is an ambiguous and high-level construction, and everything that's necessary can be expressed unambiguously in terms of the underlying basic laws that govern inertial behavior, i.e., the usual definitions of inertial coordinate systems based on the homogeneity and isotropy of inertia.

I don't see slow clock transport as a red herring since inertial motion and conservation of momentum were discovered using measurement systems that did not assume them (every definition you've given of inertial frame defines them in terms of laws of motion; so how do you discover the laws of motion?). Take my starting definition of inertial frame and use slow clock transport instead of light signals. Now you could (in principle) discover that c is isotropic and the same in all such frames; that to high precisions SR mechanics holds; and to lower precision Newtonian mechanics holds.

Basically, unless you admit there is a way to set up coordinates in which speed, distance, and time can be measured independent of laws of motion, you are saying laws of motion cannot be [directly] discovered.

 

[Samshorn]

...inertial motion and conservation of momentum were discovered using measurement systems that did not assume them.

Not true. As discussed in the vast literature on this well-worn subject, our intuitive sense of space and time has always been such that the description of phenomena is as symmetrical as possible, so we have always implicitly used inertia as the basis for resolving and coordinating events. Likewise it's well known that Newton's "laws" actually constitute the definition of an inertial coordinate system. For example, two identical particles initially adjacent and at rest, and exerting mutual repulsive force on each other, will reach equal distances in equal times. This is the "law" of action and re-action, the conservation of momentum... but of course it is not true - unless we define our system of space and time coordinates to make it true. Notice that Newton's third law implies a specific synchronization of time at separate locations. Why do we assert that synchronization? Well, because it makes Newton's third law true! So, up to this point, Newton's third law, asserting the isotropy of mechanical inertia, is tautological - just as it is tautological to assert that the speed of light is isotropic in terms of coordinates defined such that the speed of light is isotropic. But...

The significance of these seemingly tautological assertions/definitions is that, once we've defined a system of coordinates in which the given property is satisfied for one pair of particles, we find that it is satisfied for all pairs of particles. So what began as a definition, ends as an empirical law. (As Newton said, the whole burden of philosophy is to infer patterns from phenomena, and then using those patterns to predict other phenomena.) By the same token, newbies often complain that defining simultanteity as whatever it needs to be to make light speed isotropic implies that the isotropy of light speed is purely tautological, but, again, the utility of the definition is that it correlates a wide range of conceivably independent phenomena.

I don't see slow clock transport as a red herring...

Einstein himself discussed the fallacy of invoking the concept of "clock" for foundational purposes (and apologized for having done it himself in his early writings!), since it is an ambiguous high-level concept, whereas the definitions of inertial coordinates are derived from the basic unambiguous low-level concepts of isotropy and homogeneity of inertia.

Basically, unless you admit there is a way to set up coordinates in which speed, distance, and time can be measured independent of laws of motion, you are saying laws of motion cannot be discovered.

Nope. See above (and any good book on the foundation of mechanics). The "laws of motion" serve first as definitions of the coordinates, and then as laws. The epistemology of physical theories is more subtle than most people realize.

 

[PAllen]

Einstein himself discussed the fallacy of invoking the concept of "clock" for foundational purposes (and apologized for having done it himself in his early writings!), since it is an ambiguous high-level concept, whereas the definitions of inertial coordinates are derived from the basic unambiguous low-level concepts of isotropy and homogeneity of inertia.

Einstein said many different things at different times. If you can accept the a generic object can be compared to a reference object to measure length, then I see no issue with introducing a reference or ideal clock for time.

Despite your efforts, I do not find your argument at all convincing. Laws of motion were discovered using methods of measurement that did not assume them.

 

[pervect]

Not true. As discussed in the vast literature on this well-worn subject, our intuitive sense of space and time has always been such that the description of phenomena is as symmetrical as possible, so we have always implicitly used inertia as the basis for resolving and coordinating events.

I'd have to disagree. I would say that inertia, and hence inertial frames, are not a part of kinematics, but of dynamics.

