Report

0
Resolving an example of the Twin Paradox in Special
Relativity
Llyr Humphries – llh18@aber.ac.uk
MPhys - Astrophysics
Institute of Mathematics, Physics and Computer Sciences
Aberystwyth University
4th
-year Project

1
Abstract
The purpose of this report is to outline how the
apparent ‘Twin Paradox’ arises from a thought
experiment involving a pair of twins moving
with a relative relativistic speed and to
subsequently resolve said paradox. The Twin
Paradox has been thoroughly investigated and
resolved, mainly and most informatively by
expressing the significance of acceleration and
thereby exhibiting the non-inertial reference
frames experienced by the travelling twin.
However, virtually no text book nor paper on
the subject matter precisely describe the
deceleration phases or how to derive the
appropriate equations for dealing with
relativistic deceleration. One of the main intents
of this report is to outline these derivations and
summarise the required equations, applying
them to a several facsimiles of a particular
relativistic scenario consisting of a handful of
discrete accelerations and velocities.
The physics of Special Relativity Theory and
the Twin Paradox are thoroughly discussed, as
well as the physics for its resolution. Other
physical aspects and interpretations of the twin
paradox are also discussed, including (but not
limited to) the implementation of gravitational
fields via the use of General Relativity.
Two Relativistic models are used, the first of
which was derived from First Principles, the
Lorentz transformations, and aspects of
Galilean-esque relativity i.e. assuming that the
traveller’s rocket accelerates uniformly when
measured from an inertial frame of reference,
while the other was constructed using trivial
calculus in conjunction with the Lorentz
Transformations and the composition velocity
law. As is presumably immediately discernible,
the former model is based upon an erroneous
premise and is therefore less accurate than the
latter. However, it was decided that a
comparison of both would provide a unique
addition to the report.
Originally, an investigation of a more complex
version of the Twin Paradox was intended,
namely the ‘Triplet Paradox’, whereby a third
observer (second traveller) is involved. Such a
paradox would involve the third observer
experiencing a similar, anti-parallel trip to the
other travelling observer. Unfortunately, due to
time constraints the ‘Triplet Paradox’ scenario
was not explored but is nevertheless briefly
referred to in the discussion section. However,
the renouncement of the Triplet Paradox
investigation allowed for a greater focus on the
main aspect of the investigation i.e. resolving
the Twin Paradox.

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Contents
1. Introduction.........................................................................................................................................3
2. Theory.................................................................................................................................................3
2.1. Frames of Reference ....................................................................................................................3
2.2. The Physics and Philosophy of Space-Time................................................................................4
2.3. Coordinates of multiple Reference Frames..................................................................................4
2.4. Lorentz Transformations..............................................................................................................5
2.5. The Relativity of Simultaneity.....................................................................................................7
2.6. The Twin Paradox........................................................................................................................8
2.7. Resolution #1: At least 2 inertial reference frames......................................................................9
2.8. Resolution #2: Several instances of Accelerations ....................................................................10
2.9. A summary of the scenario and its phases.................................................................................10
2.10. Acceleration analysis: Method #1............................................................................................11
2.11. Acceleration analysis: Method #2............................................................................................12
2.12. A derivation of Method #2’s equations....................................................................................13
3. Experimental Method........................................................................................................................15
4. Results...............................................................................................................................................16
4.1. Method #1..................................................................................................................................16
4.2. Method #2..................................................................................................................................18
4.3. Minkowski Diagrams.................................................................................................................19
5. Discussion.........................................................................................................................................21
5.1. The equations.............................................................................................................................21
5.2. The results..................................................................................................................................22
5.3. Is acceleration necessary? ..........................................................................................................25
5.4. General Relativity......................................................................................................................25
5.5. Mach’s Principle........................................................................................................................28
5.6. Further Investigation..................................................................................................................28
6. Conclusion ........................................................................................................................................29
7. Acknowledgements...........................................................................................................................30
8. References.........................................................................................................................................30
9. Appendix...........................................................................................................................................32
Appendix A – Derivation of the Lorentz Transformations...............................................................32
Appendix B – Composition of Velocities.........................................................................................33
10. Literature Review............................................................................................................................34

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1. Introduction
Since its historic formulation in 1905, the
Theory of Special Relativity has bewildered,
fascinated and perplexed physicists and laymen
alike. Albert Einstein’s analysis of Maxwell’s
equations lead to his dismissal of common
sense notions of Galilean Relativity and
Newtonian Motion and to his eventual
formulation of both the General and Special
Relativity theories. [1]
According to Maxwell’s equations, there are no
frames of reference in which light travels at any
speed other than 𝑐. As Einstein realised, the
counterintuitive consequence of these equ-
ations is that one would determine a beam of
light to be moving away from them at a speed
equal to 𝑐, even if one travels at 𝑐. [1]
One might
be tempted to avoid this nonsensical conclusion
by assuming that the laws which dictate the
nature of light in one set of spatial and temporal
coordinates differs from those in a different
coordinate system, moving relative to the first
observer’s point of view. This left Einstein with
two options; maintaining the common sense
notion of Galilean Relativity at the expense of
altering a law of nature each time it is described
within a different reference frame, or that this
law should be maintained in all frames of
reference at the expense of violating instinctive
notions of spatial and temporal coordinates and
velocity additions. From an epistemological
and philosophical perspective (e.g. Occam’s
Razor), the latter option is the most logical.
Einstein accepted the latter alternative,
postulating the objectivity of the law that
governs the propagation of light (and by
extension, all laws of nature). In other words,
any particular law of nature as described in
different spatial and temporal coordinate
systems must correspond with each other.
The outline of the report is as follows; section 2
addresses and analyses the physics concerning
the axioms of Special Relativity Theory (STR),
time dilation, frames of reference, and the
derivation of relativistic acceleration. Section 3
is concerned with the experimental aspects of
the project, whereby the few programs and
apparatus required to perform the research are
described.
Section 4 encompasses the results of the study:
results obtained from each model are provided
in graphical form, demonstrating the
phenomenon of time dilation and depicting the
asymmetrical ageing experienced between both
observers.
Section 5 discusses the results of section 4,
whereby the data are interpreted with respect to
STR. The equations used in both methods are
inspected (particularly those used in the less
accurate method). Furthermore, alternative
explanations for the phenomenon observed in
the data are considered, including an
assessment of auxiliary documentation based
on said explanations.
Section 6 provides a concise summary of the
results and postliminary conclusions, as well as
suggestions of how to surpass this study
including alternative relativistic scenarios
worth investigating.
Sections 7, 8, 9 and 10 consists of
acknowledgements made by the author for any
form of assistance, a list of references, an
appendix, and a preliminary literature review,
respectively.
2. Theory
2.1. Frames of Reference
STR is based upon two postulates concerning
the speed of light in a vacuum and the laws of
nature;
1. The physical laws of the universe are
identical in all inertial frames of
reference
2. All observers’ measurements of the
speed of light in a vacuum must be the
same, regardless of the source of light’s
motion
It is worth emphasising the significance of
‘inertial’ reference frames in the first postulate;
visualizing this axiom is elementary. Compare
a scenario in which one is sitting in a stationary
car with a bowl of water placed firmly in one’s

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lap, with an analogous scenario in which one is
sitting in said car now travelling at 30 mph
cradling the same bowl of water. One would
likely notice that the water within the bowl
would remain stable in both of these scenarios.
The water would certainly remain stable at any
speed, whether it be 30 mph or 0.99c. These
Frames of Reference (FoR) are said to be
Inertial frames of reference i.e. a system in
which a body remains at rest or moves with a
constant linear velocity relative to another
inertial reference frame. However, if said car
were to accelerate from, say, 30 mph to 100
mph (or vice versa), the water would likely
lurch from the bowl’s periphery. These are
known as Non-inertial FoR i.e. FoR whereby
the relative speed varies. It is likely evident,
then, that the laws of nature (and apparently, the
water) act identically in all Inertial FoR, but the
same cannot be said for Non-inertial FoR [2]
.
2.2. The Physics and Philosophy of
Space-Time
When studying STR, one must do away with
familiar inclinations of motion, space, and time.
An every-day definition of space would likely
allude to the Spinozan and Aristotelian
philosophies of ‘naïve realism’, whereby space
is a three-dimensional entity we directly
perceive, the purpose of which is to be filled
with matter. [3]
Whilst the latter aspect of this
definition is still in debate, the former is worth
focusing on. Various senses allow one’s mind
to draw conclusions about the nature of space.
However, a more accurate understanding of
space stems partly from Bertrand Russell [4]
and
Gottfied Leibniz’s philosophies. [5]
Specifically, Russel states that there is are
fundamental truths of reality beyond – and
independent of – the cosmos’ verisimilitude;
that one’s impressions of reality merely provide
clues to the structure of the universe.
Additionally, Leibniz postulated that one may
only specify a point of space relative to other
spatial points, rather than Newton’s proposition
of absolute space.
Similarly, time is a quantity subject to
philosophical debate. Comparably, Spinozan’s,
Russell’s and Leibniz’s reasoning can be (and
have been) applied to the nature of time
alongside Descartes’ stance which denies the
reality of space and time (and anything else, for
that matter) independent of one’s conscious-
ness. However, Descartes’ ‘idealism’ can be
promptly discarded as it conflicts with the use
of abstract time in field theory. [6]
At the centre of STR, the principle of relativity
asserts that physical laws are classed as
objective entities independent of their
expression by some observer in any (and every)
inertial FoR, which is essentially the first axiom
listed in section 2.1. For ensuring the
objectivity of these laws it was necessary to
unify space and time into space-time, a four
dimensional coordinate system. Space and time
were now considered to be relativistic (non-
absolute) entities i.e. entirely subjective and in
full agreement with Leibniz’s philosophy [7]
. A
purely spatial/temporal interval in one FoR
would necessarily be interpreted as a
combination of spatial and temporal intervals in
another relatively moving FoR [8]
. In order to
facilitate the amalgamation of space and time, a
mutual dimension by which spatial and
temporal measurements may be expressed was
required. This is typically achieved by
multiplying the temporal unit with a conversion
factor; the numerical equivalent of the speed of
light in a vacuum, c. When describing distances
in space and time, then, the lightyear unit is
customarily used e.g. an astronaut pilots a
rocket to a planet 3 lightyears away which takes
them 4.5 lightyears; this is the convention used
throughout this report and during the
investigation.
2.3. Coordinates of multiple Reference
Frames
The above-mentioned conversion allows one to
describe and event; an instance related to a
particular spatial and temporal coordinate
within a certain FoR. Suppose one witnesses the
detonation of a star from a safe distance. With
respect to the star prior to the explosion, one can
be considered to be at rest. One may then
measure the spatial and temporal coordinates
(denoted as x. y, z and t, respectively) of the
detonation relative to one’s inertial reference

