# Twin paradox Doppler shift explanation

I was reading the wikipedia article on the twin paradox and came upon the section describing it in terms of the relativistic Doppler shift (link).

The image below illustrates the received signals from Earth to ship (left) and ship to Earth (right).

The explanation states that on the outward journey the twin on the ship sees the twin on Earth age only 1 year (illustrated by the few red signals in the left image), while on the return journey he sees the twin on Earth age by 9 years (illustrated by the many blue signals).

I understand this from the explanation, but doesn't this conflict with the concept that time seems to run slower for objects moving relative to an observer? This explanation would lead one to believe that the rate at which an observer sees a moving object travel through time depends on whether the object is moving towards or away from the observer. This should not be case according to the time dilation equation which depends on the absolute value of the velocity, not the direction.

• Don't confuse "Doppler Effect" with "Time Dilation". – robphy Nov 28 '17 at 19:54
• I don't see how I'm confusing the two. I'm questioning how the twin on the ship sees what's happening with the twin on earth. Isn't this time dilation? – DaYu1729 Nov 28 '17 at 19:57
• Seeing isn't associated with time-dilation. I've submitted a fuller answer below. – robphy Nov 28 '17 at 21:26

A possible source of confusion comes from the use of the word "see". Frequently when we talk about Special Relativity, we say that one observer "sees" another experiencing time at a certain rate or whatever. What we really mean is that the first observer calculates the time of a certain event — usually in the frame of that first observer. On the other hand, those diagrams depict what the observers literally see — as in receiving photons. That distinction is important.

Lower down on the wikipedia page, it shows what the first observer will actually calculate, and indeed it is insensitive to the direction of the velocity.

Time-Dilation and Doppler Effect are two different effects, and involve different sets of events on a spacetime diagram. (See my spacetime diagram drawn on rotated graph paper so that we can easily see the tickmarks on the worldlines.)

• Time-Dilation deals with comparing intervals to endpoint events that are simultaneous to an observer. As a triangle, it is a "right triangle in Minkowski spacetime" with a timelike hypotenuse and a timelike leg and a spacelike leg perpendicular to it.
Examples:
OCF (where C and F are simultaneous according to OC.. so CF and OC are perpendicular). The Time-Dilation factor is OC/OF=5/4.
OGB (where G and B are simultaneous according to OG.. so GB and OG are perpendicular). The Time-Dilation factor is OG/OB=5/4.
• Doppler deals with comparing intervals to endpoint events that are lightlike related.
As a triangle, it has two timelike legs and a lightlike leg.
Examples:
OAF, with future-directed light-signal AF. The Doppler factor is OF/OA=2.
(Comparing reception period to transmission period.)
OEB, with future-directed light-signal FD. The Doppler factor is OB/OE=2.

Note "simultaneity" (a spacelike relation) is not the same as "seeing" (a lightlike relation).

Mathematically...

• Time-dilation involves $\cosh\theta$ (which is an even function of the rapidity $\theta$... so coming or going with the same speed leads to the same factor).
• Doppler effect involves $\exp\theta=\cosh\theta+\sinh\theta$ (which is in neither even nor odd... so coming or going with the same speed does matter... interestingly $\exp(-\theta)=1/\exp(\theta)$, a reciprocal relation.

UPDATE
Although this diagram belongs above the one above, I'll leave it down here because this is "later" in the story of the Twin Paradox.

When they approach, one Doppler Triangle is
ZVS, with future-directed light-signal VS. The Doppler factor is SZ/VZ=1/2.
(Comparing reception period to transmission period.)

(My "Doppler factor" is a specific ratio of periods... you might have to do a little transformation to compare frequencies. SZ/VZ=1/2 corresponds to receiving a higher-frequency than what was transmitted.)

But doesn't this conflict with the concept that time seems to run slower for objects moving relative to an observer? This explanation would lead one to believe that the rate at which an observer sees a moving object travel through time depends on whether the object is moving towards or away from the observer.

Time indeed runs slower for objects moving relative to the observer. Time dilation for outward and inward journey depends only on relative velocity. The number of signals received during outward and inward journey differ due to Doppler effect. Doppler effect is about perception whereas time dilation is real. Mike's answer points out this distinction.

Also, if you refer the derivation for relativistic Doppler effect you will see that the formula is derived by applying Doppler shift and time dilation as two independent concepts.

