The Speed of Light -- Why can an object not travel faster than the speed of light?

[Jiyong Chung]

TL;DR Summary

A basic question on why an object cannot move faster than the speed of light

I apologize if this question is in the wrong forum section - but I could not find a proper place for what is a basic question or perhaps a nonsensical one.

Is the reason why an object cannot travel faster than the speed of light because the object itself is ultimately made of waves? Is the speed of the object merely the group velocity of the waves that comprise the object?

 

[Orodruin]

No.

 

[Jiyong Chung]

No.

Thanks! A few more questions (if you don't mind) - Is it even correct to think that an object is comprised of waves (e.g., at the quantum level)? Does a wave function (in the context of Schrodinger's equation) have a speed limit, and is that the speed of light also?

 

[PeroK]

Thanks! Few more questions (if you don't mind) - Is it even correct to think that an object is comprised of waves (e.g., at quantum level)? Does a wave function (in the context of Schrodinger's equation) have a speed limit, and is that she speed of light also?

The Schrodinger equation (SDE) is the basis of non-relativistic quantum mechanics. The constraints of relativity are not, therefore, present in the SDE.

The wave-function describes the dynamic properties of a particle, so it's not helpful to think of a particle as made of waves.

 

[Ibix]

Travelling at the speed of light turns out to be just a different way of saying "has zero mass". Exceeding the speed of light would, similarly, be a way of saying "has imaginary mass" (if such a thing makes sense, which I'm not sure it does). Travelling slower than light turns out to mean "has non-zero mass".

Since your mass is a real non-zero number, you are traveling slower than light.

 

[Mister T]

Summary:: A basic question on why an object cannot move faster than the speed of light

There is either a maximum speed or there isn't. We live in a universe where there is.

 

[kent davidge]

The wave-function describes the dynamic properties of a particle, so it's not helpful to think of a particle as made of waves

thanks for reminding us of this. people too often talk bout particles being waves in quantum mechanics, as if the particles were just a type of wave packet moving in space.

 

[strangerep]

Summary:: [...] why an object cannot move faster than the speed of light

Possibly, this should have been tagged as a B-level thread, but since you've tagged it I-level, I'll give you an I-level answer.

Essentially,...
If there exist inertial reference frames in which the observer feels no acceleration, so the local equation of motion is $d^2 x^i/dt^2 = 0$, and the laws of physics are essentially the same in all inertial reference frames, (no inertial reference frame is somehow distinguished as special),
Then there exists an invariant maximum relative speed constant between inertial reference frames (whose spacetime origins and spatial frame axes have been arranged to coincide). The only question (which is for experimental measurement to find out) is the value of that maximum speed constant.

The above follows mathematically by group theoretic analysis of the symmetry transformations of the equation of motion I wrote above.

For ordinary inertial observers, that maximum relative speed has to be regarded in the sense of a mathematical limit: one can boost ever closer to that limit, but not actually attain it. If something (here called "light") is traveling at that limit speed relative to an ordinary inertial observer, there is no boost operation that can change that relative speed. I.e., the maximum relative speed is a "fixed point" in the parameter space of the transformation group.

The 2nd postulate of special relativity, i.e., the so-called "light principle", is not actually necessary.

So,... to your original question: it has nothing to do with whether we're dealing with "particles" or "waves". It's simply a consequence of the notion of physical equivalence of inertial reference frames, (and some technical assumptions about continuity and differentiability on an open neighbourhood of the observer's chosen coordinate origin).

HTH.

 

[PeterDonis]

The above follows mathematically by group theoretic analysis of the symmetry transformations of the equation of motion I wrote above.

Actually, this isn't quite correct unless you include the possibility that the "maximum speed" is infinite. That is the possibility realized in Newtonian physics; the other possibility, a finite maximum speed, is the one realized in relativity.

