Proof of the Constancy of the Speed of Light

gabemaiberger

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So, in other words, a theory consists of multiple explanations whose validity can only be determined by repeatable and falsifiable tests. But that still does not make it a fact.
Science doesn't deal in facts. It is a method of knowing more than it is a body of knowledge. We can never prove hypotheses. We can only attempt to falsify them. When a hypothesis has failed to have been falsified through experimentation, it is provisionally accepted as a potential explanation for a phenomenon. When this happens enough times, the hypothesis is inductively proven due to a lack of falsification.
 
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gabemaiberger

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I don't think that he necessarily manipulated Maxwell's formulas to produce a favored outcome, but he certainly did make a few important "assumptions" as he made various substitutions/simplifications to Maxwell's equations, as sjastro rightly pointed out:



In other words, his mathematical 'proof' is only valid in a specific set of circumstances, specifically it only applies to the speed of light in free empty space where no charges or currents are present.

If one assumes a different set of conditions, like the speed of light through a dense material, or in the presence of charges or powerful currents, the speed of light will vary based on those different conditions.

It's a valid mathematical proof about the speed of light in the very simplified instance of empty space. The real "space" as it exists in reality however is full of charged plasma particles and currents. His proof is therefore a bit limited, and applies to oversimplified scenarios, but it's still mathematically valid in the specific conditions that sjastro listed. You and sjastro are correct that gabemaiberger should have included some type of verbal explanation, particularly in the instance instances where he actually deviated from (simplified) Maxwell's actual formulas.
I should have specified that I was using the 'Heaviside' form of Maxwell's Equations.
 
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gabemaiberger

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I don't think that he necessarily manipulated Maxwell's formulas to produce a favored outcome, but he certainly did make a few important "assumptions" as he made various substitutions/simplifications to Maxwell's equations, as sjastro rightly pointed out:



In other words, his mathematical 'proof' is only valid in a specific set of circumstances, specifically it only applies to the speed of light in free empty space where no charges or currents are present.

If one assumes a different set of conditions, like the speed of light through a dense material, or in the presence of charges or powerful currents, the speed of light will vary based on those different conditions.

It's a valid mathematical proof about the speed of light in the very simplified instance of empty space. The real "space" as it exists in reality however is full of charged plasma particles and currents. His proof is therefore a bit limited, and applies to oversimplified scenarios, but it's still mathematically valid in the specific conditions that sjastro listed. You and sjastro are correct that gabemaiberger should have included some type of verbal explanation, particularly in the instance instances where he actually deviated from (simplified) Maxwell's actual formulas.
Where did I deviate from the Heaviside form of Maxwell's Equations?
 
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I should have specified that I was using the 'Heaviside' form of Maxwell's Equations.

Well, more than that, you're making a series of assumptions which sjastro pointed out as you simplified the formulas. In other words, your "proof" is only valid in a very limited set of physical circumstances. As sjastro and others have suggested, you should probably explain how and why you simplified Maxwell's equations, otherwise it's bound to cause confusion.
 
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gabemaiberger

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Well, more than that, you're making a series of assumptions which sjastro pointed out as you simplified the formulas. In other words, your "proof" is only valid in a very limited set of physical circumstances. As sjastro and others have suggested, you should probably explain how and why you simplified Maxwell's equations, otherwise it's bound to cause confusion.
Okay, fair enough. For the first equation (Gauss' Law for electric fields), in a vacuum far enough away from charged objects, the divergence of an electric field (E) is zero because the electric charge density (rho) is zero. Zero divided by the nonzero permittivity of free space is zero. For the fourth equation (Maxwell's modification to Ampere's Circuital Law), in a vacuum far enough away from electrical current, magnetization fields, and time-varying electric dipoles, the curl of the magnetic field (B) is μ₀ε₀*(∂E/∂t) because the total current density (J) is zero. Zero multiplied by the nonzero permeability of free space (μ₀) is zero.
 
