Two ways to slower aging - or just one?

Abstract

In the context of teaching relativity, one encounters two situations in which two observers age at different rates. Usually, these effects are treated as if they were independent of each other. Often one is referred to as special-relativistic, the other as general-relativistic. We show that both are special cases of an effect which follows naturally from the fact that space and time form a unity. They exist in a flat space-time and therefore neither of them deserves the label general relativistic.

Full text

Education in Physics Revista Mexicana de F´
ısica E 21 010202 1–3 JANUARY–JUNE 2024
Two ways to slower aging - or just one?
F. Herrmann and M. Pohlig
Karlsruhe Institute of Technology, Faculty of Physics, 76131 Karlsruhe, Germany.
e-mail: f.herrmann@kit.edu; pohlig@kit.edu
Received 14 April 2023; accepted 14 June 2023
In the context of teaching relativity, one encounters two situations in which two observers age at different rates. Usually, these effects are
treated as if they were independent of each other. Often one is referred to as special-relativistic, the other as general-relativistic. We show
that both are special cases of an effect which follows naturally from the fact that space and time form a unity. They exist in a flat space-time
and therefore neither of them deserves the label general relativistic.
Keywords: Twin paradox; special relativity; general relativity.
DOI: https://doi.org/10.31349/RevMexFisE.21.010202
1. Introduction
When teaching relativistic physics, two situations or exper-
iments are usually addressed in which two observers, com-
paring their clocks, discover that one has aged more than the
other, two “clock effects”. Let us briefly recall them. Our
observers are Alice and Bob.
The first experiment is as follows: Alice and Bob adjust
their watches. Alice makes a long journey, moving at con-
stant speed. A comparison of the watches after Alice’s return
shows that more time has passed for Bob than for Alice. Bob
has aged more than Alice.
In the second experiment Alice and Bob start in a
skyscraper on a floor half way up. They adjust their watches.
Then Bob goes up and Alice goes down, and they stay there
for a while. Finally, they return to their starting point. Com-
paring their watches again, they find that more time has
passed for Bob than for Alice; he has aged more. Since in
the following we repeatedly refer to the two clock effects,
we will give them their own names: we call the first one the
travel effect, the second one the skyscraper effect.
One often finds the claim that the travel effect is an effect
of special relativity (SR), whereas the skyscraper effect is an
effect of general relativity (GR), see for example Wikipedia
under the keyword Global Positioning System [1] or Satellite
navigation [2]. Also in schoolbooks one can find this assign-
ment.
In fact, the two clock effects are only two special cases of
a phenomenon which should better be formulated in general
terms. In particular, the skyscraper effect has nothing to do
with the curvature of space. Therefore, it is not appropriate to
call it a GR effect. Actually, this conclusion could be drawn
from the known literature, see for instance [3].
In Sec. 2, we suggest how to address the interlinking of
space and time in the classroom. In Sec. 3 we briefly recall
how to calculate the difference in aging in both cases. In
Sec. 4 we show that the skyscraper effect can be transformed
into a kind of travel effect by simply changing the frame of
reference.
2. Spacetime in elementary education
Special relativity has consequences and makes predictions
that do not correspond to our everyday experience. The stu-
dents need to take note of them, and to accept them as a fact.
We suggest telling something like the following story.
Bob wants to go shopping, Alice wants to exercise on the
sports field. They separate and arrange to meet again later.
Before they separate, they start their pedometers and they
adjust their watches. When they meet again, they compare
their pedometers to see who has walked the longer distance,
and they compare their watches to see who has aged more.
While it is obvious that the distances traveled are most prob-
ably different, anyone would expect that the clocks indicate
the same duration for their absence. But Alice and Bob look
at their watches very carefully and discover that they also dis-
play different times. They know how to arrange it so that the
pedometers show as big a difference as possible: One of them
doesn’t move at all, while the other runs at as high a speed as
possible all the time. But they also want to know howto make
the watches show the longest possible time and the shortest
possible time. They discover the following: For the one who
does not move at all (later in the lesson we specify: the one
who is “free-floating”) the most time passes. And someone
who is moving with (almost) the terminal speed c (almost) no
time passes at all.
3. Calculation of the in aging in both cases
Both the travel effect and the skyscraper effect are special
cases of a phenomenon that we have described in general
terms with the help of our the story in Sec. 2. Other spe-
cial scenarios can also be found in the literature, such as the
one where the twins experience the same acceleration for the
same amount of time, yet age differently [4,5].
What all scenarios have in common is that two clocks are
adjusted, they are then made to move around somehow in dif-

