# Accounting for the constant speed of light

• B
I keep hearing that the relative speed of light remains constant because time and lengths change with speed, what I believe is called the Lorentz factor. At slow speeds the Lorentz factor is extremely small, so what do people believe accounts for the rest of the change?
If I travel towards a source of light at 55 miles per hour, then relative to me, the speed of light will have to be reduced 55 miles per hour for the speed to remain constant. The Lorentz factor is so small at this speed that it can be ignored, so what is currently believed to account for the 55 miles per hour change?

Dale
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If I travel towards a source of light at 55 miles per hour, then relative to me, the speed of light will have to be reduced 55 miles per hour for the speed to remain constant.
Huh? This doesn’t make any sense. Relative to you the speed of light is c. It is not reduced 55 mph.

Ok, if I were to travel towards the source of light at 55 mph, without knowing about special relativity, I would expect to measure the speed of light plus 55 mph. What is believed to cause this difference since it's obviously not the Lorentz factor?

Ok, if I were to travel towards the source of light at 55 mph, without knowing about special relativity, I would expect to measure the speed of light plus 55 mph. What is believed to cause this difference since it's obviously not the Lorentz factor?
The Lorentz factor is not used in adding velocities. If you want to ask what the velocity of something is in a frame that is moving relative to you, you need the special relativity velocity addition equation: $$u = \frac {v + u’} {1 + \frac {vu’} {c^2}}$$If I’m correctly interpreting what you are imagining, in your scenario ##u## would be the speed of light (##c##) in the Earth’s rest frame, ##v## would be the velocity of, say, the car (55mph) relative to the earth, and ##u’## would be the speed of light in the car’s rest frame. If you only start with the assumption that the speed of light in the earth’s rest frame is ##c## (##u = c##), then you can easily calculate that ##u’## must also be ##c##. This velocity addition equation can be derived from the Lorentz transformations.

Dale
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Ok, if I were to travel towards the source of light at 55 mph, without knowing about special relativity, I would expect to measure the speed of light plus 55 mph. What is believed to cause this difference since it's obviously not the Lorentz factor?
It is relativistic velocity addition. 55 mph is not a relativistic speed, but c clearly is. So relativistic velocity addition must be used.

Nugatory
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What is believed to cause this difference since it's obviously not the Lorentz factor?
That word "obviously" suggests that you're so sure about what the answer will be that you haven't bothered to check your initial assumption.

Try actually calculating the effects of length contraction and time dilation (and depending on the details of your setup, relativity of simultaneity) in this situation where you're moving towards the source at 55 mph. When you do, you will find that the results match what you get using the velocity addition formula @Pencilvester provides above. This is an unsurprising result because the velocity addition formula and the length contraction/time dilation formulas are both derived from the Lorentz transformations, Lorentz factor and all.

Pencilvester
Dale
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That word "obviously" suggests that you're so sure about what the answer will be that you haven't bothered to check your initial assumption.
I agree with this comment. Another issue for @grounded is that the Lorentz factor is indeed small at 55 mph, but 55 mph is a small fraction of c. So it is not obvious at all that the correction is not simply the Lorentz factor.

Pencilvester
Mister T
Gold Member
Ok, if I were to travel towards the source of light at 55 mph, without knowing about special relativity, I would expect to measure the speed of light plus 55 mph.

That's correct. If you were to naively believe that adding speeds is the right way to combine them, that is indeed what you'd expect.

The very small amount of length contraction associated with a speed of 55 mph, by the way, is just enough to make the speed come out to ##c## rather than the very slightly different ##c## plus 55 mph.

Grinkle
Gold Member
What is believed to cause this difference since it's obviously not the Lorentz factor?

Nothing causes the difference - this is asking why your intuition is incorrect - your intuition being that c is not constant.

It is experimentally confirmed that c is the same for all reference frames. One starts from that fact to explain other phenomena like length contraction.

