Relativity Science Calculator - Twin Clock Paradox

Twin Clock Paradox

"Relativity is a purely scientific matter and has nothing to do with religion" - Albert Einstein ( 1879 - 1955 )

The October 1971 Joe Hafele and Richard Keating Famous Experiment Using Four Cesium - Beam Atomic Clocks

testing general and special relativity time dilation on a British Airways Boeing aircraft
movie source: PBS's NOVA Science - "Time Dilation Experiment, October 1971"
source: J. C. Hafele and R.E. Keating, Science 177: 166 - 68 &168 - 70 ( 1972 )
Read: "Performance and Results of Portable Clocks in Aircraft", by J.C. Hafele
Also: "Around-the-World Atomic Clocks: Observed Relativistic Time Gains", by J.C. Hafele and Richard E. Keating

Observed Facts: Flying eastward and then westward from the stationary U.S. Naval Observatory, General Relativity predicted a time loss of 40 ± 23 nonoseconds on the eastward bound and a time gain of 275 ± 21 nanoseconds for the westward trip; the actual eastward loss was 59 ± 10 nanoseconds going eastward but gained 273 ± 7 nanoseconds for the westward trip!!

Rhetorical Question: Is the October 1971 Joe Hafele and Richard Keating clock experiment showing the effects of Special Relativity or General Relativity? Is the British Airways flight around the earth a straight - line, inertial flight subject to Special Relativity analysis or is it rather a matter for General Relativity analysis?

The Twin Clock Paradox is only inexplicable for the situation where the actual relativity mathematics are not fully understood.

The Problem: If a rocket leaves relatively stationary earth and travels a sufficiently great distance with a velocity at a significant fraction of the speed of light, and then later returns ostensibly "younger" than what was left behind, the question therefore arises if all motion is indeed relative, why then could not the earth be considered as having left the rocket and therefore upon the mutual return of earth to the rocket be shown equivalently also as "younger"?

The Solution: The solution comes in two parts:

  1. For inertial ( non - accelerating ) straight - line motion, both the rocket observer and the earth observer will see the other as being "younger" since their own respective clock will go faster than the observed passing clock of the other.

    This is a situation of pure inertial ( non - accelerating ) frames of reference passing each other and hence is not what's being questioned in the "Twin Clock Paradox" since the rocket does not accelerate as they each pass the other unaccelerated.

    As in length contraction for inertial ( non - accelerating ) straight - line motion, this type of time dilation is mutually reciprocal and therefore needs only special relativity mathematics for an explanation.

  2. In this second case, gravity force fields must be considered whenever acceleration is occurring such as the rocket in the Twin Clock Paradox turning around and returning back to earth.

    In this case, where two clocks are running in different gravitational potential fields due to acceleration, it is natural to consider that not only clocks will run at differing rates of time but also that all biologic organisms, including human heart rates and metabolisms, will equally experience differing rates of ageing.

The mathematics for accounting for differing gravitational potential fields is best understood in the general theory as opposed to the special theory of relativity

which reduces down to

in the conventional mathematics of special relativity.

And when the above Riemannian metric for General Relativity is used for an accelerating, non - inertial spacetme system of bodies in relative motion, the Twin Clock Paradox is resolved and time dilation thus becomes mutually reciprocal as between the stationary observer versus the other moving person!

Notice: this generalized Riemannian equation uses only average mean square velocities and does not utilize either the strengths or durations of any bursts of acceleration !

proper time vs. temporal time

Analyzing the Twin Clock Paradox using Doppler frequencies

However, we can still analyze the Twin Clock Paradox using Doppler frequencies that communicate between non - accelerated, relatively stationary earth, and our accelerated rocket as the following explanation provides with the added benefit of deriving the time dilation equation of special relativity using Minkowski's spacetime geometry!

One admission that must be stated is that earlier Doppler was itself derived from time dilation and, therefore, there is somewhat of a tautology being expressed here. However, Doppler can and should be derived separately by means of the original Lorentz Transformation Equations which David Bohm's "The Special Theory of Relativity" accomplishes and to which in some later recapitulation of Relativity Science Calculator this author will submit a simplified version of this derivation. In the meantime, everything shown here is both mathematically and experimentally true and valid.


Earth Observers vs. Astronauts
v/c = 0.60 and ve/c = 0.693
( Galaxy or Star )
dl ∗∗
( earth distance )
( astronaut distance )
dt [2]
( earth time )
( astronaut time )
Milky Way Center 2.5 x 104 ly [a] 2.311 x 104 ly 4.166 x 104 years 3.333 x 104 years
Proxima Centauri 4.223 ly 3.903 ly 7.0 years 5.6 years
Sirius star 8.6059 ly 7.954 ly 14.34 years 11.47 years
Procyon star 11.4043 ly 10.540 ly 19.0 years 15.2 years
Andromeda ( M31 ) 2.544 M ly [b] 2.351 M ly 4.24 x 106 years 3.39 x 106 years
∗∗ caveat lector! reader, please beware of these unsettled observations and numbers.
a,b  ly = light year; M ly = million light years

Using our upper limit for velocity of a photon rocket consisting almost entirely of photon energy in order to maximize deep interstellar Milky Way space travel towards, say, Proxima Centauri dwarf star [ see chapter: Photon Rocket - § boundary condition 100% photon energy ], then let


note: this calculation assumes inertial ( non - gravitational, non - accelerating ) roundtrip motion which on its face is not possible since any "return motion" is gravitational; unless, of course, you wish to consider a sort of "slow motion turns" as an infinite series of inertial ( non - accelerating ) turns for the return trip.

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