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frame, which for convenience will be denoted
as 𝑆. Suppose that a second observer witnesses
the destruction of this star and that their spatial
and temporal axes coincide with that of 𝑆 at the
moment of obliteration (i.e. at 𝑡 = 0), but is
moving with a velocity 𝑣 realtive to 𝑆, which
will be denoted 𝑆′
(see figure 1). The second
observer may also measure the spatial and
temporal coordinates (detonated as x’, y’, z’ and
t’, respectively) of the explosion with respect to
their FoR. An event (such as the stellar
detonation) can be entirely expressed using
these eight space-time coordinates. [9]
Incidentally, it is worth mentioning that
‘observer’ in this context has no anthro-
pomorphic focus; it is merely a mathematical
context from which events are evaluated. [11a]
Therefore, ‘observer’ may be applicable to
anything from a creature on a distant planet to a
high-speed electron darting through a particle
accelerator. Also, considering that there are no
absolute reference frames in STR, anything can
be considered to be moving with respect to
anything else, which, to reiterate, renders
obsolete the concept of space and time as
separate, objective entities.
In any case, a given observer must be capable
of expressing their own temporal and spatial
coordinates of an event as well as those
experiences by a relatively moving observer.
2.4. Lorentz Transformations
As previously stated, relatively moving
observers require transformations between the
subjective temporal and spatial expressions of
an event with respect to their corresponding
coordinate systems. Galileo Galilei is
documented as the first to have reached this
conclusion while studying the motion of both
celestial and terrestrial objects, consequently
attaining a crude understanding (with respect to
our current apprehension of the Relativity
Theory) of the principle of relativity. [12]
Consider a scenario in which person A is seated
within an airplane moving at a certain speed 𝑣
relative to the ground. Far below the airplane is
person B, sitting on the ground watching the
airplane move overhead. Person A sits at the
back of the airplane, x distance from the
cockpit. From B’s reference frame (who’s said
to be stationary relative to A), A’s position
would alter from 𝑥 to 𝑥′
= 𝑥 + 𝑣𝑡 in time 𝑡.
However, from A’s reference frame, their
distance 𝑥 from the cockpit remains unchanged
for the duration of the flight. Ignoring engine
noise and turbulence, A would be entirely
unaware of any motion if they were to avoid
gazing out the window [13]
. If A were to peer out
of the plane and spot person B on the ground,
they would affirm that B’s position would have
shifted by 𝑥 = 𝑥′
− 𝑣𝑡′ in time 𝑡′, since B is
viewed to be travelling at −𝑣 relative to the
airplane. Only when 𝑣 = 0 would A’s spatial
expression 𝑥 coincide with B’s spatial
expression 𝑥′ (see figure 2). Galileo asserted
that A’s and B’s statements are equivalent,
provided that one can convert from A’s (or B’s)
statement to B’s (or A’s) statement by
producing B’s (or A’s) statement from a
rearrangement of A’s (or B’s) statement. This
may be done by setting 𝑡 = 𝑡′, which is an
expression of Galileo’s assertion that time is
absolute i.e. independent of an observer’s
expression within any FoR.
Figure 1: Diagram of the S and S’ reference
frames, whereby S’ moves with a velocity v
with respect to S along the x-axis only. The
FoR share a common origin at 𝑡 = 0 [10]

6
Similarly, suppose that person A throws a stone
from a cliff face with an initial velocity 𝑣0
(whereby the positive axis points from the top
to the bottom of the cliff); the equation
governing the stone’s displacement in time 𝑡 is
𝑥 = 𝑣0 𝑡 +
1
2
𝑎𝑡2 (2.4.1)
where 𝑎 represents the acceleration experienced
by the stone. Suppose now that B is in
possession of a jetpack and uses it to scale the
cliff face with a constant velocity of – 𝑣 as the
stone plummets to the ground. B’s observation
of the stone’s motion would correspond to 𝑥′
=
𝑣0
′
𝑡′
+
1
2
𝑎𝑡′2
. Considering Galileo’s assertion
of absolute time, B’s observation becomes:
𝑥′
= 𝑣0
′
𝑡 +
1
2
𝑎𝑡2 (2.4.2)
where 𝑣0
′
= 𝑣0 + (−𝑣). By substituting the
expression for 𝑣0
′
and 𝑥′
= 𝑥 + 𝑣𝑡 into
equation 2.4.2, it should be evident that this
results in a reduction to equation 2.4.1.
An extrapolation of this result to every physical
law implied that a one-to-one correspondence
in all relatively moving reference frames was a
necessity for determining the validity of any
particular natural law. [11b]
However, during the late 19th
century while
studying Maxwell’s equations, Hendrik
Lorentz discovered that Galileo’s principle of
relativity didn’t apply to electromagnetic laws;
as mentioned in the introduction, it was found
that the speed of light in a vacuum would equal
c, regardless of the FoR from which it is
measured. Lorentz discovered that in order to
demonstrate a conformity of electromagnetic
laws in different FoRs, Galileo’s expressions
for spatial and temporal transformations
required alterations [15]
:
𝑥′
= 𝛾(𝑥 − 𝑣𝑡) (2.4.3)
𝑥 = 𝛾(𝑥′
+ 𝑣𝑡′) (2.4.4)
𝑡′
= 𝛾 (𝑡 −
𝑣𝑥
𝑐2
) (2.4.5)
𝑡 = 𝛾 (𝑡′
+
𝑣𝑥′
𝑐2 )
(2.4.6)
where 𝑐 represents the speed of light in a
vacuum, 𝑣 represents the relative uniform
motion between reference frames 𝑆 and 𝑆′
, and
𝛾 represents the Lorentz factor [16]
, which may
be expressed in the following form:
𝛾 =
1
√1 −
𝑣2
𝑐2
(2.4.7)
These are known as the Lorentz
Transformations. For a derivation of said
transformations using Einstein’s reasoning, see
Appendix A. [17]
Initially, Lorentz attempted to interpret these
transformations as physical results of the
luminiferous aether (see section 10). Einstein
instead interpreted them as conversions
between subjective spatial and temporal
parameters, thus doing away with the Galilean
notion of absolute time and the Newtonian
notion of absolute space as an assertion of the
objectivity of all physical laws. However,
Einstein’s and Lorentz’s coordinate transform-
Figure 2: Configuration of two coordinate
systems, S and S’ moving with relative speed
𝑉. The origin represents a point at which 𝑣 =
0 and where 𝑥 = 𝑥′ [14]

7
ations must encapsulate those expressed by
Galileo and Newton if one is to welcome them
with certainty. In Galileo’s experiments, the
relative speed of a moving object would have
been very much slower than the speed of light
i.e. 𝑣/𝑐 ~ 0. Applying this to the Lorentz
Transformations, it’s fairly easy to see that they
reduce to the following approximations:
𝑥′ = 𝑥 + 𝑣𝑡
𝑥 = 𝑥′
− 𝑣𝑡′
𝑡 = 𝑡′
whereby lim
𝑣/𝑐→0
𝛾 = 1, which correspond to
Galilean transformations and the notion of
objective time.
All equations described in this section assume
uniform, rectilinear motion along the x-axis.
For such an assumption, the y and z coordinates
remain unchanged in both Galileo and
Lorentz’s expressions of space/time
transformations, i.e. 𝑦 = 𝑦′ and 𝑧′
= 𝑧. While
the main concern of this analysis most certainly
focuses on varible motion, the assertion of
rectilinear motion (along the x-axis) is
maintained for the entirety of this report.
Incidentally, from this point forward, ‘A’, ‘𝑆’
and ‘earthbound’ are interchangeable when
describing the stationary twin, while ‘B’, ‘𝑆′
’
and ‘rocket-bound’ are interchangeable when
describing the travelling twin.
2.5. The Relativity of Simultaneity
As stated throughout sections 2.2-2.4, (𝑥, 𝑡)
(𝑥′
, 𝑡′
) do not refer to any objective definition
of space and time; they merely expedite their
description. [11c]
From the Lorentz transform-
ations, 𝑥 and 𝑡 represent conceptual coordinates
used by the stationary observer A within some
reference frame, whereby the 𝑥 coordinate is
measured relative to some arbitrary point within
the A’s FoR, and the 𝑡 coordinate relates to the
rate at which some physical clock ticks within
same FoR. [11c]
Additionally, 𝑥′
and 𝑡′
are also
abstract coordinates, whereby 𝑥′
is also used by
A but refers to B’s relatively moving coordinate
frame as B moves away from A with velocity 𝑣
while 𝑡′
relates to the rate at which some
physical clock within B’s FoR ticks (as
measured by A). While the coordinates (𝑥, 𝑡)
and (𝑥′
, 𝑡′
) are interpreted from A’s
perspective, one may certainly define these
coordinates from B’s perspective; alternatively,
B may be considered to be the stationary
observer while A moves away from them with
velocity −𝑣). As previously stated, doing so
should not vary the structure of natural laws,
whether they’re expressed from A’s FoR or
from B’s FoR.
One may find that manipulations of the Lorentz
transformations yield counter-intuitive conclu-
sions, particularly concerning the relative
passage of time. Consider the following: an
event occurs at (𝑥1, 𝑡1) while some other,
separate event occurs at (𝑥2, 𝑡2). For relatively
moving FoRs, the appropriate Lorentz
transformation (equation 2.4.5) is as follows:
𝑡2
′
− 𝑡1
′
= 𝛾 [(𝑡2 − 𝑡1)
+
𝑣
𝑐2
(𝑥2 − 𝑥1)]
(2.5)
If two spatially separate events occur
simultaneously in A’s reference frame, then one
may state that 𝑡2 = 𝑡1. However, equation 2.5
suggests that the two events do not occur
simultaneously from B’s FoR (as measured by
A) i.e. 𝑡2
′
≠ 𝑡1
′
which clearly demonstrates the
objectivity of the temporal coordinate i.e. 𝑡′
≠
𝑡, and the phenomenon known as time dilation.
In summation, two events which appear to
occur simultaneously within a particular frame
of reference would appear not to occur
simultaneously from a relatively moving frame
of reference; this is the nature of the relativity
of simultaneity. At first, arriving at an absolute
conclusion of the simultaneity of certain events
may seem impossible if said events occur at
different points in space (following the
conclusion drawn from equation 2.5). Suppose
now that two asteroid collisions occur on the
surface of the Earth, separated by an arbitrary
distance (see figure 3). From an earthbound
observer’s (A) FoR, the impacts occur
simultaneously.

