# Special Relativity - Self Contradiction?

From his original thesis, Einstein said light speed is always constant (c). There is very clear evidence for this.
Every ray of light moves in the “stationary coordinate system” with the same velocity c, the velocity being independent of the condition whether this ray of light is emitted by a body at rest or in motion.

However, later in the paper, he writes:
we take into consideration the fact that light when viewed from the stationary system, is always propagated along those axes with the velocity √{c2-v2}.
and
Now the ray of light moves relative to the origin of k with a velocity c-v, measured in the stationary system

This appears self-contradictory... light speed cannot be all three:
1. constant (c)
2. non-constant (√{c2-v2})
3. non-constant (c-v)

PeterDonis
Mentor
From his original thesis

Please provide a link. We need to know precisely which English translation (the actual "original thesis" was written in German) you are reading.

Good point! I'm quoting: On the Electrodynamics of Moving Bodies (Annalen der Physik, 1905). Translation by Megh Nad Saha in The Principle of Relativity: Original Papers by A. Einstein and H. Minkowski, University of Calcutta, 1920, pp. 1–34:

The other common translation is as follows:
"it being borne in mind that light is always propagated along these axes, when viewed from the stationary system, with the velocity √{c2-v2}"
and
"But the ray moves relatively to the initial point of k, when measured in the stationary system, with the velocity c-v"

Andrew Mason
Homework Helper
From his original thesis, Einstein said light speed is always constant (c). There is very clear evidence for this.
Every ray of light moves in the “stationary coordinate system” with the same velocity c, the velocity being independent of the condition whether this ray of light is emitted by a body at rest or in motion.

However, later in the paper, he writes:
we take into consideration the fact that light when viewed from the stationary system, is always propagated along those axes with the velocity √{c2-v2}.
and
Now the ray of light moves relative to the origin of k with a velocity c-v, measured in the stationary system

This appears self-contradictory... light speed cannot be all three:
1. constant (c)
2. non-constant (√{c2-v2})
3. non-constant (c-v)
Hi Jonathan. Welcome to PF!

You have be read Einstein's paper very carefully. It is not easy to follow. You are better off reading an introductory text first before trying to follow Einstein's paper.

Einstein: "An analogous consideration—applied to the axes of Y and Z—it being borne in mind that light is always propagated along these axes, when viewed from the stationary system, with the velocity ##\sqrt{c^2 - v^2}##"

Explanation: This is the speed of a light wave front propagation in the direction the Y or Z axis (ie. perpendicular to the velocity of the frame which is moving along the X and x axis at speed v) relative to the moving origin as measured by the stationary observer. It is just an application of Pythagoras' theorem. If you imagine the moving origin at speed v along the x axis and a light ray traveling from that origin in the Y direction, it travels relative to the stationary observer at speed c. So you have a velocity triangle in which the hypotenuse is c, the near side (along the x axis) is v and, therefore the opposite side, the speed of propagation relative to the moving origin in the y direction, is ##\sqrt{c^2 - v^2}##". In practical terms, what this means is that a photon directed along the y axis in the moving frame will appear to be travelling at a forward angle such that the Y component is ##\sqrt{c^2 - v^2}## and the x component v. This is actually what is observed in, for example, a synchrotron where photons emitted from high speed electrons (or positrons) in all directions (in the moving frame of the electron/positron) appear to be all angled forward in the laboratory frame.

Einstein: "But the ray moves relatively to the initial point of k, when measured in the stationary system, with the velocity c-v, so that ##\frac{x'}{c-v} = t##"

Expanation: ##c - v## is the speed that the observer in the stationary frame measures of the ray of light relative to the origin in the moving frame (i.e moving along the x axis, which is the direction of the light ray)

AM

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Bandersnatch
PeterDonis
Mentor
The other common translation

This one looks like a better one to me.

Regarding your original question, you are confusing different quantities that play different roles in Einstein's argument.

constant (c)

This is the speed of light, i.e., the magnitude of its velocity vector. It is indeed always c in any frame (provided we transform the light's velocity vector correctly into that frame).

non-constant (√{c2-v2})

This is the component of the velocity vector of a light beam, in the stationary frame, along the Y (or Z) axis. Here Einstein is not very clear about exactly how the light beam in these cases is supposed to be propagating; he only says "a similar conception being applied to the y-axis and the z-axis"--in the 1920 translation--or "an analogous consideration applied to the axes of Y and Z"--in the other translation. What I think he means is that we imagine a light beam that is propagated purely along the Y or Z axis in the moving frame, so that in the stationary frame, it has velocity components along both the Y or Z axis and the X axis. The X axis component will be ##v##--the relative velocity between the frames, which is defined to be purely in the X direction--so the Y or Z component will be as Einstein says.

non-constant (c-v)

Here the system ##k## is the moving system, which is moving at speed ##v## in the positive X direction relative to the stationary system, so what Einstein is saying is that, if we emit a ray of light in the positive X direction, then in the stationary system, the light will catch up with the origin of the moving system at a speed ##c - v##.