• I would discourage phrases like "Doppler effect is about perception whereas time dilation is real.". The Doppler effect is also real: photons from an approaching source are more energetic than they would be if the source was at rest, a fact which has experimental consequences in, for instance, Mössbauer spectroscopy. – dmckee --- ex-moderator kitten Dec 2 '17 at 23:09

I understand this from the explanation, but doesn't this conflict with the concept that time seems to run slower for objects moving relative to an observer? This explanation would lead one to believe that the rate at which an observer sees a moving object travel through time depends on whether the object is moving towards or away from the observer.

I think it is worth to mention , that Relativistic Doppler Effect itself says nothing about clock rate, although includes time dilation. That depends on chosen reference frame whether “time seems to run slower for object moving relative to an observer” of “time runs faster” or even at the “same rate”.

To demonstrate that we can consider Relativistic Doppler Effect in different frames of reference.

Let’s consider that case, when a source of radiation approaches an observer with velocity close to $c$. In this case the observer will see a blueshift of frequency of “infinite intensity”, as Albert Einstein justly noted in 1905 article:

https://www.fourmilab.ch/etexts/einstein/specrel/www/ - § 7. Theory of Doppler's Principle and of Aberration

Measured Doppler blueshift of frequency is absolute effect; it cannot depend on chosen reference frame. However, relative contributions of time dilation are frame - dependent.

Let’s assume that an observer considers himself being “at rest” and a source of radiation approaches him.

„Classic” Doppler effect in this case will be:

$$f_0= \frac{f_s}{1-\cos\theta_s\cdot v/c}$$

Hence, in ”classic” case, if $v=c$ all wavefronts will gather straight in the front of the source into one and would hit the observer at once, like fighter jet’s sonic boom.

However, there is dilation of the source’s clock. The source oscillates $1/\sqrt {1-v^2/c^2}$ times slower, so relativistic formula looks like that: $$f_0=f_s \frac {\sqrt {1-v^2/c^2}}{1-\cos\theta_s\cdot v/c}$$ Hence, dilation of source’s clock “slows down” oscillations. Measured frequency will tend to infinity as velocity of the source approaches that of light, but at “slower” rate than classic one.

On the other hand, the observer may say, that he was in motion himself in the reference frame of the source, so his own clock slowed down.

In classic non - relativistic case, if an observer moves towards a source, measured Doppler shift will be:

$$f_0=(1+\cos\theta_s\cdot v/c)f_s$$

According to this formula, if an observer approaches the source with velocity $v=c$, maximum measured frequency $f_o$ would be equal to $2f_s$, since wavefrons and the observer approach each other with equal velocities.

Since the observer’s clock slows down, now we must divide this frequency by $\sqrt {1-v^2/c^2}$. That means, due to time dilation, the observer turns into a “dawdler”. While he lives just one minute, the world around him may live one year. So, he sees “usual” processes around him as very fast, as if someone plays movie in fast forward mode.

$$f_0= \frac {(1+\cos\theta_s\cdot v/c)}{\sqrt {1-v^2/c^2}}f_s$$

We see, that frequency $2f_s$ turns into infinitely intense, as velocity of observer approaches $c$.

Though these formulas (for moving source and moving observer) at first glance look different, actually they are the same.

http://www.feynmanlectures.caltech.edu/I_34.html - 34.6 - The Doppler Effect.

This way an observer can interpret Relativistic Doppler Effect as he wishes. He can say anything he wishes about a rate of a clock that approaches him. If he considers himself being “at rest”, he will say that clock is ticking slower. If he considers himself being “in motion”, he will say that the clock is ticking faster. If he thinks, that he himself and the source approach each other with equal velocities, he will say, that source’s clock is ticking at the same rate as his own.

http://www.mathpages.com/home/kmath587/kmath587.htm “Since both emitter and receiver have the speed v relative to this system of reference, there is no differential time dilation”

Quite interesting, that Albert Einstein considers Relativistic Doppler Effect in the reference frame of “stationary” source. His formula shows, that frequency $2\nu’$ not decreases, but increases $1/\sqrt{1-v^2/c^2}$ times, i.e. moving observer "sees" or better to say "interprets" that the clock “at rest” is ticking not slower, but faster.

The same thing happens if an observer measures time dilation by means of synchronized clocks. If an observer employs Einstein synchrony convention, he will measure, that moving clock is ticking slower.

However, if an observer considers as moving himself in the reference frame of the source, he must synchronize clocks in his frame according to time of source’s reference frame. This synchronization will be equivalent to Reichenbach synchrony condition, which keeps two – way speed of light isotropic while one – way speeds of light anisotropic. In this case this observer will see, that a clock “at rest” is ticking not slower, but faster, exactly as in the case with Doppler Effect above.

https://en.wikipedia.org/wiki/One-way_speed_of_light