If we are simply specifying the motion of bodies, without regards to the cause of motion, we do not yet need inertia. Whether or not there are forces acting on a body to hold it at rest doesn't matter to the kinematical expression of its state of motion (in this case, "at rest").

This seems pretty clear to me from the dictionary definition of the terms. You seem to be convinced that we need inertia to do kinematics, I remain unconvinced.

You mention"vast and well-worn literature on the topic", but I don't see any citations that would convince me otherwise.

An abstract way of making my point would be to note that physical bodies move along time-like worldlines, and that the relative velocity between any two timelike worldlines, (assuming the Minkowskii metric of special relativity), is less than or equal to "c".

I would say that the statement that physical bodies move along time-like worldlines is still a kinematical statement, as it only describes the motion.

 

[Samshorn]

If you can accept the a generic object can be compared to a reference object to measure length, then I see no issue with introducing a reference or ideal clock for time.

Well, as a matter of fact, Einstein applied the same apology to the use of "measuring rods" as to the use of "clocks", and for the same reason. I'm not sure why you are having trouble grasping this. Ask yourself how you define a "clock". Obviously you don't have in mind a sun-dial, or even a pendulum clock (in most cases), but you might be thinking of a spring and balance clock (as long as it isn't shaken too violently) - which works based on the principles of mechanical inertia so it is entirely redundant to the coordinates defined using the isotropy and homogeneity of mechanics, and hence introducing a rube-goldberg contraption like a spring-balance clock is just a red herring, with needless ambiguities and caveats. Or you might be thinking of a "light clock", but that is obviously redundant to the coordinates defined using the assumption of isotropic light speed, so again the rube-goldberg contraption is a red herring. Now, you might even be thinking of a cesium clock, or some other quantum phenomenon, but this just relies on the basic "equation of motion" of quantum mechanics, according to which the advance of the phase of the quantum wave function is taken to be proportional to proper time. So yet again the rube-goldberg contraption is a red herring.

The point is that a "clock" will give a suitable measure of time only if the basic laws governing the behavior of the parts of the clock already entail a suitable measure of time, so it makes no sense to introduce all the ambiguities and complexities and caveats associated with some notion of a "clock" to define your measures of space and time.

Despite your efforts, I do not find your argument at all convincing. Laws of motion were discovered using methods of measurement that did not assume them.

Well, I just finished explaining why that's not true. Indeed the method that you yourself have proposed for defining coordinates (using the propagation of inertial objects consisting of pulses of electromagnetic radiation) does the very same thing (i.e., you define your measures of space and time based on the very phenomena that the laws of motion are intended to describe), so it's hard to understand why you persist in denying it. This really isn't a controversial point (except, apparently, in this forum): Newton's laws of motion serve first as effectively the definition of inertial coordinate systems. You haven't really offered anything to support your position, other than to continue repeating the weird and obviously false claim that our systems of measure are somehow 'a priori'.

Look, the "discovery of the laws of motion" consisted of people like Galileo and Newton noticing that it is possible to assign space and time coordinates to events in such a way that physical phenomena obey a simple set of laws. The laws and the coordinate systems in which they are true are part of the same discovery. If this is unfamiliar to you, you might be interested in reading some books about the foundations of mechanics.

 

[PAllen]

I'm not sure why you are having trouble grasping this. Ask yourself how you define a "clock".

The issue isn't grasping, the issue is disagreement. I grasp what you say and still disagree with it.

 

[Samshorn]

I'd have to disagree. I would say that inertia, and hence inertial frames, are not a part of kinematics, but of dynamics.

Just to be clear, you're agreeing with me, and disagreeing with yourself, right? Remember, your original claim was that inertial frames are part of kinematics, not dynamics. I disagreed, pointing out that inertial frames are obviously part of dynamics, not kinematics. So we're now in agreement (except for that fact that you interpret it as disagreement!)

If we are simply specifying the motion of bodies, without regards to the cause of motion, we do not yet need inertia. Whether or not there are forces acting on a body to hold it at rest doesn't matter to the kinematical expression of its state of motion (in this case, "at rest").