8
However, an observer in a relatively moving
frame of reference (B) may not observe the
impacts to occur simultaneously [20]
. If both
events can be causally connected, the order of
events can be conserved; B must calibrate their
clock to coincide with the Earth’s reference
frame and an observer within said reference
frame. In other words, if B uses the Lorentz
transformations to resolve their motion-
dependent time coordinate and appreciates that
these physical actions (the asteroid impacts)
occur within a co-moving reference frame (with
respect to said physical actions) – which in this
scenario is A’s FoR – they may then agree with
A that the events do in fact occur
simultaneously within A’s FoR [11c]
(see figure
4 for a visual aid). This conclusion coincides
somewhat with Spinozan realism and the
assertion of the existence of a real universe, the
physical characteristics of which are
independent of the motion of observers and
their relative descriptions of said character-
istics.
2.6. The Twin Paradox
Of all the apparent paradoxes fomented by
Relativity Theory, the ‘Twin’ Paradox (or
‘Clock’ Paradox) is arguably the most famous
and controversial. The twin paradox is a
thought experiment whereby a stationary twin
(twin A) remains on Earth, while their identical
twin (twin B) makes a journey into space
aboard a high-speed rocket. [21]
Each twin
clutches an atomic pocket watch capable of
perfect time-keeping i.e. unaffected by intense
acceleration and no loss of accuracy. A’s watch
remains within an (apparent) ‘inertial’
reference frame, while B’s watch, which
initially was in agreement with A’s watch, is
taken along a space-time path corresponding to
the travelling rocket’s journey. Upon B’s
return, B’s clock will have lost time compared
to A’s. The asymmetric ageing between both
twins is determined from equations 2.4.5 and
2.4.6.
If there is indeed a one-to-one correspondence
between the time interval of a clock in motion
relative to A (as concluded by A) with
physically evolving processes within B’s
(moving) reference frame e.g. the decay of an
arbitrarily heavy atomic nucleus or the
biological decay of organic tissue, then A
would determine that B ages at a slower rate
Figure 3: Image crudely depicting two asteroid
impacts separate by an arbitrary distance. The
events are said to occur simultaneously from
an earthbound observer’s reference frame [18]
Event 2
Event 1
Figure 4: Minkowski diagram of three
relatively moving reference frames. Event A is
simultaneous with B in the green FoR, precedes
B in the red FoR, and follows B in the blue FoR
[19]

9
than the aging process experienced within A’s
own reference frame. [11d]
One would then
conclude that twin B would be younger than
twin A upon B’s return to Earth. However,
according to the principle of relativity, motion
is considered only as a subjective aspect of a
description of the laws of nature. [11d]
In this
sense, B may be considered to be stationary
while A moves away with a relative velocity
−𝑣. A’s analysis would now be carried out by
B, leading B to the conclusion that A has aged
less than themselves when A returns. Both
observers determine that their respective twin
has aged less than them; a logical paradox [22]
.
Paradoxes of this nature typically appear due to
incomplete or erroneous inferences of a
particular theory.
2.7. Resolution #1: At least 2 inertial
reference frames
While the resolution of the twin paradox has
been widely disputed, one particular resolution
emerges from an analysis of the paradoxical
scenario’s premises. A closer inspection of B’s
journey in its simplest form reveals that B
undergoes (at least) three distinct events:
1. B begins their journey, reaching a
constant velocity 𝑣 within a negligibly
short time
2. After journeying for some distance, B
suddenly reverses their velocity
3. B returns to their starting point on
Earth, immediately stopping upon
arrival
The paradox arises from an ignorance of the
basic tenets of relativity; it is assumed that the
travelling twin remains within a single inertial
reference frame for the entire trip. However, the
apparent paradox crumbles upon appreciating
that the earthbound twin remains in a single
inertial FoR throughout the journey while the
travelling twin switches at the midpoint of the
trip from being at rest within an inertial frame
with velocity 𝑣 (away from Earth) to being at
rest within another inertial frame with velocity
−𝑣 (towards Earth). This would result in an
asymmetry path between the space-time paths
of both twins, and would therefore also result in
asymmetric ageing between both twins during
the trip [23]
.
Considering the relativity of simultaneity, a
moment-by-moment understanding of the
traveller’s voyage unfolds. As stated earlier,
event’s that occur simultaneously in one inertial
frame do not occur simultaneously within
another, relatively moving inertial reference
frame; different sets of events occur
simultaneously within different frames of
reference. One would deduce the simultaneity
of events via a Minkowski space-time diagram;
a two-dimensional graph used to portray world-
lines as curves in planes corresponding to
motion along a spatial axis. [24]
The x-axis typically depicts spatial
displacement, while the y-axis is usually
temporal. Each point on such a diagram
represent an event, regardless of whether
something significant occurs at this particular
position and time. Figure 5 depicts a
Minkowski diagram of two relatively moving
reference frames; the resting frame (𝑥, 𝑡) and
the moving frame (𝑥′
, 𝑡′). The diagram also
demonstrates how an event at A is assigned a
different spatial and temporal coordinate
depending on the observer.
The angle 𝛼 between 𝑥 and 𝑥′
(or 𝑡 and 𝑡′
) is
determined by the frames’ relative velocity;
𝛼 = arctan(𝑣/𝑐). The 𝑥′
-axis may be used to
determine which events an observer in 𝑆′
may
interpret to occur simultaneously i.e. any two
events which lie on the 𝑥′
-axis occur
Figure 5: Minkowski diagram of two relatively
moving FoRs [25]

10
simultaneously in terms of 𝑡′
. Figure 6
demonstrates how 𝑥′
-axes may be used as
planes of simultaneity.
Upon B’s switch from one inertial frame of
reference to another, an adjustment of these
simultaneity planes is required. Figure 6 is
drawn from the earthbound twin’s frame of
reference, such that their world-line is parallel
with the vertical axis i.e. their spatial coordinate
remains unchanged while they move forward
through time.
During the first half of the trip, B’s
displacement from Earth increases, moving
them to the right on the space-time diagram
shown above. During the second half, B makes
their way back to Earth, thus decreasing their
displacement and moving them to the left on the
space-time diagram. The blue lines represent
the planes of simultaneity for the travelling twin
during B’s outward journey, while the red lines
represent the planes of simultaneity during B’s
return. Immediately before and after the point
of turnaround, B would calculate their twin’s
age by measuring the interval along the
temporal axis from the origin to the top-most
blue line and from the origin to the lowest red
line, respectively. During the turnaround phase
the plane of simultaneity alters from blue to red,
bypassing a large portion of the earthbound
twin’s world-line, which accounts for the
asymmetrical ageing experienced between both
twins. The paradox is resolved, whereby the
earthbound twin ages more than their rocket-
bound counterpart. [27] [28]
2.8. Resolution #2: Several instances of
Accelerations
A similar resolution may be applied which
involves accelerations. The impossibility of
such a journey is likely obvious; an
instantaneous change from 0 𝑚𝑠−1
(or any
speed, for that matter) to any other speed
(relativistic or otherwise) demands an infinite
acceleration. The resolution, therefore, stems
from the consideration of B undergoing at least
four acceleration stages:
1. Initial acceleration from Earth towards
some distant point, achieving a
maximum velocity 𝑣
2. Deceleration towards some distant
point until a velocity of 0 is achieved
3. Acceleration back towards Earth until a
velocity of −𝑣 is achieved
4. Deceleration towards Earth, stopping at
the starting point
The Lorentz Transformations are applicable
only to measurements of time from reference
frames moving uniformly with respect to each
other. Therefore, another resolution of the
paradox is to attribute the asymmetrical ageing
to the acceleration experienced by B at various
stages of their journey; this is the primary focus
of this study and will be discussed further in
sections 2.9 – 2.12.
2.9. A summary of the scenario and its
phases
For the purpose of this study, the traveller’s
journey was divided into the following phases,
whereby 𝑆 denotes the earthbound reference
frame and 𝑆′ denotes the Rocket’s series of
reference frames (the rocket’s journey from
Earth to some distant planet is considered to lie
along the positive x-axis):
1. The rocket accelerates for a distance
𝑋 𝑎 = 3 𝑙𝑦𝑟𝑠 (as measured from 𝑆) until
it reaches some velocity 𝑣 (also
measured by 𝑆)
Figure 6: Minkowski diagram of the twin
‘paradox’, depicting the difference between the
world-lines of both twins. The diagonal lines
represent the two reference frames experienced
by the travelling twin [26]

11
2. The rocket coasts at 𝑣 across a distance
𝑋 𝑏 = 4 𝑙𝑦𝑟𝑠 (as measured from 𝑆)
3. The rocket fires its engines in the
opposite direction, such that it now
accelerates along the negative x-axis
with equal yet opposite magnitude as
phase 1 for some distance 𝑋 𝑎 as
measured by 𝑆 until the rocket is at rest
with respect to 𝑆
4. The Rocket remains stationary at some
distant planet for some time 𝑇𝑃 =
1 𝑦𝑒𝑎𝑟
5. The rocket reignites its engines and
accelerates along the negative x-axis
for some distance 𝑋 𝑎 (according to 𝑆),
until 𝑆′ reaches some velocity −𝑉 (also
according to 𝑆)
6. The Rocket coasts towards 𝑆 at velocity
−𝑉 for some distance 𝑋 𝑏 (according to
𝑆)
7. The Rocket fires its engines in the
direction of 𝑆, accelerating along the
positive x-axis with equal magnitude as
phase 1 for some distance 𝑋 𝑎
(according to 𝑆), until A and B are
reunited
While conducting research into the equations
required for an analysis of the non-inertial
phases of B’s journey, it was found that two
dissimilar methods may be incorporated, which
are discussed below.
2.10. Acceleration analysis: Method #1
The first method assumes that 𝑆′
’s rocket
accelerates at a uniform rate as measured by 𝑆
during phases 1, 3, 5 and 7, such that 𝑆′
’s
velocity (as measured by 𝑆) may be described
in the following fashion:
𝑣(𝑡) = 𝑣0 + 𝑎𝑡 (2.10.1)
whereby 𝑣0 represents the initial velocity at the
beginning of any phase, 𝑎 represents 𝑆′
’s
acceleration (as measured by 𝑆), and 𝑡
represents time (as measured by 𝑆). Equation
2.10.1 may also be expressed in the following
form:
𝛽(𝜏) = 𝛽0 + 𝛼𝜏 (2.10.2)
whereby 𝛽0 = 𝑣0/𝑐, 𝜏 = 𝑐𝑡, 𝛼 = 𝑎/𝑐2
, and 𝑐
represents the speed of light in a vacuum. From
this, the Lorentz factor (equation 2.4.7)
becomes:
𝛾 =
1
√1 − 𝛽(𝜏)2
By using equation 2.4.6 and the fact that from
𝑆′
’s perspective, 𝑆′
doesn’t move i.e. Δ𝑥′
= 0,
then [29]
:
d𝜏 = 𝛾(d𝜏′
+ 𝛽d𝑥′) = 𝛾d𝜏′
⇒ 𝑑𝜏′
=
𝑑𝜏
𝛾
(2.10.3)
Using equation 2.4.1, the time of flight for the
first phase of the journey as measured by 𝑆 may
be determined. If one applies the same
manipulations performed on equation 2.10.1,
equation 2.4.1 becomes:
𝑑𝑥 = 𝑥2(𝜏2) − 𝑥1(𝜏1) = 𝛽0 𝜏 +
1
2
𝛼(𝑑𝜏)2
During phase 1, 𝑥1 = 𝛽0 = 0, therefore:
𝑑𝑥 =
1
2
𝛼(𝑑𝜏)2 (2.10.4)
and
𝛽 = 𝛽0 + 𝛼𝑑𝜏 = 𝛼𝑑𝜏
Combining these, one may define the time of
flight during any acceleration phase – due to
the rocket accelerating/decelerating at equal
rates during each acceleration/deceleration
phase – with respect to the travel distance and
the rocket’s velocity (all of which are measured
by 𝑆):
𝑑𝑥 =
1
2
𝛼(𝑑𝜏)2
=
1
2
𝛽𝑑𝜏
⇒ 𝑑𝜏 =
2𝑑𝑥
𝛽
As previously mentioned, any frame of
reference which is moving with some constant
speed 𝑉 with respect to some other frame of
reference is considered to be an inertial
reference frame. Within these reference frames,
one may use the Lorentz transformations as a