It might be that we lived in a universe where c was not constant for all reference frames - physics does not provide an answer for why we live in a universe where c is constant. We do, we can confirm that, and we go from there.

phinds
Ok, if I were to travel towards the source of light at 55 mph, without knowing about special relativity, I would expect to measure the speed of light plus 55 mph. What is believed to cause this difference since it's obviously not the Lorentz factor?

The times of your clocks are off by a 'large' amount without you knowing it. That's the answer.

Next question is then: How do the clocks get so much out of sync, if the relativistic effects are very small?

For the people that have difficulties understanding the question:

Observers in two frames disagree about some things:
Disagreement about light's closing speed = 55 mph = large
Disagreement about lengths of rulers and ticking rates of clocks = small

I keep hearing that the relative speed of light remains constant because time and lengths change with speed, what I believe is called the Lorentz factor. At slow speeds the Lorentz factor is extremely small, so what do people believe accounts for the rest of the change?
If I travel towards a source of light at 55 miles per hour, then relative to me, the speed of light will have to be reduced 55 miles per hour for the speed to remain constant. The Lorentz factor is so small at this speed that it can be ignored, so what is currently believed to account for the 55 miles per hour change?
There is not cause that physics can answer. It's just the way the universe is. But maybe a reason can be given for you? What follows is a overly simplistic historical retracing of some of the events and reasoning that led to this realization, and hopefully some understanding.

(1) Galileo realized (with Newton eventually filling in all the fine details derived from this) that if you are moving smoothly, there is no experiment you can perform that will determine exactly how fast you are moving, or if you are even moving at all. His quintessential example of being inside the cargo hold of a smoothly sailing ship* illustrates this: water dripping from a bucket does not suddenly fly to the back of the ship as you go forward; butterflies still fly the same; etc. You can see this yourself in an airplane once you reach cruising altitude and there is no turbulence. Simply flip a coin. If it flies to the back of the airplane, Galileo is wrong (Hint: it doesn't. It does up and down, just like on the surface). In physics parlance, the laws of physics are the same for all inertial reference frames. This is the principle of relativity, and it leads to the assertion that all speeds are meaningless unless specified what the speed is relative to (e.g., 30 MPH is meaningless, but 30 MPH with respect to the Eiffel Tower means something).

*What Galileo actually wrote (not really needed but it provides some historical enrichment):
Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other. The cause of all these correspondences of effects is the fact that the ship's motion is common to all the things contained in it, and to the air also. That is why I said you should be below decks; for if this took place above in the open air, which would not follow the course of the ship, more or less noticeable differences would be seen in some of the effects noted.​

(2) Hundreds of years of experiments with electricity and magnetism led to four crucial equations called Maxwell's equations (you will recognize some of the names that contributed: Faraday- of the unit for capacitance, the "farad"; Volta- of the unit for electric potential, the "volt"; Ohm of "Ohm's law" and the unit of resistance, the "ohm"; Ampère, who's name is used for the unit of electrical current the "almpere," usually shortened to "amp"; Joule (you guessed it- the unit of energy, the "joule," is named after him); Galvani (of "galvanization" and "galvanometer"); Gauss (even if you don't take physics, if you take calculus you'll hear of him); and countless others (I could make an entire thread full of these guys).

These equations happened to combine into a wave equation. This wave equation describes the electromagnetic wave (and shows that light is part of this wave). This wave also has a characteristic speed- but one that is not described as being with respect to anything.

(3) Physicists assumed this speed was with respect to the medium through which the electromagnetic wave traveled. After all, all waves need a medium, right? Except experiments failed to find this medium. Experiments suggested that this speed was with respect to everything, not just some medium. Of course, this directly contradicted (1), the principle of relativity. This was a big problem, because the principle of relativity was experimentally true, but so were Maxwell's equations AND the medium through which the electromagnetic wave traveled was undetectable.