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sjastro

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And in such cases, you will fail to convince anyone of what you are hoping to convince them of because what you are presenting them will not make any sense unless your evidence can be presented in such a way as to make sense to everyone, and not just to a handful of elitists.
Argument from personal incredulity.
 
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And in such cases, you will fail to convince anyone of what you are hoping to convince them of because what you are presenting them will not make any sense unless your evidence can be presented in such a way as to make sense to everyone, and not just to a handful of elitists.


You think people are trying to convince you? Be ignorant, that is your own choice and no one is going to stop you. :wave:
 
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sjastro

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Entanglement, for example, categorically must exchange information faster than the speed of light (superlunimar).
No it doesn't as it violates the no communication theorem which states information exchange cannot occur in an entangled state.
Only when the measurement is made can information exchange at or below the speed of light occur when there is longer an entangled state.
Quantum mechanics can violate locality but not causality which is explained by the use of Minkowski space time diagrams if the speed of information exchange exceeds the speed of light.
 
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Kaon

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No it doesn't as it violates the no communication theorem which states information exchange cannot occur in an entangled state.
Only when the measurement is made can information exchange at or below the speed of light occur when there is longer an entangled state.
Quantum mechanics can violate locality but not causality which is explained by the use of Minkowski space time diagrams if the speed of information exchange exceeds the speed of light.


The independence and invariance of the speed of EM quanta for the relatively small scale (micro) are due to the math evolution of the rest mass and reduction. This is from measuring the speed of objects with respect to the critical speed as it appears. Using the Lorentz factor to prove the speed maximum of light is akin to using the very word you are looking to define to define the word - it is circular. It is "imprudent" to calculate the properties of objects with respect to light by saying they are measured with respect to light and naturally produce infinity when v approaches c. It is theoretically source-independent (with the highest limit with respect to itself).

The constant c is the inertial referential velocity, but its not actually the limit velocity, and we would only be able to vindicate either side by categorically measuring from outside (like an entanglement experiment) that would need to be superluminal. Source-independence shouldn't be enough for us because it isn't enough. If we really want to test superluminal speeds, we need some other superluminal inertial frame as a reference. Souce-independence accepts light quanta as its own metric, but that is part of what is making SR hard to coalesce with QM.

Transmitting a signal means transmitting momentum and energy (allegedly inseparable in the TOR) which are capable of switching off/on a certain "device or trigger; it works out that there is no preferred reference frame (but there may be one velocity in all of them), but there is a problem if we forget that finite-velocity EM quanta are preferred reference frames themselves in special relativity.


The ERP problem is solved if wave packets are localized not only with respect to one frame of reference but to others as well. When wave/particles are thus allowed to have superluminal
sped, it is because it already existed in nature before our scope. If causation is continuous wherever activity exist in space, then this is acceptable. Geometrical moments can be non-casual, non-localized and/or absolutely continuous at some points, so a physical wave/particle's discreteness may be proof of erroneous application of QM theory to real geometric discontinuities - but "casually" continuous space-time moments.

In other words, saying the speed of light is a maximum of c just shows the greenness of our understanding of physical phenomena in our universe.
 
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Ophiolite

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And in such cases, you will fail to convince anyone of what you are hoping to convince them of because what you are presenting them will not make any sense unless your evidence can be presented in such a way as to make sense to everyone, and not just to a handful of elitists.
I suggest there are four categories of people in matters such as this:
1. Individuals smart enough and committed enough to develop an understanding of such complex matters through experiment and hypothesis formation.
2. Individuals smart enough and committed enough to understand and, perhaps, explain the work done by the first group.
3. Individuals smart enough to recognise they don't belong in the first two groups and yet still smart enough to recognise the strength of the scientific method employed by the first two groups and so are ready to, provisionally, accept their findings.
4. Individuals who figure if they can't understand it, it's probably not true and who aren't smart enough to understand selection bias, Dunning-Kruger effect and the like.