12
method of translation between subjective
coordinate systems. It naturally follows, then,
that any reference frame which is moving with
variable speed i.e. an accelerating reference
frame, is defined as a non-inertial reference
frame, whereby the Lorentz transformations do
not apply.
However, it is possible to define an object’s
velocity at any time during its acceleration. It is
also possible to define an object within an
accelerating reference frame as an object
accelerating through several inertial reference
frames [30]
. In order to derive some form of
translation between 𝑆 and 𝑆′
’s coordinate
systems and 𝑆′
’s temporal coordinate, an
integration across all inertial reference frames
is required:
Δ𝜏′
= ∫ 𝑑𝜏′
𝜏
0
= ∫
𝑑𝜏
𝛾
𝜏
0
⇒ Δ𝜏′
= ∫ √1 −
𝑣2
𝑐2
𝜏
0
𝑑𝜏 = ∫ √1 − 𝛽2
𝜏
0
𝑑𝜏
⇒ Δ𝜏′
= ∫ √1 − (𝛽0 + 𝛼𝜏)2
𝜏
0
𝑑𝜏
(2.10.5)
⇒ Δ𝜏′
=
1
2𝛼
[√1 − (𝜇)2(𝜇)
+ sin−1(𝜇)]
𝜏1
𝜏2
(2.10.6)
whereby 𝜏′
= 𝑐𝑡′
and 𝜇 = 𝛽0 + 𝛼𝜏. The
equation above may be applied to each
acceleration phase, while the Lorentz
Transformations may be applied to the constant
velocity phases. The equation above simplifies
to the following form provided that 𝜏0 = 0 at
the beginning of each acceleration phase:
Δ𝜏′
=
1
2𝛼
[√1 − (𝛽0 + 𝛼𝜏)2(𝛽0 + 𝛼𝜏)
+ sin−1(𝛽0 + 𝛼𝜏)
− 𝛽 𝑜√1 − 𝛽0
2
+ sin−1(𝛽0)]
Incidentally, if the rocket ceases accelerating
i.e. 𝛼 = 0 during phases 2, 4 and 6, equation
2.10.5 reduces to the following form:
Δ𝜏′
= ∫ √1 − (𝛽0 + 0)2
𝜏
0
𝑑𝜏
⇒ Δ𝜏′
= ∫ √1 − 𝛽0
2
𝜏
0
𝑑𝜏 = √1 − 𝛽0
2
𝑑𝜏 =
𝑑𝜏
𝛾
which resembles the result determined via the
Lorentz transformations (equation 2.10.3)
provided that 𝑥′
= 0.
2.11. Acceleration analysis: Method #2
The second method also assumes that 𝑆′
accelerates uniformly during each acceleration/
deceleration phase (throughout this report,
deceleration refers to an acceleration along the
negative x-axis). However, the acceleration is
said to be constant as measured from 𝑆′
’s
perspective within Momentary Co-moving
Reference Frames (MCRFs); inertial frames
moving at the same velocity as the traveller at a
particular moment. This acceleration may be
defined in the following manner [31]
:
𝑑𝑣′
𝑑𝜏′
= 𝛼′
Once again one may derive 𝑑𝑡/𝑑𝑡′ = 𝛾 by
assuming that 𝑥′
= 0 in the Lorentz
transformations. The rocket’s velocity it
considered to be zero from the rocket’s frame
of reference, which provides one with the
following result:
𝑣1
′
= 𝑣0
′
+ 𝑑𝑣′
= 𝑑𝑣′
By using the composition law of velocities:
𝑣′ =
𝑣 − 𝑉
1 − 𝑉𝑣
Whereby 𝑉 represents the velocity of the
MCRF relative to Earth’s FoR, and 𝑣1
represents the rocket’s current velocity (as
measured from Earth’s FoR) [20] [32]
together
with the previous result, we arrive at the
following equation [33]
:
𝑑𝑣′
=
𝑣 − 𝑉
1 − 𝑉𝑣
=
𝑑𝑣
1 − 𝑉2
= 𝛾2
𝑑𝑣
whereby 𝑉 = 𝑣1 i.e. the MCRF’s relative
velocity is equal to the rocket’s current velocity,
logically. The derivation of the composition

13
law is not necessary here, but is included in the
appendix.
The rocket’s acceleration from the Earth’s
frame of reference, therefore, may be
determined in the following manner [34]
:
𝑑𝑣′
𝑑𝑡′
=
𝛾2
𝑑𝑣
1
𝛾 𝑑𝑡
⇒
𝑑𝑣
𝑑𝑡
=
1
𝛾3
𝑑𝑣′
𝑑𝑡′
=
𝛼′
𝛾3
⇒ 𝛼 =
𝛼′
𝛾3
(2.11.1)
From this, an expression for the rocket’s
velocity (as measured from 𝑆) as a function of
time (as measured from 𝑆) follows:
𝑣1 =
𝛼′
𝑡 + √ 𝜀
√1 + (𝛼′ 𝑡 + √ 𝜀)
2
whereby 𝑣1 represents the current relative
velocity (as measured from 𝑆), respectively,
and
𝜀 =
𝑣0
2
1 − 𝑣0
2
whereby 𝑣0 represents the previous relative
velocity. Also determined were expressions for
the Lorentz Factor:
𝛾 = √1 + (𝛼′ 𝑡 + 𝜀)2 (2.11.2)
displacement (as measured from 𝑆):
Δ𝑥 =
1
𝛼′
[√ 𝛼′2
𝑡2 + 𝜀 + 2𝛼′
√ 𝜀𝑡 + 1
− √𝜀 + 1]
(2.11.3)
and time (as measured from both 𝑆 and 𝑆′
,
respectively):
Δ𝑡
= −
√ 𝜀
𝛼′
± √
𝜀
𝛼′2 + 𝑥2 +
2𝑥
𝛼′
√𝜀 + 1
(2.11.4)
Δ𝑡′
=
1
𝛼′ [sinh−1
(𝛼′
𝑡 + √ 𝜀)
− sinh−1
(√ 𝜀)]
(2.11.5)
Within the 𝜀 expression, 𝑣0 represents the
initial velocity during each individual phase i.e.
𝑣0 = 0 during phase 1, 𝑣0 = 𝑣 𝑚𝑎𝑥 during
phase 2 etc. Each of these equations are
applicable to any and all acceleration phases,
while the Lorentz Transformations are
applicable to phases involving constant relative
motion. The derivation of the above equations
are demonstrated in the following sub-section.
2.12. A derivation of Method #2’s
equations
We begin with an expression which connects
the accelerations from 𝑆 and 𝑆′
’s FoR and
MCRFs, respectively;
𝛼 =
𝛼′
𝛾3
From this, we may find the following
expression:
𝛼′
=
1
(1 − 𝑣2)
3
2
𝑑𝑣
𝑑𝑡
By assuming that 𝛼′
is constant, the equation
above can be integrated;
𝛼′
∫ 𝑑𝑡
𝑡1
𝑡0
= ∫
𝑑𝑣
(1 − 𝑣2)
3
2
𝑣1
𝑣0
(2.12.1)
By assuming that 𝑣(0) = 𝑣0 = 0 and 𝑡0 = 0,
then equation 2.12.1 becomes
𝛼′
𝑡 =
𝑣
√1 − 𝑣2
Incidentally, the assumption 𝑡0 = 0 will be
implemented in each phase.
However, if 𝑣0 ≠ 0, then equation 2.12.1
becomes
𝛼′
𝑡 =
𝑣1
√1 − 𝑣1
2
−
𝑣0
√1 − 𝑣0
2
(2.12.2)
Here, 𝑣0 equals the maximum speed achieved
during the previous phase i.e. 0 for phase 1,

14
𝑣 𝑚𝑎𝑥 during phases 2 and 3 etc). From equation
2.12.2, 𝑣1 – the rocket’s velocity at any time 𝑡
– may be determined;
𝑣1
2
1 − 𝑣1
2 = (𝛼′
𝑡 +
𝑣0
√1 − 𝑣0
2
)
2
⇒ 𝑣1
2
= (1 − 𝑣1
2) (𝛼′
𝑡 +
𝑣0
√1 − 𝑣0
2
)
2
⇒ 𝑣1
2
[1 + (𝛼′
𝑡 +
𝑣0
√1 − 𝑣0
2
)
2
]
= (𝛼′
𝑡 +
𝑣0
√1 − 𝑣0
2
)
2
⇒ 𝑣1
=
𝛼′
𝑡 +
𝑣0
√1 − 𝑣0
2
√1 + (𝛼′ 𝑡 +
𝑣0
√1 − 𝑣0
2
)
2
(2.12.3)
Also, if 𝑣0 = 0 (during phases 1, 4 and 5),
equation 2.12.3 reduces to
𝑣1 = 𝑣 =
𝛼′
𝑡
√1 + 𝛼′2
𝑡2
By applying equation 2.12.3 to the Lorentz
Factor, we find that
𝛾 =
1
√1 − 𝑣1
2
=
1
√
1 −
(𝛼′ 𝑡 +
𝑣0
√1 − 𝑣0
2
)
2
1 + (𝛼′ 𝑡 +
𝑣0
√1 − 𝑣0
2
)
2
Which may be simplified to
𝛾 = √1 + (𝛼′ 𝑡 +
𝑣0
√1 − 𝑣0
2
)
2
If 𝑣0 = 0 then the equation above reduces to
𝛾 = √1 + 𝛼′2
𝑡2
To determine Δ𝑥 (the rocket’s displacement
from Earth as measured from 𝑆) equation 2.12.3
must be integrated with respect to 𝑡;
Δ𝑥 = ∫ 𝑣1 𝑑𝑡
𝑡1
𝑡0
= ∫
𝛼′
𝑡 +
𝑣0
√1 − 𝑣0
2
√1 + (𝛼′ 𝑡 +
𝑣0
√1 − 𝑣0
2
)
2
𝑑𝑡
𝑡1
𝑡0
= [
1
𝛼′
√(𝛼′ 𝑡 +
𝑣0
√1 − 𝑣0
2
)
2
+ 1]
𝑡0
𝑡1
=
1
𝛼′ [√𝛼′2
𝑡2 +
𝑣0
2
1 − 𝑣0
2 +
2𝛼𝑣0 𝑡
√1 − 𝑣0
2
+ 1]
𝑡0
𝑡1
Once again 𝑡0 = 0, therefore:
Δ𝑥
=
1
𝛼′
[√ 𝛼′2
𝑡2 + 𝜀 + 2𝛼′
√ 𝜀𝑡 + 1
− √𝜀 + 1]
(2.12.4)
whereby, once again;
𝜀 =
𝑣0
2
1 − 𝑣0
2
If 𝑣0 = 0, then the above equation reduces to
Δ𝑥 =
1
𝛼′
[√ 𝛼′2
𝑡2 + 1 − 1]
The journey’s duration for each individual
phase as measured from the stationary
observer’s FoR may be determined by
rearranging equation 2.12.4;
(𝛼′
Δ𝑥 + √
𝑣0
2
1 − 𝑣0
2 + 1)
2
= 𝛼′2
𝑡2
+
𝑣0
2
1 − 𝑣0
2 +
2𝛼′
𝑣0 𝑡
√1 − 𝑣0
2
+ 1
From here, Δ𝑥 may be referred to as 𝑥, since
𝑥0 = 0 ⇒ 𝑥1 − 𝑥0 = 𝑥1 = 𝑥.
⇒ 𝛼′2
𝑥2
+ 2𝑥𝛼′√
𝑣0
2
1 − 𝑣0
2 + 1 = 𝛼′2
𝑡2
+
2𝛼′
𝑣0 𝑡
√1 − 𝑣0
2