2F. HERRMANN AND M. POHLIG
ferent ways, thereby changing the velocities arbitrarily (and
thus also the accelerations) and going up and down arbitrar-
ily. When they are brought together again, they will display
different durations. The reason for this behavior is that space
and time form a unity: spacetime. It is obvious that, apart
from a few exceptions, accelerations occur during the move-
ments. These can be transformed away, but thereby a gravi-
tational field shows up. We can also express it the other way
round: If a gravitational field appears in a somehow chosen
reference frame, it can be transformed away by a change of
the reference frame. In any case, the existence of a gravita-
tional field in itself is not a manifestation of GR [6].
We remind how the term “GR effect” is commonly used.
It is an effect for whose description one needs the Einstein
field equation, or in other words: for which the Riemann ten-
sor is not equal to zero.
Regarding the skyscraper effect it occurs in a homoge-
neous gravitational field, i.e. in the case where the Riemann
tensor is equal to zero everywhere. Therefore, it is not appro-
priate to call it the GR effect.
But why are these two special cases, i.e. the travel ef-
fect and the skyscraper effect so popular? Because in each of
the two cases a reference frame can be found, in which the
description and the corresponding calculation becomes par-
ticularly transparent.
For the travel effect, this is the reference frame in which
Bob is at rest. In the case of the skyscraper effect, it is the
frame in which the skyscraper is at rest. Let us briefly recall
the calculation of the age difference ∆Tin the two cases.
We start with the travel effect. Let T0be the duration that
Bob measures: from the time of the separation from Alice un-
til the time of her return. From Bob’s point of view, Alice’s
clock - the moving clock - is running slower. He gets the du-
ration Tthat her clock measures by means of the well-known
formula for the “time dilation”:
T=T0
q1−v2
c2
.(1)
Here vis the constant velocity of Alice and cis the terminal
velocity (speed of light). In order to facilitate the compar-
ison of the result with that of the Skyscraper situation, we
slightly transform the equation, assuming that the durations
measured by Bob and Alice differ only by a small amount,
i.e. that ∆T=T−T0¿T
We obtain ∆T
T=1
c2
v2
2.(2)
Now to the Skyscraper effect: It is usually presented as a rel-
ative frequency change ∆f/f of a light signal that Alice is
sending to Bob [7]:
∆f
f=1
c2g∆h. (3)
Here gis the gravitational acceleration, ∆hthe height of our
skyscraper. From this follows the relative time difference
∆T
T=1
c2g∆h. (4)
Let us compare the two results, Eqs. (2) and (4).
In the case of the travel effect, the relative time differ-
ence ∆T/T depends only on the velocity. The reason is that
we had chosen the world lines of Alice and Bob in such a
way that the velocity (more precisely: the velocity difference
between Alice and Bob) is constant during their separation,
except for the short reversal phase.
In the case of the Skyscraper effect, the relative time dif-
ference ∆T/T depends only on the height of the skyscraper.
The reason is that the world lines of Alice and Bob have been
chosen in such a way that the height difference (more pre-
cisely: the gravitational potential difference g·∆hbetween
Alice and Bob) is constant during their separation, except for
the short time intervals in which they ascend or descend in
the building.
The calculation of ∆T/T would be more complicated in
both cases, if one had chosen another, less suitable reference
frame. However, the result would not change.
In the following section we will describe the skyscraper
effect, despite the expected mathematical difficulties, in a ref-
erence frame in which gravitational forces no longer exist.
4. The skyscraper effect in unfamiliar refer-
ence frame
In the reference frame of the earth we have a (nearly) ho-
mogeneous gravitational field. It is responsible for many ev-
eryday phenomena. However, this gravitational field can be
easily transformed away by describing the world in a free-
falling reference frame. In such a reference frame, much of
what we observed before now requires a new interpretation.
FIGURE 1. Carol’s world line between the events P and Q is that
of a free-falling body. In her frame of reference there are no grav-
itational forces. Alice’s and Bob’s clocks show that Bob has aged
more than Alice. (The vertical axis corresponds to Carol’s time, so
neither to Bob’s nor to Alice’s).
Rev. Mex. Fis. E 21 010202