I suggest that, in the matter of science, the members of group 4 could be safely disregarded had we not already made the mistake of giving them the vote. :)

(For the record I clawed my way into group 3. Does that make me an elitist? If so, may I wear the T-shirt?)
 
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Hans Blaster

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Okay, fair enough. For the first equation (Gauss' Law for electric fields), in a vacuum far enough away from charged objects, the divergence of an electric field (E) is zero because the electric charge density (rho) is zero. Zero divided by the nonzero permittivity of free space is zero. For the fourth equation (Maxwell's modification to Ampere's Circuital Law), in a vacuum far enough away from electrical current, magnetization fields, and time-varying electric dipoles, the curl of the magnetic field (B) is μ₀ε₀*(∂E/∂t) because the total current density (J) is zero. Zero multiplied by the nonzero permeability of free space (μ₀) is zero.

It wasn't so much the choice of equations, but the lack of description that was missing. When I saw you title then the vector calculus with E & B, I knew what you were up to and skipped to the end. Sure enough there was the speed of light!

What you posted in this message was a good description to go with the first four equations (Maxwell's equations in free space). Then followed by the rest "large" equations: "I do some substitutions and use some vector identities to get the wave equation in free space...". At that point the answer just falls out, since this is a well known form of the wave equation and were the terms "μ₀ε₀" is 1/c^2 and c is the wave propagation speed. (So, solving the wave equation wasn't strictly necessary.)

Since μ₀ε₀ has been measured, this shows that electromagnetic waves travel at the measured speed of light. Light is an electromagnetic wave, yay!

If I remember my relativity history correctly, it was from this that Einstein considered his postulate "The speed of light is the same for all observers." then using the Lorentz transformation, applied it and determined the consequences. The predictions of Einstein's work were then experimentally verified showing that indeed, Maxwell's equations and the speed of light are the same for all observers.

Carry on!
 
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Hans Blaster

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I suggest there are four categories of people in matters such as this:
1. Individuals smart enough and committed enough to develop an understanding of such complex matters through experiment and hypothesis formation.
2. Individuals smart enough and committed enough to understand and, perhaps, explain the work done by the first group.
3. Individuals smart enough to recognise they don't belong in the first two groups and yet still smart enough to recognise the strength of the scientific method employed by the first two groups and so are ready to, provisionally, accept their findings.
4. Individuals who figure if they can't understand it, it's probably not true and who aren't smart enough to understand selection bias, Dunning-Kruger effect and the like.

I suggest that, in the matter of science, the members of group 4 could be safely disregarded had we not already made the mistake of giving them the vote. :)

(For the record I clawed my way into group 3. Does that make me an elitist? If so, may I wear the T-shirt?)

Welcome to group 3! It is well deserved. (I thought it was a polo shirt with a logo, but if you have the T-shirt wear it with pride!) I hope that I can properly function in group 3 (at least where I'm not group 1) and am working on my group 2 status as appropriate. Group 4's are a confounding bunch, some of the won't even learn the right notation for the things they incorrectly think they understand. (*cough* "SN Ia" not "SN1A" *cough*)
 
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sjastro

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The independence and invariance of the speed of EM quanta for the relatively small scale (micro) are due to the math evolution of the rest mass and reduction. This is from measuring the speed of objects with respect to the critical speed as it appears. Using the Lorentz factor to prove the speed maximum of light is akin to using the very word you are looking to define to define the word - it is circular. It is "imprudent" to calculate the properties of objects with respect to light by saying they are measured with respect to light and naturally produce infinity when v approaches c. It is theoretically source-independent (with the highest limit with respect to itself).

The constant c is the inertial referential velocity, but its not actually the limit velocity, and we would only be able to vindicate either side by categorically measuring from outside (like an entanglement experiment) that would need to be superluminal. Source-independence shouldn't be enough for us because it isn't enough. If we really want to test superluminal speeds, we need some other superluminal inertial frame as a reference. Souce-independence accepts light quanta as its own metric, but that is part of what is making SR hard to coalesce with QM.