15
⇒ (𝛼′2
)𝑡2
+ (
2𝛼′
𝑣0
√1 − 𝑣0
2
) 𝑡
− (𝛼′2
𝑥2
+ 2𝑥𝛼′√
𝑣0
2
1 − 𝑣0
2 + 1)
= 0
Next, the Quadratic Formula can be used by
defining the following;
𝑎 = 𝛼′2
𝑏 =
2𝛼′
𝑣0
√1 − 𝑣0
2
𝑐 = − (𝛼′2
𝑥2
+ 2𝑥𝛼′√
𝑣0
2
1 − 𝑣0
2 + 1)
⇒ 𝑡 =
−𝑏 ± √𝑏2 − 4𝑎𝑐
2𝑎
= −
2𝛼′
𝑣0
2𝛼′2
√1 − 𝑣0
2
±
√
4𝛼′2
𝑣0
2
1 − 𝑣0
2 + 4𝛼′2
(𝛼′2
𝑥2 + 2𝑥𝛼′√
𝑣0
2
1 − 𝑣0
2 + 1)
2𝛼′2
⇒ 𝑡
= −
𝑣0
𝛼′√1 − 𝑣0
2
± √
4𝛼′2
𝑣0
2
4𝛼′4
(1 − 𝑣0
2)
+
4𝛼′4
𝑥2
4𝛼′4 +
8𝑥𝛼′3
4𝛼′4
√
𝑣0
2
1 − 𝑣0
2 + 1
⇒ 𝑡
= −
𝑣0
𝛼′√1 − 𝑣0
2
± √
𝑣0
2
𝛼′2
(1 − 𝑣0
2)
+ 𝑥2 +
2𝑥
𝛼′
√
𝑣0
2
1 − 𝑣0
2 + 1
Which, if 𝑣0 = 0, simplifies to
𝑡 = +√ 𝑥2 +
2𝑥
𝛼′
In order to determine the journey’s duration for
each individual phase from the rocket-bound
twin’s MCRFs, the following equation is used:
Δ𝑡′
= ∫
𝑑𝑡′
𝑑𝑡
𝑑𝑡
𝑡 𝑓
𝑡0
= ∫
1
𝛾
𝑑𝑡
𝑡 𝑓
𝑡0
= ∫
1
√1 + (𝛼′ 𝑡 +
𝑣0
√1 − 𝑣0
2
)
2
𝑑𝑡
𝑡 𝑓
𝑡0
Once again, 𝑡0 = 0;
⇒ Δ𝑡′
=
1
𝛼′
[sinh−1
(𝛼′
𝑡 +
𝑣0
√1 − 𝑣0
2
)]
0
𝑡
⇒ Δ𝑡′
=
1
𝛼′
[sinh−1
(𝛼′
𝑡 +
𝑣0
√1 − 𝑣0
2
)
− sinh−1
(
𝑣0
√1 − 𝑣0
2
)]
If 𝑡 = 0, Δ𝑡′
= 0. Also, if 𝑣0 = 0, then the
above equation simplifies to
Δ𝑡′
=
sinh−1(𝛼′
𝑡)
𝛼′
(2.12.5)
3. Experimental Method
The investigation itself was performed using
Microsoft Excel, whereby the appropriate
equations were utilized for both methods and
for several distinct maximum speeds e.g. 0.85c,
0.95c, etc.
The derivations of the required equations were
mostly performed by hand, utilizing Wolfram
Alpha as an essential tool for calculating
derivations of more complicated integrations.
For each maximum-velocity scenario, the time
of journey’s duration as measured from both 𝑆
and 𝑆′
were compared and plotted. This was
performed for both methods. Minkowski space-
time diagrams describing the world-lines of a
single maximum velocity example from both
methods were also plotted, as well as an
adapted space-time diagram which includes
several planes of simultaneity.

16
4. Results
4.1. Method #1
Figures 7: A graph demonstrating the rate at which the Earth and the Rocket age as
measured from the Earth’s FoR. Green indicates acceleration phases (along the positive
x-axis), red indicates phases in which the Earth and Rocket are moving with some
relative speed, and purple indicates the phase whereby the rocket remains at the distant
planet. Blue represents the Earth’s age, while orange represents the rocket’s age. The
colours of the deceleration phases (along the positive x-axis) have been left unchanged
i.e. orange represents the deceleration phases
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20 25 30 35 40 45
TIME(YRS)
EARTH'S AGE (YRS)
Measurements from the Earth's FoR
0.999c
0.99c
0.95c
0.85c

17
Figure 9: A comparison of multiple phase 1 results as calculated using method #1 from
distinct values of maximum velocity i.e. the velocity achieved as phase 1 ends and at
which the rocket will coast throughout phase 2
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7 8
TIME(YRS)
ROCKET'S AGE (YRS)
Method #1, phase 1
0.999c
0.85c
0.95c
0.99c
0.75c
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20 25 30 35
TIME(YRS)
ROCKET'S AGE (YRS)
Measurements from the Rocket's MCRFs
Figure 8: A graph demonstrating the rate at which the Earth and the Rocket age as
measured from the Rocket’s MCRFs. Green indicates acceleration phases (along the
positive x-axis), red indicates phases in which the Earth and Rocket are moving with
some relative speed, and purple indicates the phase whereby the rocket remains at the
distant planet. Blue represents the Earth’s age, while orange represents the rocket’s age.
The colours of the deceleration (along the positive x-axis) phases have been left
unchanged i.e. blue represents the deceleration phases
0.85c0.95c0.99c
0.999c

18
4.2. Method #2
Figures 10 and 11: Graphs demonstrating the rate at which the Earth and the Rocket age from the Earth’s
FoR and from the Rocket’s MCRFs, respectively. Green indicates acceleration phases (along the positive
x-axis), red indicates phases in which the Earth and Rocket are moving with some relative speed, and
purple indicates the phase whereby the rocket remains at the distant planet. Blue represents the Earth’s
age, while orange represents the rocket’s age. The colours of the deceleration phases (along the positive x-
axis) have been left unchanged i.e. orange represents the deceleration phases in figure 10, while blue
indicates the deceleration phases in figure 11
0
5
10
15
20
25
30
35
0 5 10 15 20 25 30 35
TIME(YEARS)
EARTH'S AGE (YEARS)
Measurements from the Earth's FoR
0.999c
0.99c
0.95c
0.85c
0
5
10
15
20
25
30
35
0 5 10 15 20 25
TIME(YEARS)
ROCKET'S AGE (YEARS)
Measurements from the Rocket's co-moving FoR
0.999c
0.99c
0.95c
0.85c

19
4.3. Minkowski Diagrams
0
5
10
15
20
25
30
0 5 10
TIME(YRS)
DISPLACEMENT FROM EARTH (LYR)
Minkowski Diagram from Earth's
FoR
Traveller's
World-line
Photon World-
Line
0
5
10
15
20
25
30
35
40
0 5 10
TIME(YRS)
Minkowski diagram from Earth's
FoR
Photon
World-line
Traveller's
World-line
Figures 12 and 13: Minkowski Space-Time diagrams produced from Method #1 and
Method #2, respectively, whereby the maximum relative speed achieved is 0.85c

20
0
5
10
15
20
25
30
35
40
45
0 2 4 6 8 10 12
TIME(YRS)
Earth's FoR, Method #1
Traveller's World-line Photon World-line 0.425c
0.85c 0.34 0.085c
0.0c -0.085c -0.34c
-0.85c -0.425c
Figure 14: A version of figure 12 which includes planes of simultaneity. The squares and circles
of similar colour represent two events which appear to occur simultaneously from 𝑆′
’s MCRFs,
whereby each plane of simultaneity corresponds to a MCRF moving at some speed according to
the diagram’s legend

21
5. Discussion
5.1. The equations
During the investigation, it was determined that
while several examples of relevant literature
and publications describe the acceleration
stages of similar scenarios [35] [36] [37] [38]
, a
comprehensive analysis of the deceleration
phases proved elusive. While equations 2.11.2
– 2.11.5 cannot be considered conclusions
unique to this report, their derivation would
likely prove useful for any investigation
involving non-inertial reference frames and
scenarios involving distant, relativistic travel.
One particular example of such a description –
a description which fails to focus on the
deceleration stages – can be found in Lagoute
(1995). [39]
Lagoute describes a scenario in which one twin
remains earthbound while the second twin
embarks on a 6-phase journey whereby the
twin’s rocket accelerates for some time 𝑇𝑎 until
it reaches some maximum velocity 𝑣
(according to 𝑆 along the positive x-axis). The
rocket then coasts for some time 𝑇𝑐,
subsequently decelerates to 0, accelerates
(along the negative x-axis) to −𝑣, coasts at −𝑣
for some time 𝑇𝑐, and finally decelerates (along
the negative x-axis) to 0. This scenario differs
from that described in section 2.8; in Lagoute’s
scenario, the traveller doesn’t remain at the
distant planet for 1 year. Instead, the rocket’s
engines continue to fire along the positive x-
axis as the rocket ends its deceleration (phase
3), causing the rocket to accelerate back
towards Earth (along the negative x-axis). This
scenario also differs from the one in section 2.8
whereby the rocket’s acceleration phases are
described across some time 𝑇𝑎 whereas the
scenario examined in this study describes the
rocket’s accelerations across some distance 𝑋 𝑎.
Lagoute uses the following integral to
determine the travelling twin’s experience of
the initial acceleration phase’s duration:
Δ𝑡′
= ∫ √1 − (
𝑣(𝑡)
𝑐
)
2
𝑑𝑡 (5.1.1)
whereby 𝑣(𝑡) describes the velocity of a MCRF
as a function of time (measured from 𝑆):
𝑣(𝑡) =
𝑎𝑡
√1 + (
𝑎𝑡
𝑐 )
2 (5.1.2)
By substituting 5.1.2 into 5.1.1 Lagoute
determines the following description of time (as
measured from 𝑆′
) during phase 1:
Δ𝑡′
=
𝑐
𝑎
sinh−1
(
𝑎𝑇𝑎
𝑐
) (5.1.3)
Assuming 𝑐 = 1 and Lagoute’s definition of 𝑎
as the ‘proper’ acceleration i.e. the acceleration
experienced by the rocket-bound twin as they
accelerate from one inertial frame to the next,
the above equation resembles equation 2.12.5.
Lagoute also states that equation 5.1.3 may
equally be applied to each subsequent
acceleration/deceleration phase provided that
the time 𝑇𝑎 remains the same for each
acceleration/deceleration phase. While this may
be true, the equation is not appropriate for
calculating moment-by-moment measurements
of 𝑆′
’s time (from 𝑆’s FoR) for each
acceleration/deceleration phase. As is
demonstrated in figure 15, equation 5.1.3
produces identical results for acceleration and
deceleration phases (except that the results of
the deceleration phase are displaced along the
y-axis relative to the results of the acceleration
phase by the total duration of the initial
acceleration phase). It is evident from figure 10,
however, that equations 2.11.2 – 2.11.5
accurately describe each acceleration and
deceleration phase.
Incidentally, for Method #2, it may be worth
mentioning that while the distances across
which the rocket would travel were fixed, the
magnitude of acceleration required to reach a
required velocity was calculated by rearranging
equation 2.11.3:
𝛼′
=
−1 ± √
1
1 − 𝑣 𝑚𝑎𝑥
2
Δ𝑥
whereby 𝑣 𝑚𝑎𝑥 represents the required final
velocity. The above equation was implemented