Thanks Andrew and Peter. Good thoughts. I think I must concede ##c-v## is not the contradiction it appeared to be. However, it appears you both are affirming the contradiction on ##\sqrt{c^2 - v^2}##

We know from Einstein's definition, light's speed is:
independent of the condition whether this ray of light is emitted by a body at rest or in motion.

Light emitted at a perpendicular angle (or any other angle) from the direction of travel must not be affected whatsoever in its propagation velocity. Even if light is focused directly along the x-axis, it still must propagate at c. In other words: this remains a self-contradiction.

I see Andrew pointed out
This is actually what is observed in, for example, a synchrotron where photons emitted from high speed electrons (or positrons) in all directions (in the moving frame of the electron/positron) appear to be all angled forward in the laboratory frame.
Just to be clear, I'm not disputing this. I'm simply working to apply the logic of Einstein's work exactly as written. As defined, light doesn't get benefit or penalty from being emitted from a moving body. So we're still left with a self-contradiction.

Bandersnatch
There's no self-contradiction. Both ##\sqrt{c^2-v^2}## and ##c-v## are the exact same case, only in a different direction.
In either one, the light beam travels at c - as measured in the stationary system of coordinates K.
One then simply asks what is the closing speed between the light beam and the moving system k. Again, as measured by K.
The closing speed between the light beam and anything else is not restricted to c. One can ask, for example, what is the closing speed between a space ship leaving Earth at 0.5c and a light beam emitted towards that space ship (it's 0.5 c). Or what is the closing speed between two light beams emitted towards K from opposite directions (it's 2c). Or between a light beam emitted perpendicular to the direction of travel of k and k (that's the Pythagorean theorem expression).
In all of those cases, the speed of the beam of light itself is always measured as c by the stationary observer K.

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PeroK
Homework Helper
Gold Member
2021 Award
However, it appears you both are affirming the contradiction on ##\sqrt{c^2 - v^2}##

We know from Einstein's definition, light's speed is:
independent of the condition whether this ray of light is emitted by a body at rest or in motion.

Light emitted at a perpendicular angle (or any other angle) from the direction of travel must not be affected whatsoever in its propagation velocity. Even if light is focused directly along the x-axis, it still must propagate at c. In other words: this remains a self-contradiction.

So we're still left with a self-contradiction.

It's funny that no one ever noticed that before. Well then, that's 113 years of physics out the window!

Sorcerer and russ_watters
PeterDonis
Mentor
it appears you both are affirming the contradiction on ##\sqrt{c^2 - v^2}##

No, we aren't. Go read my response to that in post #5 again, carefully.

One then simply asks what is the closing speed between the light beam and the moving system k. Again, as measured by K.

Thanks; I appreciate the analogy. Einstein's wording is
propagated along those axes with the velocity ##\sqrt{c^2 - v^2}##

He does not say that this is relative to the origin k, so I'm unclear how you are interpreting this to be relative to k?

Nugatory
Mentor
He does not say that this is relative to the origin k, so I'm unclear how you are interpreting this to be relative to k?
That understanding goes all the way back to Galileo. Einstein was writing for an audience of turn-of-the century physicists; he's making the altogether reasonable presumption that that audience is going to default to the conventions of Galilean relativity.

You will find some introductory treatments that assume less of their audience. That's one of many reasons why it's generally easier to learn from a modern textbook than from the pioneering papers.

m4r35n357, Sorcerer, russ_watters and 1 other person
Janus
Staff Emeritus
Gold Member
Thanks Andrew and Peter. Good thoughts. I think I must concede ##c-v## is not the contradiction it appeared to be. However, it appears you both are affirming the contradiction on ##\sqrt{c^2 - v^2}##

We know from Einstein's definition, light's speed is:
independent of the condition whether this ray of light is emitted by a body at rest or in motion.

Light emitted at a perpendicular angle (or any other angle) from the direction of travel must not be affected whatsoever in its propagation velocity.
Light speed is independent of the relative motion of the source, but its velocity (which includes the direction) is not. This results is a well known phenomena called the aberration of light.
Even if light is focused directly along the x-axis, it still must propagate at c. In other words: this remains a self-contradiction.
Yes it propagates at c. But if a source emits light in the Y direction as perceived by itself, then for an observer for which this source is moving at some velocity v in the in the X direction, that light will travel at c at an angle between the X and Y axis.
I see Andrew pointed out
Just to be clear, I'm not disputing this. I'm simply working to apply the logic of Einstein's work exactly as written. As defined, light doesn't get benefit or penalty from being emitted from a moving body. So we're still left with a self-contradiction.