Transmitting a signal means transmitting momentum and energy (allegedly inseparable in the TOR) which are capable of switching off/on a certain "device or trigger; it works out that there is no preferred reference frame (but there may be one velocity in all of them), but there is a problem if we forget that finite-velocity EM quanta are preferred reference frames themselves in special relativity.


The ERP problem is solved if wave packets are localized not only with respect to one frame of reference but to others as well. When wave/particles are thus allowed to have superluminal
sped, it is because it already existed in nature before our scope. If causation is continuous wherever activity exist in space, then this is acceptable. Geometrical moments can be non-casual, non-localized and/or absolutely continuous at some points, so a physical wave/particle's discreteness may be proof of erroneous application of QM theory to real geometric discontinuities - but "casually" continuous space-time moments.

In other words, saying the speed of light is a maximum of c just shows the greenness of our understanding of physical phenomena in our universe.
Let me make some corrections to your post.
Firstly the Lorentz factor is not used to prove the speed of light is a maximum.
Its origin predated SR and was used as an ad hoc explanation to explain the null result of the MM experiment while preserving an absolute ether frame by suggesting that objects shrank in the direction of motion.
Einstein derived the Lorentz transformations in SR based on the postulate the speed of light is the same to all inertial observers, not as a speed limit.
Note in GR the speed of light is not an upper limit as the recession velocities of high z galaxies indicate.

Secondly combining SR with QM was a problem in the 1930s not now as explained in this post.

For the rest of your post there is absolutely no experimental evidence for the existence of a “superluminal inertial frame”.
Every single phenomenon that may have pointed to superluminal velocities ranging from a practical application of the lighthouse paradox to the phase velocity of an EM wave exceeding c has found no information transfer is possible.

Here is a video that goes into somewhat more detail as to why the no communication theorem prevents superluminal velocities for entangled particles.
 
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sjastro

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If you can't explain it simply, you don't understand it well enough.

Albert Einstein :)
Yet comparing this to Feymann's quote................
"Hell, if I could explain it to the average person, it wouldn't have been worth the Nobel prize."
 
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Ophiolite

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If you can't explain it simply, you don't understand it well enough.

Albert Einstein :)

Yet comparing this to Feymann's quote................
"Hell, if I could explain it to the average person, it wouldn't have been worth the Nobel prize."
I suspect both statements are simultaneously valid.

With a thorough understanding of a concept, and the requisite communication skills, it should be possible to pitch explanations at various levels, appropriate for the audience. However, simple explanations are not "proofs", they are not "true" representations of the concept, therefore their acceptance is in the hands of the audience.

An audience demanding a "convincing" explanation places the expert in the position of Feynman, unable to explain it because the audience lack the background to absorb the complexities. They have had the simple explanation - they just don't like it. That's a problem with the audience, not the concept, nor the teacher.
 
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Kaon

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Let me make some corrections to your post.
Firstly the Lorentz factor is not used to prove the speed of light is a maximum.
Its origin predated SR and was used as an ad hoc explanation to explain the null result of the MM experiment while preserving an absolute ether frame by suggesting that objects shrank in the direction of motion.
Einstein derived the Lorentz transformations in SR based on the postulate the speed of light is the same to all inertial observers, not as a speed limit.

The context is the speed of light in a vacuum - which is implied to be not only constant, but a maximum of c in all inertial frames. The math being used to get to part of that conclusion is the Lorentz Factor - which is source-independent. That is the problem, not whether or not it is a limit - the self-independence of the critical velocity.


Note in GR the speed of light is not an upper limit as the recession velocities of high z galaxies indicate.

The Hubble Constant is seriously flawed. But I admire GR.


Secondly combining SR with QM was a problem in the 1930s not now as explained in this post.

For the rest of your post there is absolutely no experimental evidence for the existence of a “superluminal inertial frame”.
Every single phenomenon that may have pointed to superluminal velocities ranging from a practical application of the lighthouse paradox to the phase velocity of an EM wave exceeding c has found no information transfer is possible.