TWO WAYS TO SLOWER AGING - OR JUST ONE? 3
FIGURE 2. For Carol, the skyscraper is small at first. Then it gets
taller and assumes its normal height at the reversal point. There-
after it appears smaller again. For Carol, Alice’s velocity is greater
than Bob’s at every instant, see the two tangents to the world lines.
Here is an example for such a situation that is familiar
to everyone: A body hanging on a spring extends the spring
due to the force of gravity acting on the body - at least this is
how we earthlings describe the phenomenon. A free-falling
observer sees it differently: the spring is extended, but now
because one is accelerating the body by means of the spring.
Let us now describe the Skyscraper effect in such a
gravity-free reference frame.
As already mentioned, Alice and Bob, after having ad-
justed their clocks, start at a middle floor. Then Bob goes up
and Alice goes down, Fig. 1.
Besides Alice and Bob, however, we need a third partici-
pant, Carol. She also takes part in the two clock adjustments.
At the very same moment when Alice and Bob separate in
the middle of the skyscraper (event P in Fig. 1), Carol jumps
up into the sky in such a way that she lands again just for the
second clock adjustment with Alice and Bob (event Q). First,
what the three of them notice when they do the second clock
comparison: Again, Bob has aged more than Alice. In addi-
tion, it turns out that for Carol most time has passed, about
which nobody is surprised, because her reference frame, (the
reference frame in which she is at rest) was an inertial frame,
and in an inertial frame the elapsed time is known to be the
greatest.
We now describe the situation in Carol’s reference frame.
Carol’s world line is a straight line. Since she is in free
fall (also during the upward movement), there are no grav-
itational forces. But how does she describe the motion of
Alice and Bob? Immediately after she jumps up, for her the
skyscraper has a high speed. This means that the skyscraper
is not as high as it used to be before she jumped off. This
is the well-known length contraction, which is an SR effect.
But then the speed of the skyscraper becomes smaller for
Carol, and thus the skyscraper becomes higher, and when
the speed is zero, it takes on its original height. Thereafter
the skyscraper moves back again towards Carol, and does so
faster and faster, whereby its height decreases again.
We can see from the two tangents in Fig. 2 that (in Carol’s
frame of reference) Alice moves faster than Bob at each in-
stant of time. From this, Carol, who remembers the length
contraction effect, concludes that Alice ages less than Bob.
A scenario equivalent to this is described by Thorne [8].
We thus have transformed the skyscraper effect, i.e. the
effect based on gravity, into a travel effect making the field
strength of the gravitational field zero by describing the situ-
ation in Carol’s reference frame. We see that the difference
in aging has nothing to do with general relativity. It is just
another version of the travel effect. The only difference with
the travel effect is that the motion does not occur at a constant
speed.
1.Wikipedia, Global positioning system, “Special and general rel-
ativity predicted that the clocks on GPS satellites, as observed
by those on Earth, run 38 microseconds faster per day than
those on the Earth.” (2023, April 14)
2.Wikipedia, Satellite navigation, “Einstein’s theory of general
relativity is applied to GPS time correction, the net result is that
time on a GPS satellite clock advances faster than a clock on the
ground by about 38 microseconds per day.” (2023, April 14).
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