Consider the rest frame of the source first. It fires a light pulse to a mirror in the Y direction, which reflects the light back to it. As along as the source and mirror are at rest with respect to each other, the light as measured from this frame, travels straight out to the mirror and back at c relative to source and mirror.

Now consider a frame for which the source and mirror are moving in the X direction. The fact that the source_mirror combination is moving relative to this frame has no effect on what happens according to the source and mirror. The light must still travel from source to mirror and back.

This must hold true for the frame in which the pair are moving also. ( if it didn't, then you would have a real contradiction) The light must leave the source, hit the mirror and reflect back to the source. But since in this frame, the position of the pair on the X axis changes as the light travels, the light has to follow an angled path in order to accomplish this. But it still restricted to moving at c relative to the this frame. This results in the time it takes for light to leave the source and return to be longer as measured in this frame than it is as measured in the rest frame of the source and mirror. This is the essence of time dilation.

You get something like this:

Here we have the dots showing the light pulses, while the expanding circles show how the light would propagate at c equally in all directions in this frame. (for comparison we have a second pair of mirrors at rest with respect to the animation frame.)

#### Attachments

• time_dil.gif
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Andrew Mason
stevendaryl
Staff Emeritus
Thanks Andrew and Peter. Good thoughts. I think I must concede ##c-v## is not the contradiction it appeared to be. However, it appears you both are affirming the contradiction on ##\sqrt{c^2 - v^2}##

Einstein is trying to figure out the relationship between two different inertial coordinate systems, namely the one of the "stationary" system, ##(x,y,z,t)## and the one of the "moving" system, ##(\xi, \eta, \zeta, \tau)##. He introduces an auxiliary coordinate, ##x'## which is defined by: ##x' \equiv x - vt##. This is not the coordinate used by the moving frame, but Einstein is anticipating that it will be convenient for the derivation. The significance of ##x'## is that if an observer is at "rest" in the moving frame, then that implies that ##x'## is constant.

The following is all from the point of view of the "stationary" frame:

No matter what direction light is traveling, it has to satisfy ##c^2 = (\frac{dx}{dt})^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2##. In terms of the auxiliary coordinate ##x'##, we have: ##x' = x - vt## so ##\frac{dx'}{dt} = \frac{dx}{dt} - v##. So the constancy of the speed of light in terms of the coordinate ##x'## tells us that:

##c^2 = (\frac{dx'}{dt} + v)^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2##

We look at two special cases:

Case 1: ##\frac{dy}{dt} = \frac{dz}{dt} = 0##. Then we have:
• ##c^2 = (\frac{dx'}{dt} + v)^2##
• ##\frac{dx'}{dt} + v = \pm c##
• ##\frac{dx'}{dt} = \pm c - v##
• ##|\frac{dx'}{dt}| = c \mp v##
Case 2: ##\frac{dx'}{dt} = \frac{dz}{dt} = 0##. Then we have:
• ##c^2 = v^2 + (\frac{dy}{dt})^2##
• ##\frac{dy}{dt} = \pm \sqrt{c^2 - v^2}##
• ##|\frac{dy}{dt}| = \sqrt{c^2 - v^2}##
These calculations are not about the speed of light as measured in the moving frame. They are about the values ##\frac{dx'}{dt}, \frac{dy}{dt}, \frac{dz}{dt}## as measured in the stationary frame.

Sorcerer
Andrew Mason
Homework Helper
Thanks; I appreciate the analogy. Einstein's wording is
propagated along those axes with the velocity ##\sqrt{c^2 - v^2}##

He does not say that this is relative to the origin k, so I'm unclear how you are interpreting this to be relative to k?
The light ray is propagated in a direction perpendicular to the direction of motion in the moving frame. But it doesn't matter from which origin you measure the speed in the direction perpendicular to the direction of motion, so long as it is measured by the stationary observer in the stationary frame. That perpendicular speed is ##\sqrt{c^2 - v^2}##.

Consider a light ray that is emitted when the two origins coincide (x = x' = 0). In the moving frame, k, the ray travels along the y axis, perpendicular to the direction of travel. Since the light ray propagates in the perpendicular direction in the moving frame it must travel in a forward direction as observed in the stationary frame. The moving observer will measure the light ray to be moving vertically at the speed of light, c. The stationary observer will measure the light ray to be moving at a forward angle α at the same speed, c, such that ##\cos(α) = v## and ##\sin(α) = \frac{\sqrt{c^2 - v^2}}{c} = \sqrt{1 - v^2/c^2}##.

The stationary observer, adhering to the postulate of relativity that the speed of light is measured by all observers as c, concludes that the moving observer measures time more slowly than the stationary observer. This is because in the stationary frame the light covers less distance from the moving origin per unit time than the light travels from the stationary origin.

AM