That there is no experimental evidence for the existence of superluinal inertial frame is a testament to our crude physics - not to its impossibility. Have we been able to reproduce a galaxy in the lab to test its dynamics in the same manner it would occur in real-time nature? That doesn't stop the mathematical theory from evolving things we will/hope to see later on (things we are incapable of realizing now, for whatever reason).

Here is a video that goes into somewhat more detail as to why the no communication theorem prevents superluminal velocities for entangled particles.

Ok. Let's create an example that is in the middle - an alternative (where one front is superluminal, and one is not); we don't even need the particle to be superluminal for this example. Define alternative as superluminal if both fronts are superluminal, and semi-superluminal if only one front is superluminal.

Lets say in "world" M1 there is a photon sent from the Earth (call this event E1) to arrive at a distant star at some moment T1 by the clock of that star. Let M2 be the world that was initially the same as M1 but instead of the photon assume the "Enterprise" is sent (the start of the spaceship is event E2). On its way to the star the spaceship "warps" and tears spacetime by travelling very fast passing stars, merging binary black holes and triggering other imaginable powerful processes.

We are still assuming no tachyonic matter.

Despite all of this, the spaceship arrives at the star later than the photon emitted in E2. However, we can still entertain that the spaceship arrives in time T2 less than T1. So, the speed of the
spaceship in one world (M2) would "exceed" the speed of light in another (M1), which
would not contradict the non-tachyonic nature of the spaceship. It also wouldn't break the ‘light barrier’ in M1: the inequality T2 < T1 implies the front (call it N1) is superluminal, but no matter signal in M1 corresponds to the front. Particularly no spaceship in that space-time is associated with N1.


snap_232d33bdfdcd99ae75f076bab2298c91.png



These two "worlds" would be considered the alternative of semi-superluminar speed, and if one can accept these conditions it shows such an alternative allows "superluminal" signalling without tachyons.

Its "superluminal" character does not contradict the principle of causality in M1, because in the space M1 (Minkowski space) the surface N1 does not correspond to any signal. The front N2 is not superluminal, so the alternative is semi-superluminal. The spaceship reaches the destination at a moment preceding the arrival of any photon emitted in E1, but no tachyons are involved. Even though the photons are in M1, the spaceship belongs to the universe M2, where its trajectory is timelike.

If/When we get the opportunity to build a spaceship like this and test it, we would be inclined to confirm what we would have already known.
 
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What exactly is proven here? That a photon travelling in free space, free of external forces travels at a constant speed, according to classical physics? Was that ever in doubt? With all the assumptions taken to derive the solution, you could have just made one assumption: that light in free space travels at a constant rate.
 
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Kaon

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What exactly is proven here? That a photon travelling in free space, free of external forces travels at a constant speed, according to classical physics? Was that ever in doubt? With all the assumptions taken to derive the solution, you could have just made one assumption: that light in free space travels at a constant rate.

Was this for the OP or my post immediately before?
 
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The speed of light in a vacuum, for classical phenomena (i.e. non quantum) is considered a constant because of the boundary conditions (constraints) of the Wave finction. We find (through those solutions to the Wave equation) that the speed of light evolves out of the math as a costant and maximum of c ~ 300,000,000 m/s, dependent on magnetic and electric permeability.

Special relativity attempts to juxtaposition general relativity and quantum mechanics. This is where the speed of light c as a constant and maximum begins to break down, and where SR and even basic QM loses some of its substance.


Sounds like the speed of light can be affected by a number of variables and circumstances and therefore what the speed of light may appear to be under one set of circumstances may be different under circumstances that are different.


Electrodynamics can be thoroughly proven in the lab, but there are still issues with classical electroydynamics. Depending on your method of information transfer, c can be greater or smaller than the vacuum value.


Sounds subjective. Overall, an informative response post.
 
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