22
during the investigation to determine the
constant acceleration (experienced by 𝑆′
)
necessary for the rocket to achieve 𝑣 𝑚𝑎𝑥 over a
distance Δ𝑥 = 𝑋 𝑎 = 3 𝑙𝑦𝑟.
Concerning the two methods used during the
study; while both provide compelling results
which suggest that the travelling twin ages less
than the earthbound twin, method #1 is based
on a fundamental flaw. The first method
assumes that the earthbound twin would
‘witness’ the travelling twin accelerating away
(during phase 1, at least) at a constant rate.
However, this isn’t the case. From the
earthbound twin’s frame of reference, the
rocket will appear to gain mass as its kinetic
energy increases; a conclusion of the mass-
energy equivalence. The rocket’s engines are
stated to provide a constant thrust, and would
therefore result in progressively smaller
acceleration due the rocket’s increasing mass.
A description of 𝑆′
’s acceleration from 𝑆’s FoR
is described in equation 2.11.1; if 𝑣 → 1, 𝛾 →
∞ and 𝛼 → 0 i.e. the rocket’s acceleration
(from 𝑆’s FoR) decreases as the rocket gains
speed (also as measured from 𝑆’s FoR). The
second method doesn’t encounter this problem,
and accurately explains the acceleration phases
of the travelling twin.
5.2. The results
Figures 7, 8, 10, and 11 clearly demonstrate the
asymmetric ageing between both twins. Figures
7 and 10 demonstrate the ageing of each twin
from the Earth’s FoR from Method #1 and
Method #2 respectively, while figures 8 and 11
represent the ageing of both twins from the
rocket’s MCRFs from Methods #1 and #2
respectively. An immediately obvious
discrepancy between each observer’s point of
view is the rate at which their respective twin
ages. From the earthbound twin’s FoR, the
travelling twin ages at a slower rate during the
entire trip except during phase 4, at which point
both twins are at rest relative to each other and
age at the same rate (ignoring the gravitational
influence of the distant planet and Earth).
However, from the rocket-bound twin’s point
of view, the earthbound twin ages at a slower
rate during the acceleration and constant speed
phases, and then, rather bizarrely, the travelling
twin measures the stationary twin’s age to
increase at a tremendous rate, much faster than
their own rate of ageing. From 𝑆’s FoR the
acceleration and deceleration phases suggest an
inverse symmetry with respect to each other,
while from 𝑆′
’s MCRFs, this is certainly not the
case. It seems as though the deceleration phases
are the decisive events, events which dictate the
asymmetric ageing. However, a conundrum
remains; can a ‘travelling’ twin truly be
considered to be the traveller in a twin paradox
scenario? Could it be considered that the
earthbound twin is accelerating away from the
rocket while the rocket remains stationary, as
the rocket-bound twin would likely experience?
Such a topic is inspected in sections 5.3 – 5.5.
While Method #1’s inaccuracy (due to its false
premises) was previously asserted, a closer
inspection of figure 8 would likely reinforce
this statement. Figure 9 provides a zoomed-in
view of phase 1 (as calculated using method #1)
from figure 8. It is immediately obvious that the
results do not appear to be logical. According to
figure 9, as the observer in 𝑆′
approaches their
maximum speed (except 0.75𝑐) and provided
that the observer is in possession of an
instantaneous internet connection (allowing
them to video chat with the observer in 𝑆), then
0
2
4
6
8
10
12
0 5 10
AGEING(YRS)
EARTH'S AGE (YRS)
Measurements from Earth's FoR
Rocket's Age
Earth's Age
Figure 15: A graph demonstrating a 2-phase
journey (using equation 5.1.3) whereby the
traveller accelerates along the x-axis and then
subsequently decelerates. The graph
demonstrates that equation 5.1.3 is
inappropriate for phase 2 (and therefore any
phase involving deceleration)

23
the observer in 𝑆′
would witness their twin
ageing in reverse! This anomalous result was
investigated using the concept of a Minkowski
space-time diagram and the use of equation
2.12.3. During phase 1;
𝑥 = 0.5𝛼𝜏2
, ⇒ 𝜏 = √2𝑥/𝛼
From this, we find that
𝑑𝜏
𝑑𝑥
=
1
√2𝛼𝑥
, ⇒ 𝑚 = √2𝛼𝑥
where 𝑚 is the gradient 𝑆′
’s plane of
simultaneity. As demonstrated in figure 16 and
via the use of straightforward geometry, the
following equation describes the Earth’s age as
measured from 𝑆′
:
𝑡 𝐸 = √2𝛼 (
√ 𝑥
𝛼
− √ 𝑥3)
According to the results from figure 8, the
following must be true:
𝑑𝑡 𝐸
𝑑𝑥
=
1
√ 𝛼𝑥
− 3√ 𝛼𝑥 < 0
(5.2.1)
which corresponds to a decrease in the
earthbound twin’s age as the rocket’s
displacement from Earth (as measured by 𝑆)
increases. We may also assert that the rocket’s
acceleration may not exceed the speed of light;
𝛼𝜏 < 𝑐 = 1
which is a valid assumption as the inverse
hyperbolic sin function in equation 2.10.6
would produce a numerical error if the above
inequality were false. It logically follows, then,
that
𝛼√
2𝑥
𝛼
< 1
⇒ 𝑥 <
1
2𝛼
(5.2.2)
Combining equations 5.2.1 and 5.2.2, we find
the following inequality:
1
3𝛼
< 𝑥 <
1
2𝛼
(5.2.3)
As an example, we shall focus on 0.95c. In
order to reach such a speed, the rocket would
require an acceleration of ~ 0.1504 𝑙𝑦𝑟/𝑦𝑟𝑠2
(according to Method #1). Phase 1 occurs
across a distance of 3 𝑙𝑦𝑟𝑠, and by using the
left-hand side of equation 5.2.3 we may find the
following value of acceleration; 𝛼 = 0.11̇ <
0.1504, which implies that at this acceleration,
the time-reversal anomaly occurs. A similar
inspection of the remaining speeds listed in
figure 9 will reveal that this anomaly presents
itself for all speeds bar 0.75c, whereby the
rocket would require an acceleration of
~ 0.938 𝑙𝑦𝑟/𝑦𝑟𝑠2
< 0.11̇. This anomaly
evidently doesn’t occur when using Method #2.
While this result may not be definitively
dismissed as a non-physical phenomenon, 𝑆′
’s
experience of ageing and 𝑆′
’s conclusion that 𝑆
ages in reverse are not causally linked in any
way. In other words, for 𝑆′
to actually witness
their twin ageing in reverse is not possible due
to the time delay between the earthbound twin
sending information and the travelling twin
receiving said information i.e. information
follows photon world-lines, not planes of
simultaneity.
Figure 16: An arbitrary Minkowski diagram,
whereby the black dotted axes represent 𝑆,
while the black and and red lines represent 𝑡′
and 𝑥′
, respectively for some instantaneous
MCRF of 𝑆′
. The purple arrow represents a
plane of simultaneity from 𝑆, while the orange
arrow represents a plane of simultaneity from
𝑆′
; by following the orange arrow to the
vertical axis, we may find 𝑡 𝐸
[40]

24
Figures 12 and 13 are Minkowski Space-Time
diagrams produced from Method #1 and #2,
respectively. Both represent the traveller’s
world-line from the Earth’s FoR, as well as the
world-line of a photon. The slight discrepancy
between both graphs’ y-axes is due to the
difference in each method’s calculation of the
journey’s total duration. It may be worth very
briefly drawing attention to the hyperbola
present in both Minkowski diagrams during the
acceleration phases; the mathematical
description of an observer’s ‘felt’ acceleration
is
𝛼′
=
1
(1 −
𝑣2
𝑐2)
3
2
𝑑𝑢
𝑑𝑡 (5.2.4)
whereby 𝑣 represents 𝑆′
’s speed (as measured
from 𝑆), c represents the speed of light, and 𝑡
represents time (also as measured from 𝑆). [37]
Solving for this equation results in
𝑥2
− 𝑐2
𝑡2
=
𝑐4
𝛼′2
which describes a hyperbole in time and space.
However equation 5.2.4 is only truly applicable
to figure 13 as it was produced from Method #2
which assumes a similar definition of
acceleration i.e. the traveller’s ‘felt’
acceleration is constant as they move from one
inertial FoR to another.
Figure 14 demonstrates how Resolution 1 may
be implemented to resolve the paradox. Rather
than the twin experiencing two inertial FoRs –
one during both the outward and return halves
of the journey – as suggested in section 2.6, the
twin accelerates through several inertial FoRs
as suggested in sections 2.8 – 2.12. The planes
which cut across the traveller’s world-line
represent planes of simultaneity from the
traveller’s MCRFs at various velocities.
The planes of simultaneity (from 𝑆′
) were
determined using the linear equation
𝑡 𝑆′ = 𝛽𝑥 + 𝑏
whereby 𝑡 𝑆′ represents the time at which an
event on Earth appears to occur simultaneously
with some event on board the rocket (according
to 𝑆′
), 𝑥 represents the displacement of the
rocket at any time 𝑡 (measured from 𝑆), 𝑏
represents some undefined constant, and
𝛽 =
𝑣
𝑐
= tan 𝛼 ≈ 𝛼
represents the rocket’s normalised velocity and
the angle between the 𝑥- and 𝑥′
-axes (see figure
17). The following linear equation was also
required:
𝑡 𝑆 = 𝛽𝑥 𝑆 + 𝑏, ⇒ 𝑏 = 𝑡 𝑆 − 𝛽𝑥 𝑆
i.e. the 𝑆 simultaneity plane, whereby 𝑡 𝑆
denotes the time on Earth at which an event
occurs on the rocket i.e. an event on Earth
which occurs simultaneously with an event on
the rocket according to 𝑆 (parallel to the 𝑥-
axis), and 𝑥 𝑆 represents the location of the on-
board event as measured by 𝑆 (see figure 17).
Combining both of these equations, one finds
the following description of any 𝑆′
simultaneity
plane:
𝑡 𝑆′ = 𝑡 𝑆 + 𝛽(𝑥 − 𝑥 𝑆)
The traveller’s supposed switch from one
inertial reference frame to another (as
demonstrated in figure 6) is more evident in
figure 14 as the gradient of subsequent
simultaneity planes transitions from a positive
gradient during the outward half of the journey
to a gradient of zero which is parallel with 𝑥 (as
it should be when the twins’ relative velocity is
0) to a negative gradient during the return half
of the journey. It is also apparent that the
gradient of a plane of simultaneity depends
upon the MCRF’s velocity i.e. a negative
velocity corresponds with a negative gradient
(and vice versa), and that a greater relative
velocity results in a more extreme gradient.
However, the magnitude of the gradient of the
photon world-line is greater than that of each
and every plane of simultaneity displayed in
figure 14, which is an expected result
considering that the traveller never exceeds the
speed of light.