 

[strangerep]

Actually, this isn't quite correct unless you include the possibility that the "maximum speed" is infinite. That is the possibility realized in Newtonian physics; the other possibility, a finite maximum speed, is the one realized in relativity.

Counter-actually ( ), it is correct if one includes a little more detail. In particular, the constant that emerges naturally from the analysis -- I'll call it $\lambda_v$ -- has dimensions of inverse velocity squared. Hence most people (a bit carelessly) take the next step of defining $\lambda_v = 1/c^2$.

The possible value $\lambda_v = 0$ is well-defined (unlike velocity = $\infty$), and allowed by the analysis.

So this analysis does indeed allow for the Galilean case directly, all in a well-defined manner, unless one tries to write $c = 1/\sqrt{\lambda_v}$ which is $1/0$. Instead, one should simply put $\lambda_v = 0$ to find the transformation equations for that case. Then all is well.

 

[Jiyong Chung]

The wave-function describes the dynamic properties of a particle, so it's not helpful to think of a particle as made of waves.

This seems like a compelling statement when it comes to the math.

When you look at different experiments, though, such as the double slit experiment, oil-droplet experiments (in support of the pilot-wave theory) and see visual effects of the wave functions, there seems to be something physical about the wave functions - they "exist."

This brings me to the next question. The fact that wave functions physically exist - this seems to support the notion that the free space SUPPORTS the waves of the wave function. But then, the free space also supports the waves of the wave equation.

So, let us say that there is a photon - Is there any physical relationship between the wave function of the photon and its wave equation? Do they describe the same or similar aspect of the "same thing"? Based on what you said above, the answer is no, right?

 

[PeroK]

This seems like a compelling statement when it comes to the math.

When you look at different experiments, though, such as the double slit experiment, oil-droplet experiments (in support of the pilot-wave theory) and see visual effects of the wave functions, there seems to be something physical about the wave functions - they "exist."

This brings me to the next question. The fact that wave functions physically exist - this seems to support the notion that the free space SUPPORTS the waves of the wave function. But then, the free space also supports the waves of the wave equation.

So, let us say that there is a photon - Is there any physical relationship between the wave function of the photon and its wave equation? Do they describe the same or similar aspect of the "same thing"? Based on what you said above, the answer is no, right?

First, in physics you have physical phenomena, which is what you observe in experiments. Diffraction, for example. Those phenomena may be explained by a mathematical model. It's not then generally logical to assume that the entities in the model exist. For example:

In classical EM you have the electric and magnetic fields - you might assume, therefore, that these fields must physically exist. After all, you can see the effects of the electric and magnetic fields in experiments. But, then you have QED (quantum electrodynamics), which describes the same things through Feynman diagrams and the interaction of electrons exchanging virtual photons. Then, do you assume that the Feynman processes really happen and the virtual photons must exist? And now the electromagnetic fields no longer exist?

It doesn't make a lot of sense to me to say that when you are using classical EM to describe phenomena that the electric and magnetic field must exist because you can see their effects. And then when you use QED you say that the theoretical elements of QED really exist.

What if in ten years a new, better theory (perhaps encompasing quantum gravity) is developed that doesn't use the machinery of wavefunctions and photons? Is it the case that all of sudden these things no longer exist? Does the wavefunction cease to exist if a better mathematical model comes along?

In general, all these things (photons, wavefunctions, gravitational potential) are all theoretical tools to help describe what we observe. It doesn't make a lot of sense to start claiming that these things "really exist" in some sense.

 

[Jiyong Chung]

If there exist inertial reference frames in which the observer feels no acceleration, so the local equation of motion is $d^2 x^i/dt^2 = 0$, and the laws of physics are essentially the same in all inertial reference frames, (no inertial reference frame is somehow distinguished as special),
Then there exists an invariant maximum relative speed constant between inertial reference frames (whose spacetime origins and spatial frame axes have been arranged to coincide). The only question (which is for experimental measurement to find out) is the value of that maximum speed constant

That the speed of light is the same in all inertial frames - this is essentially an empirical fact, and not a theoretical one, correct?