(4) Einstein realized the two ideas were not mutually exclusive. All that had to change was the way in which the results from the principle of relativity were formulated- specifically, the way coordinates are transformed from one inertial frame to another (some have already posted the addition of velocity formula. That is derived from the corrected transformation equations). Fortunately, the math was already there (in part thanks to Lorentz, of the Lorentz factor). Einstein simply realized what that math really meant: that our instinctual notions about how velocities add, how coordinates transform, how time and space "behave," and that simultaneity is absolute, were all mistaken. With this also died the theoretical need for a medium for light to move through.

(5) Then comes the special theory of relativity, which puts it all together, unifying the principle of relativity (and mechanics along with it) with Maxwell's equations. The way this was done by Einstein was he took the experiments at face value, and from them made two assumptions:

I - The laws of physics are the same for all inertial reference frames (the principle of relativity)
II- The speed of light in a vacuum is the same for all inertial reference frames, regardless of the speed of the source of the light (with respect to anything. That is, the speed of light on an airplane going 500 mph is the same as the speed of light on the ground, and anyone on the ground will measure a beam of light coming from the airplane moving at c, not 500+c).​

And that is a brief summary of the reason as it is usually presented (historically). The tl;dr version is this: We observe both the principle of relativity to be true AND that the speed of light is the same regardless of its source. The logical (mathematical) conclusion of those two things is that the correct transformation between reference frames is the Lorentz transformation (which includes the Lorentz factor), and from that comes all the interesting things of special relativity.

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Zman and Imager
Mister T
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Disagreement about light's closing speed = 55 mph = large

That "large" disagreement is less than one part in 10 million!

Thank you all for the replies…

To save some time, I know it’s a fact that the speed of light is constant, I am not arguing or questioning that. I am not pushing my theory (been here a while and I know the rules) or arguing with yours, just trying to understand it better.

I am not adding velocities or comparing two people’s observations, so the velocity addition formula and the Lorentz transformation do not really apply. I am simply calculating the speed of light by measuring the frequency and wavelength of an unknown beam of light, then I repeat the test while traveling towards the light and again measure the frequency and wavelength. As I travel towards the light at different speeds, I will measure a change in the frequencies and wavelengths, but they will always multiply together to equal the same speed. I have my own explanation for this, but I’d like to understand what is currently believed and some of it is not clear to me. I keep hearing that the change in frequency and wavelength is due to time dilation and length contraction, but they do not have a linear effect and the Lorentz factor alone does not account for the total reduction to the wavelength.

Keeping the math simple, if the speed of light is 300MPH and I travel towards it at 25MPH, the time dilation and length contraction (Lorentz factor) would reduce the relative wavelength to .996521728% of its original length, however the measured wavelength will be .91986621% of its original size. How is the difference explained, or am I overlooking something obvious?

Well, the relativistic Doppler effect is just the standard Doppler effect with the added Lorentz factor. And if the standard Doppler effect can change the frequency AND the wavelength, then the speed of light would remain constant even without the Lorentz factor, so it could not be the cause of it. This would mean that time and space being relative has nothing to do with the constant speed of light, it just causes an additional change. I don’t think that is what’s currently believed which is why I’m asking how it is currently accounted for?

Grinkle
Gold Member
@grounded I may not be following you, perhaps you are pointing out something deeper than what I have below. In any case -

c is constant - start from there. If the frequency changes, the wavelength will also change at least because it is an axiom that c is constant.

Grinkle
Gold Member
the speed of light would remain constant even without the Lorentz factor, so it could not be the cause of it.

I re-read your post. The Lorentz factor is a result of c being constant, it is in no way whatsoever a cause of c being constant. There is no cause of c being constant.

Dale
Mister T
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And if the standard Doppler effect can change the frequency AND the wavelength, then the speed of light would remain constant

If you use the nonrelativistic Doppler effect formulas you will find that the speed of light is not constant.

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PeroK
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Well, the relativistic Doppler effect is just the standard Doppler effect with the added Lorentz factor. And if the standard Doppler effect can change the frequency AND the wavelength, then the speed of light would remain constant even without the Lorentz factor, so it could not be the cause of it. This would mean that time and space being relative has nothing to do with the constant speed of light, it just causes an additional change. I don’t think that is what’s currently believed which is why I’m asking how it is currently accounted for?