25
5.3. Is acceleration necessary?
Despite the resolutions focused upon in the
theory section, many others exist, some of
which are worth discussing here. A minority of
physicists contend that certain prerequisites of
the primary resolutions – as well as those that
will be discussed shortly – are erroneous or
worth reconsidering from either a physical or
philosophical standpoint.
While most texts assert the significance of the
acceleration experienced by the travelling twin
as the key to resolving the paradox, it has been
noted that the role of acceleration may play no
direct role, per se. [41] [42] [43] [44]
Consider a set of
triplets, whereby the earthbound and rocket-
bound triplets are denoted as 𝑆 and 𝑆′
respectively, as before, and that the last triplet
is denoted as 𝑆′′
. 𝑆′
embarks on a journey
consisting of 4 acceleration/deceleration phases
between Earth and a distant planet. Suppose
that 𝑆′′
waits for 𝑆′
on board a second space-
craft at the distant planet, and that the planet’s
gravitational influence is ignored. At the point
of 𝑆′
’s turnaround, 𝑆′
and 𝑆′′
synchronize their
clocks and both proceed towards Earth. E.
Minguzzi [42] suggests that the acceleration
experienced by 𝑆′
and 𝑆′′
plays no direct role in
the time dilation experienced by both 𝑆′
and 𝑆′′
.
The paper demonstrates that a quantity 𝑇(𝜏) –
the travel duration with respect to 𝑆 of an
accelerated round-trip that ends at 𝜏 (the round-
trip’s duration as measured by 𝑆′
) – is Lorentz
invariant i.e. 𝑇(𝜏) doesn’t depend upon the
choice of reference frame, the choice of an
initial velocity, or acceleration. While the result
proves interesting, the details of the paper’s
analysis are likely unnecessary and in any case
beyond this study’s degree of investigation.
A similar proposition is suggested by T.
Maudlin [45] whereby the ‘length’ of each
observer’s world-line is responsible for any
time dilation experienced, rather than how
curved it becomes during acceleration. The
length referred to here is the proper time
interval of a trajectory, which is defined as a 4-
vector corresponding to a change in proper time
during a traveller’s journey as measured by a
clock which follows said traveller’s trajectory
[46]
i.e. a clock which resides within the 𝑆′
moving frame. By definition, a proper time
interval is Lorentz invariant i.e. invariant under
coordinate transformations and therefore
independent of coordinates, which follows a
similar connotation described in [42].
5.4. General Relativity
Another method of resolution involves the use
of the General Theory of Relativity (GTR).
While the focus of this study primarily involves
STR, a brief inspection of an implementation of
GTR would likely prove intriguing. As one
likely knows, Special Relativity may be derived
from General Relativity provided that a
Vacuum Solution is applied to GTR; such a
solution arises from the assumption that the
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4
TIME(YRS)
DISPLACEMENT FROM EARTH
(LYR)
Earth's FoR, Method #1
Traveller's World-line
Photon World-line
S' simultaneity plane
S simultaneity plane
𝑡 𝑆′
𝑡 𝑆
𝛽
Figure 17: A zoomed-in view of the bottom-left
corner of figure 14 with the addition of a plane
of simultaneity from 𝑆’s FoR which coincides
with the point at which the plane of
simultaneity from one of 𝑆′
’s MCRFs crosses
the traveller’s world-line. 𝑡 𝑆′, 𝑡 𝑆 and 𝛽 are
clearly labelled

26
stress-energy tensor is zero in the Einstein field
equation, implying that no matter nor non-
gravitational fields are present. While several
examples of explicit vacuum solutions exist, the
Minkowski space-time solution is the most
relevant, which describes empty space and
ignores cosmological expansion. While this
may seem initially irrelevant, it may be
necessary to gain some insight as to how GTR
may be applicable to the twin paradox. Vacuum
solutions reveal that such vacuous regions
contain energy as a consequence of
gravitational fields’ potential to do work; an
implication that gravitational fields can induce
displacement in the direction of some force,
regardless of the presence of matter. [47]
Additionally, an appreciation of the
equivalence principle is required – a concept
which states that an observer cannot distinguish
between experiencing a gravitational force and
the pseudo-force experienced in a non-inertial
frame of reference, and that these two are in fact
equivalent.
Einstein wrestled with the idea of the twin
paradox around the time of his publications of
both relativity theories. In 1918, a paper was
produced which briefly described how the
paradox may be resolved by defining the
travelling twin’s journey in a similar fashion
described in section 5.1, whereby A. Einstein
[48] invokes gravitational time dilation as an
explanation for the asymmetrical ageing as a
direct consequence of acceleration (see figure
18).
Suppose that a set of twins experience a similar
scenario to that described in section 5.1. 𝑆
views 𝑆′
accelerating away from the Earth
(phase 1) until it reaches some maximum speed
𝑉. 𝑆′
then coasts for some time (phase 2) before
decelerating to a halt at the distant planet (phase
3). 𝑆′
then repeats phases 1-3 but in reverse,
eventually bringing them back to Earth.
However, according to STR and GTR, 𝑆′
can
also be considered to reside within a stationary
reference frame, while 𝑆 does the accelerating.
Einstein proposed that during the acceleration
phases, 𝑆 deduces that an external force acts on
𝑆′
(most likely the rocket’s engines),
accelerating them along the positive and
negative x-axes. On the other hand, 𝑆′
will
believe themselves to be stationary during each
acceleration phase. From 𝑆′
’s perspective, an
entirely different impression of the journey
presents itself, which is described below [48]
:
1. Suppose once again that the positive x-
axis stems from Earth to some distant
planet. During phase 1, a gravitational
field directed towards the negative x-
axis appears. 𝑆 accelerates in free-fall
along this gravitational field until it
achieves some velocity −𝑉, at which
point the gravitational field disappears.
The external force exerted by the
engines on 𝑆′
prevent it from being set
in motion by said field, which results
in 𝑆′
’s lack of motion.
2. During phase 2, 𝑆′
views 𝑆 as moving
at constant velocity – 𝑉 away from
them.
3. During phases 3 and 4, a homogenous
gravitational field appears (from 𝑆′
’s
perspective) directed towards the
positive x-axis. This gravitational field
accelerates 𝑆 along the positive x-axis
until they achieve some speed 𝑉, at
which point the gravitational field will
disappear. During the time this field
‘exists’, 𝑆′
experiences an external
force from the rocket’s engines acting
upon them along the negative x-axis,
Figure 18: Diagram representing the
viewpoint of an earthbound observer, 𝐾,
and a ‘moving’ observer, 𝐾′
, used by
Einstein to demonstrate the
implementation of gravitational fields to
the twin paradox [48]

27
preventing them from being set in
motion by the field.
4. During phase 5, 𝑆′
views 𝑆 as moving
with some constant velocity 𝑉 towards
them.
5. Finally, during the final phase,
𝑆′
percieves another gravitational field
directed towards the negative x-axis. 𝑆
accelerates as free fall until it is
brought to a halt, at which point the
field disappears. The rocket’s engines
exert an external force on 𝑆′
along the
negative x-axis, retaining 𝑆′
’s rest
frame.
One may realise that during phases 2 and 5 𝑆′
views 𝑆 as moving with some velocity, which
would imply that according to 𝑆′
, 𝑆’s clock
would run slower. While this is true, phases 3
and 4 more than make up for 𝑆′
’s ‘lost time’ as
agreed upon by both twins at 𝑆′
’s return to
Earth. According to GTR, a clock within a
region of high gravitational potential will run
faster relative to a clock which resides within a
region of low gravitational potential. During
phases 3 and 4 𝑆′
would measure 𝑆’s clock
ticking at a faster rate by a factor of
(1 +
𝛼ℎ
𝑐2
)
The expression above represents the weak-field
approximation in GTR, where 𝛼 denotes the
acceleration of 𝑆 (as measured by 𝑆′), and ℎ
denotes the distance between 𝑆′
and 𝑆 (also as
measured by 𝑆′
). [49]
During phases 3 and 4, 𝑆
appears to have travelled a great distance from
𝑆′
, which results in a drastic increase in 𝑆’s age
according the expression above, which
supposedly more than accounts for 𝑆′
’s
measurements of 𝑆’s slowed ageing during
phases 2 and 5. During phases 1 and 6, 𝑆′
and 𝑆
are relatively close, therefore the asymmetry in
ageing between both twins would not be as
apparent as during other phases (see figure 19).
This seems to be an agreeable explanation for
the substantial dissimilarity between phases 1
and 6 and phases 3 and 5 in figures 8 and 11.
Incidentally, it is worth mentioning that the
apparent appearance of the gravitational fields
during various phases of the journey appear to
be focused on 𝑆′
’s ‘position’ i.e. the
gravitational potential of any point is measured
with respect to its distance from 𝑆′
. While this
isn’t explicitly stated in Einstein’s paper, it is
most likely the case.
It should be no surprise that an application of
GTR can account for the simultaneity shifts and
asymmetric ageing described in sections 2.6 –
2.12 and 5.1; as previously mentioned, an
appropriate GTR solution involving static,
homogenous gravitational fields produces an
identical set of outcomes as STR when
considering finite accelerations. [50]
One may be tempted to interpret the
gravitational fields measured by the traveller as
merely fictitious, considering that the stationary
twin doesn’t experience gravitational fields of
any kind (ignoring Earth’s, of course) and that
no matter was introduced to induce said fields.
Einstein argues that said fields may not be
considered real or fictitious as gravitational
fields are coordinate dependent and therefore
do not correspond to something “physically
real”, but may be considered to be ‘real’ with
respect to “other data” i.e. the coordinate
system 𝑆′
. [48]
By this, Einstein invokes the
induction of an electric field by accelerated
charges as an analogy for the induction of said
gravitational fields by the acceleration of each
ℎ2
ℎ1
Figure 19: A crude image depicting two
instances in which gravitational fields appear
from 𝑆′
’s perspective. The small gravitational
field (left, near Earth) is focused on the rocket
as it accelerates along the positive x-axis while
the much larger gravitational field (right, near
the distant planet) is focused on the rocket as it
accelerates along the negative x-axis. During
the deceleration phases the earthbound twin
resides within a region of high gravitational
potential relative to the acceleration phases
since ℎ2 > ℎ1