I suppose I'm asking - do the relativistic equations follow from the postulate that the speed of light is the same in all inertial frames, or is there another theoretical underpinning from the postulate is derived?

 

[Ibix]

That the speed of light is the same in all inertial frames - this is essentially an empirical fact, and not a theoretical one, correct?

Yes. As strangerep explains, you can derive most of the way to special relativity from just the principle of relativity. You are left with one constant (the one he calls $\lambda_v$) that you may set to zero (in which case you get Newtonian physics) or non-zero (in which case you get relativity).

Ultimately all of science is underpinned by observation. That's what makes it science, rather than philosophy.

 

[Jiyong Chung]

In general, all these things (photons, wavefunctions, gravitational potential) are all theoretical tools to help describe what we observe. It doesn't make a lot of sense to start claiming that these things "really exist" in some sense.

Thanks for the reply. Actually, I understand what you are saying.

 

[Jiyong Chung]

All - thanks for your replies and your patience. I know how frustrating or annoying a non-layperson's questions can be. I'm happy I learned few things!

 

[strangerep]

do the relativistic equations follow from the postulate that the speed of light is the same in all inertial frames, or is there another theoretical underpinning from the postulate is derived?

That "light postulate" is only used in the modern era as a shortcut to get to the equations of special relativity (SR) quicker. As I explained earlier, only the Relativity Postulate (that inertial observers all perceive the same laws of physics), together with an assumption of spatial isotropy and the technical assumptions I mentioned. Here's Rindler's statement of the Relativity Postulate:

The Relativity Principle: The laws of physics are identical in all inertial frames, or, equivalently, the outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial frame.

Moving to your next question,

When you look at different experiments, though, such as the double slit experiment, oil-droplet experiments (in support of the pilot-wave theory) and see visual effects of the wave functions, there seems to be something physical about the wave functions - they "exist."

That's a question that probably belongs in the QM foundations forum.

This brings me to the next question. The fact that wave functions physically exist - this seems to support the notion that the free space SUPPORTS the waves of the wave function. But then, the free space also supports the waves of the wave equation.

This is not as clear-cut as you might think. (See also the Einstein quote in my signature block below.)

A proper description of multiple quantum "particles" requires a tensor product of their individual state spaces. Each state space corresponds to what's known as a "unitary irreducible representation" of the Poincare group (i.e., the symmetry group of SR).

For multiple quantum particles in the relativistic case, it's best to pass over to quantum field theory, where (eventually) one finds that "particles" are not really particles, but neither are they conventional waves. Rather, one works with excitations of a new concept called a "quantum field".

So, let us say that there is a photon - Is there any physical relationship between the wave function of the photon and its wave equation? Do they describe the same or similar aspect of the "same thing"? Based on what you said above, the answer is no, right?

There are serious difficulties in the mathematical modeling of "a photon" -- usually the mere mention of the word "photon" is kinda banned in the relativity forum. In particular, the position operator for the photon field is problematic. So my layman-level answer to that question is that both concepts are subsumed under the more general concept of "quantum field".

 

[PeterDonis]

That's a question that probably belongs in the QM foundations forum.

More than probably--it definitely does.

 

[vanhees71]

I think the most logical mathematical statement is the following: Given the equivalence of all inertial frames (the special principle of relativity) together with the assumption Euclicity of space for all inertial observers leads to two possible spacetime models: the Galilei-Newton spacetime (a fiber bundle describing Newton's postulates of absolute space and absolute time) and the Einstein-Minkowski spacetime (a pseudo-Euclidean affine manifold). The latter implies a universal "limiting speed", $c$.

Whether or not $c$ is "the speed of light in vacuo" is a matter of empirical evidence. It's tested in terms of measuring the (invariant) mass of the photon. The today accepted upper bound is $m_{\gamma}<10^{-18} \text{eV}$.