This is not correct. The standard Doppler effect, e.g. for sound in air, results in a different speed if the receiver is moving toward the source.

The relativistic Doppler effect is consistent with the Lorentz Transformation, hence time dilation and length contraction. And results in no change in the speed of light between reference frames.

I suspect your confusion is based on an error in applying the formulas.

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Dale
Keeping the math simple, if the speed of light is 300MPH and I travel towards it at 25MPH, the time dilation and length contraction (Lorentz factor) would reduce the relative wavelength to .996521728% of its original length, however the measured wavelength will be .91986621% of its original size. How is the difference explained, or am I overlooking something obvious?

Only things whose speed is 25 MPH, when you are moving towards them at speed 25 MPH, are contracted that amount. (0 .996521728%) (assuming light speed 300 MPH)

Contraction is not linear as you have said.

We need some other formula than the usual one for contraction of things that are initially moving. ('change of length' might be better than 'contraction')

When an observer changes his velocity slightly, he observes things to change lengths, fast moving things change lengths more than slowly moving things.

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Janus
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Keeping the math simple, if the speed of light is 300MPH and I travel towards it at 25MPH, the time dilation and length contraction (Lorentz factor) would reduce the relative wavelength to .996521728% of its original length, however the measured wavelength will be .91986621% of its original size. How is the difference explained, or am I overlooking something obvious?

Your original question revolves around how you would use the Lorentz transformations to get a constant value for c.
Consider the following scenario:
You have two clocks which, in their own rest frame, synchronized to each other and 1 light sec apart. A pulse of light leaving clock A at t=0 arrives at clock B when it reads 1 sec in the rest frame of the clocks.
Now these clocks are moving at 0.6c relative to you such that B is the leading clock and A is the trailing clock.
Due to length contraction, the distance between A and B according to you is 0.8 light sec, and due to time dilation, both clocks tick at a rate of 0.8 as fast as your own. Also, due to the Relativity of Simultaneity, clock A will always be 0.6 sec ahead of clock B.
Thus when the light leaves clock A as it reads 0, by your frame of reference clock B reads -0.6 sec The light reaches it when it reads 1 sec, so it has to have advanced by 1.6 sec between the moment the light leaves A and arrives at B. Due to time dilation, this will take 2 sec by your clock. In 2 sec, clock B will have moved 1.2 light sec at 0.6c. Since it started 0.8 light sec ahead of clock A as measured in your frame, it will be 2 light sec from where A was when the light left A. So for you, the light traveled 2 light sec in 2 sec, or at c, while in the frame of the clocks it moved 1 light sec in 1 sec, also at c.

Dale
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I am not adding velocities or comparing two people’s observations, so the velocity addition formula and the Lorentz transformation do not really apply.
You seem to have some misconceptions about when the various formulas apply.

The Lorentz transform applies any time we have two inertial frames, in this case we have your frame and the frame of the source. Any time you can use length contraction or time dilation you can use the Lorentz transform since those simplified formulas are derived from the Lorentz transform. Additionally, you can use the Lorentz transform in many situations where the time dilation and length contraction formulas do not apply.

The relativistic velocity addition formula applies any time you are looking at the speed of something in two different frames. Here you are explicitly asking about the speed of light in your frame and in the sources frame. Using the relativistic velocity addition formula immediately and easily gets you the correct answer.

Both formulas apply here.

I am simply calculating the speed of light by measuring the frequency and wavelength of an unknown beam of light, then I repeat the test while traveling towards the light and again measure the frequency and wavelength.
The frequency you obtain through the relativistic Doppler equation. The wavelength is obtained by dividing c by the Doppler equation. Both can be derived from the Lorentz transform.

I keep hearing that the change in frequency and wavelength is due to time dilation and length contraction
Time dilation and length contraction cannot be applied to light. Can you think why?

You would need to use the Lorentz transform.