You're missing the point: You can't even talk about "speeds" (such as the speeds appearing in the composition formula) unless you specify an inertial frame (including the required simultaneity) in terms of which those speeds are defined (as you yourself previously agreed), and this (as we just now agreed) involves dynamics.

 

[Samshorn]

The issue isn't grasping, the issue is disagreement. I grasp what you say and still disagree with it.

But you haven't articulated any rational reason, or countered (or even addressed) any of the explanations I provide. That's what makes me characterize it as "failing to grasp" rather than "rationally disagreeing".

 

[PAllen]

But you are unable to articulate any rational reason. That's what makes me characterize it as "failing to grasp" rather than "rationally disagreeing".

No, I do articulate reasons that I could claim you do not grasp. The world could work that if rods of different materials are compared to have the same length in one orientation, they will not in a different orientation. We don't find that - instead we find that we can define length independent of material and laws of motion. Similarly, if we set up time using 5 different types of clocks, we get the same laws motion measured. This leads to the natural (except to you) notion that time and distance, and a rest frame (see earlier posts) can be defined that is independent of specific laws of motion and matter.

You will again disagree. To me, I could accuse you of not grasping. But I won't. It doesn't matter. We simply disagree.

 

[mfb]

Samshorn, PAllen: Please continue this discussion in a separate thread or via PM. Your private discussion is beyond the level of this thread.

That's why I brought up the particle going in a circle at a constant speed of 0.99c. Is it's inertia growing due to it's circular path?

No, it just depends on the velocity of the particle.

So would γ increase to a change in direction?

As you can see from the formula, it does not depend on the direction (v^2 is a scalar value).

 

[Agerhell]

I know, but a change in velocity doesn't have to be a change in speed.

That's why I brought up the particle going in a circle at a constant speed of 0.99c. Is it's inertia growing due to it's circular path?




So would γ increase to a change in direction?

For the case of electromagnetism and a small particle of charge q and mass "invariant mass" m accelerated in an electromagnetic field classically we have:

\frac{d}{dt}(m\bar{v})=q(\bar{E}+\bar{v}\times\bar{B})

Assuming "m" not to vary with time we get:

\frac{d\bar{v}}{dt}=\frac{q}{m}(\bar{E}+\bar{v}\times\bar{B})

Instead assuming that the classical momentum m\bar{v} in the first expression above should be replaced by its relativistic counterpart m\gamma\bar{v} where

\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}

and solving the differential equation we get:



\frac{{\rm d}\bar{v}}{{\rm d}t}=\frac{q(\bar{E}\cdot\hat{v})}{m\gamma}\left(1-\frac{v^2}{c^2}\right)\hat{v}-\frac{q}{m\gamma}((\bar{E}+\bar{v}\times\bar{B})\times\hat{v}) \times \hat{v}.

As you can se there is a factor of 1/\gamma also in the second term on the right side. This means that it is harder to accelerate a particle in an electromagnetic field also in the direction transverse to its velocity, such as for a circular orbit.

 

[harrylin]

No, that's why Sommerfeld added the famous clarifying footnote to that statement in Einstein's paper, saying "to the first approximation", by which he meant quasi-statically, which correctly establishes the physical basis, i.e., the coordinate systems Einstein is referring to are those in which mechanical inertia is homogeneous and isotropic.[..]

Sorry for this little off-topic question, but are you sure that it was Sommerfeld?
I assumed that it was Einstein who authorised that translation footnote, and so I have been wondering if Einstein had forgotten what he meant, as I was pretty sure that Einstein meant reference coordinate systems as used for Newton's mechanics, just as you also explain (and there is nothing "approximate" about that); according to me it should have been "are supposed to hold good". So, if it was not Einstein but someone else who put that footnote there, then that mystery is solved for me.

 

[harrylin]

[..]
That's why I brought up the particle going in a circle at a constant speed of 0.99c. Is it's inertia growing due to it's circular path?
[..]
So would γ increase to a change in direction?

γ is a function of speed, so that a particle that moves at constant speed in a circular orbit has constant γ and thus constant inertia.
Moreover, a particle can be deflected sideways in such a way that its speed remains constant. In such cases its kinetic energy remains constant, and thus also its inertia.