Keeping the math simple, if the speed of light is 300MPH and I travel towards it at 25MPH, the time dilation and length contraction (Lorentz factor) would reduce the relative wavelength to .996521728% of its original length, however the measured wavelength will be .91986621% of its original size. How is the difference explained, or am I overlooking something obvious?
Yes, you are overlooking the Lorentz transform. From it you can derive the correct expressions for the relativistic velocity addition, the relativistic Doppler, and the relativistic wavelength transformation.

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Sorcerer
Dale
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So, I had some free time today and this seemed like an interesting problem to try "the hard way" (calculating the wavelength and the frequency instead of just transforming the velocity). I will use units where c=1 and four-vectors (t,x,y,z), but I will keep everything on the x axis so I will drop y and z so (t,x)=(t,x,0,0) is understood. The source will be at the origin (in the source's frame) and the observer will start at a distance of d from the source moving at a speed of v towards the source. The source will emit a single cycle of a wave with a period of 1 starting at t=0.

The worldline of the source in the source's frame is ##(t,0)##.
The worldline of the beginning of the wave is ##(t,t)##.
The worldline of the end of the wave is ##(t,t-1)##.
The worldline of the observer is ##(t,d-vt)##.

Transforming to the observer's (primed) frame, solving for the worldlines in terms of the primed coordinates, and simplifying we get.

The worldline of the source in the observer's frame is ##(t',v t')##.
The worldline of the beginning of the wave is ##(t',t')##.
The worldline of the end of the wave is ##(t',t'-\sqrt{\frac{1-v}{1+v}})##
The worldline of the observer is ##(t',\frac{d}{\sqrt{1-v^2}})##

Of these, the only important one is the third one, but since I calculated the others before realizing that I went ahead and retained them. From the third one we see that in the primed frame the frequency is ##\sqrt{\frac{1+v}{1-v}}## and the wavelength is ##\sqrt{\frac{1-v}{1+v}}##, so the speed of the light is the product of the frequency and the wavelength, which is 1.

QED

Klystron, m4r35n357, PeroK and 1 other person
How is the constant speed of light accounted for when the different between the reference frames is the size of the gravity well? like someone on earth versus someone in a spaceship? Each would observe the speed of light to be C in their reference frame, but what about when the measure it in the other reference frame that has a different gravity well?

Grinkle
Gold Member
How is the constant speed of light accounted for

Your phrasing confused me for a bit. You mean "kept track of" and not "explained with some fundamental root cause".

Am I correct?

Ibix
How is the constant speed of light accounted for when the different between the reference frames is the size of the gravity well? like someone on earth versus someone in a spaceship? Each would observe the speed of light to be C in their reference frame, but what about when the measure it in the other reference frame that has a different gravity well?
How are you planning to measure the speed of light remotely? Generally, the remote speed of light is not constant, but depends on how you define distance - see Shapiro delay, for example. Locally, the speed of light is always c.

PeroK
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How is the constant speed of light accounted for when the different between the reference frames is the size of the gravity well? like someone on earth versus someone in a spaceship? Each would observe the speed of light to be C in their reference frame, but what about when the measure it in the other reference frame that has a different gravity well?

In GR, i.e. where gravity is involved and spacetime is not flat, the speed of light is invariant as measured locally.

@PeroK so speed of light isn't invariant when space-time is curved due to gravity and you are measuring the speed of light in an area with different curvature then your local area?

PeroK
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@PeroK so speed of light isn't invariant when space-time is curved due to gravity and you are measuring the speed of light in an area with different curvature then your local area?

In SR you have the nice concept of a global inertial reference frame, which gives the global concept of time and distance; but, in GR you only have local inertial reference frames.

In SR, in fact, the speed of light is invariant across inertial reference frames. Once you move to GR, this implies that the speed of light is invariant when measured locally, as there are no global inertial reference frames.

Also, as @Ibix mentioned, it's not clear how you define measurements of length and time remotely. In a sense in GR the only valid measurements are local. The energy of a particle, say, can only really be measured locally.

SiennaTheGr8 and Ibix
Ibix