28
and every star in the universe relative to 𝑆′
; an
interpretation of Mach’s Principle, which will
be discussed shortly. Also, as discussed and
elaborated upon earlier, GTR has no
prerequisite for the presence of matter to induce
gravitational fields. Perhaps more easily
grasped is Einstein’s comparison between the
appearance of said gravitational fields due to an
arbitrary choice of a coordinate system and
kinetic energy. From Classical Mechanics,
kinetic energy is dependent upon the motion of
the coordinate system from which it is
measured i.e. with a suitable choice of
coordinate system one may arrange for the
kinetic energy of a moving body to assume
some positive value or zero, similar to how a
gravitational field may assume some positive
magnitude or a magnitude of zero depending on
the coordinate system from which it is
measured, in this case from 𝑆′
and 𝑆,
respectively [48]
. While [48] provides no
equations for its numerous analogies,
interpretations and elaborations, its contents are
nonetheless interesting and would provide a
fascinating compliment to this study if further
investigation were to be conducted.
5.5. Mach’s Principle
As a final remark regarding alternative
explanations, it would be interesting to
investigate Mach’s Principle. Mach’s principle
is a hypothesis which suggests that local frames
may be defined as inertial or non-inertial
relative to the large-scale distribution of matter
throughout the universe, or that the large-scale
structure of the universe determines local
physical laws. [51]
A minority of physicists
favour invoking some version of Mach’s
Principle to define accelerated motion or
inertial motion relative to the fixed stars [52]
i.e.
the rest of the matter in the universe. An
example of such an invocation may be found in
A. P. French [21]; French states that it is
possible to regard 𝑆′
as the travelling twin by
referring to 𝑆′
’s view of the apparent motion of
stars. French elaborates on this by explaining
that 𝑆′
is indeed the twin which undergoes
acceleration by emphasising 𝑆′
’s observation of
inertial forces associated with said acceleration.
French concludes the argument by proposing
that an “ultimate definition” of an inertial frame
is a reference frame with zero acceleration with
respect to the large-scale distribution of matter
in the universe. [9]
On the other hand, M. Sachs [11] states that an
application of Mach’s Principle to the twin
paradox is in philosophical disagreement with
some fundamental aspect of relativity. Briefly,
Sachs states that a scenario in which the Earth
(and the rest of the universe) are considered to
reside within an inertial reference frame
(alongside 𝑆) while 𝑆′
experiences acceleration
and enters a non-inertial reference frame is in
contradiction with Relativity’s definition of
Space and Time as no more than “subjective
language elements” used to describe objective
laws of physical processes. Sachs also describes
how Relativity Theory should apply to all
physical scenarios, whether they exist or not; a
universe in which nothing exists barring a pair
of twins each residing within a rocket of their
own is suggested. In such a universe, if one of
the twins were to accelerate away and
eventually return to the other twin, asymmetric
ageing would still occur without the large-scale
distribution of matter to appeal to! If there is
any conclusion to be drawn from this ongoing
conundrum, it is far beyond the requirements
and far removed from the primary purpose of
this study.
5.6. Further Investigation
If the experiment were to be repeated or the
investigation improved upon, there are various
other scenarios in which the twin paradox may
arise:
i. A scenario in which one twin remains
on a rotating space station while
another twin leaves the station,
decelerates to a halt, and then
accelerates to dock with the station.
Such a scenario would also require a
resolution for the Erhenfest paradox.
ii. Another possible scenario whereby the
Erhenfest paradox must be considered
is demonstrated in figure 20. One twin
remains on Earth, while another
accelerates to some distant celestial
body. Upon arrival the travelling twin

29
will have achieved some velocity 𝑉.
Instead of decelerating, remaining on
the x-axis, or spending a year at the
distant object, the travelling twin will
perform a looping manoeuvre around
the distant object. Following this, the
travelling twin will decelerate back
towards Earth. By definition, the rocket
will accelerate along two spatial axes,
which is an example of the Erhenfest
paradox.
iii. Finally, an alternative scenario
involving gravitational fields may be
worth investigating. The inclusion of
masses capable of producing
gravitational fields would likely prove
to be a mathematical challenge – the
reason for which gravitational effects
were ignored throughout this study –
but would nevertheless prove
rewarding if some conclusion may be
drawn.
Other possible methods of improving the
investigation include: considering several more
values of maximum velocities; the inclusion of
more intermittent acceleration phases; or
simply increasing the number of data points on
each graph from each phase to increase the
results’ accuracy.
An interesting possibility would be to consider
the inclusion of a third participant; a scenario
involving triplets, rather than twins. The third
triplet would depart from the Earth along the
positive x-axis with an acceleration −𝛼′
(or 𝛼′
along the negative x-axis), the second triplet
would depart along the positive x-axis with an
acceleration 𝛼′
, whilst the first triplet remains
on Earth. From the viewpoint of the second and
third triplets, their measurements of each
other’s ageing would likely prove to be
peculiar. It’s possible that an investigation of
this sort may yield a result akin to that described
in section 5.3, whereby the necessity of
acceleration as a direct explanation for
asymmetric ageing may come into question.
6. Conclusion
The equations describing constantly
accelerated motion were derived and
implemented along with the Lorentz
transformations (for rectilinear, constant-
velocity motion) to describe the motion of a
‘travelling’ twin. Two dissimilar methods –
distinguishable by their definition of from
which twin’s perspective the acceleration is
considered to be constant – were implemented
to plot the ageing experienced by each twin as
measured from both 𝑆 and 𝑆′
. It was
demonstrated that the deceleration phases
proved to be the decisive stages of the travelling
twin’s journey in determining the twins’
asymmetric ageing. While both methods
produced expected results, the first method was
based upon an erroneous assumption; that the
rocket’s acceleration was constant from the
reference frame of an inertial, earthbound
observer. It was determined that the derivation
of the relativistic acceleration formulae
implemented in each method – most
importantly, the second method – were quite
elementary, requiring only some concepts of
calculus. Based on equations intended to
describe the acceleration phases of such a trip,
equations which accurately describe
acceleration as well as the deceleration phases
were derived and implemented. Minkowski
space-time diagrams were produced using both
methods, whereby planes of simultaneity were
Figure 20: Diagram of scenario ii. The green
arrow represents 𝑆′
’s acceleration from Earth
to the distant object, the red arrow represents
the looping manoeuvre, and the black arrow
represents 𝑆′
’s deceleration back to Earth

30
used to demonstrate the sudden leap in the
earthbound twin’s age from the traveller’s
perspective described in section 2.6. Several
alternative resolutions were discussed,
including the implementation of gravitational
fields to explain the twins’ asymmetric ageing
via the use of General Relativity Theory.
Several alternative interpretations of the direct
cause of the asymmetric ageing were also
discussed, such as Mach’s Principle and the
concept of world-line lengths, as well as some
of their philosophical and physical pitfalls. If an
attempt to build upon this study were to be
made, an analysis of the twin-Erhenfest
paradox scenario (see section 5.6) would likely
prove to be a challenging but unique
investigation.
7. Acknowledgements
I’d like to thank Dr Balázs Pintér for his
unparalleled support during the investigation,
his patience with respect to my inexperience
with Relativity Theory, and for his enthusiasm
to help, regardless of the challenge. Dr Pintér
has helped enkindle a new-found fascination
with Relativity Theory, an intrigue which will
surely linger long after my time at university.
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9. Appendix
Appendix A – Derivation of the Lorentz
Transformations
Einstein argued that there must be two non-zero
constants the purpose of which was to provide
a translation between the primed and non-
primed reference frames corresponding to a
photon propagation along the positive and
negative x-axis, such that
𝑥′
− 𝑐𝑡′
= 𝜆(𝑥 − 𝑐𝑡) (A1)
𝑥′
+ 𝑐𝑡′
= 𝜇(𝑥 + 𝑐𝑡) (A2)
whereby equations A1 and A2 describe motion
along the positive and negative x-axis,
respectively. By adding and subtracting
equations A2 and A1, we find
2𝑥′
= 𝜆𝑥 − 𝜆𝑐𝑡 + 𝜇𝑥 + 𝜇𝑐𝑡
−2𝑐𝑡′
= 𝜆𝑥 − 𝜆𝑐𝑡 − 𝜇𝑥 − 𝜇𝑐𝑡
One may define 𝑥 = 𝑐𝑡 as a description of the
motion of light if and only if 𝑥′
= 𝑐𝑡′
. [12]
Following these definitions, and by introducing
the following constants for convenience;
𝛾 =
𝜆 + 𝜇
2
, 𝜎 =
𝜆 − 𝜇
2
The following equations are reached;
𝑥′
= 𝛾𝑥 − 𝜎𝑐𝑡 (A3)
𝑐𝑡′
= 𝛾𝑐𝑡 − 𝜎𝑥 (A4)
From 𝑆′
’s perspective, their displacement never
changes i.e. 𝑥′
= 0. Substituting this into
equation (3), we find that
𝑥 =
𝜎𝑐
𝛾
𝑡
If 𝑥 = 𝑉𝑡 where 𝑉 corresponds to the relative
velocity of the two systems, then 𝑉 =
𝜎𝑐
𝛾
. If one
were to insert a particular value of 𝑡 (a
particular point in time as measured from 𝑆) e.g.
𝑡 = 0 into equation (A3), one would find that
𝑥′
= 𝛾𝑥
⇒ 𝑥 =
𝑥′
𝛾
Now that 𝛾 and 𝜎 are defined, they may be
substituted into equations A3 and A4, which
provides us with equations 2.4.3 and 2.4.6:
𝑥′
= 𝛾(𝑥 − 𝑉𝑡)
𝑡′
= 𝛾 (𝑡 −
𝑉
𝑐2
𝑥)
Suppose that two points along the 𝑥′
-axis are
separated by the distance Δ𝑥′
= 1 (measured
from 𝑆′
). The two point would then be
separated along the x-axis (measured from 𝑆)
by the distance
Δ𝑥 =
1
𝛾
(A5)
If one were now to insert a particular value of
𝑡′
(s particular point in time as measured from
𝑆′
) e.g. 𝑡′
= 0 into equation A4, one would
determine the following:

33
𝛾𝑐𝑡 = 𝜎𝑥
By eliminating t from both equation A3 and A4
via substitution i.e.
𝛾𝑥′
𝜎
=
𝛾2
𝜎
𝑥 − 𝜎𝑥
⇒ 𝑥′
= 𝛾𝑥 −
𝜎2
𝛾
𝑥
and by recalling that 𝑉 = 𝜎𝑐/𝛾;
𝑥′
= 𝛾 (1 −
𝑉2
𝑐2 ) 𝑥
From this equation, we may conclude that two
points separated by Δ𝑥 = 1 along the x-axis
(relative to 𝑆) are separated by the distance
Δ𝑥′
= 𝛾 (1 −
𝑉2
𝑐2 )
(A6)
as measured from 𝑆′
. However, a unit
measuring-rod at rest within 𝑆′
must be exactly
the same length as a unit measuring-rod at rest
within 𝑆 (as measured from 𝑆′
), which implies
that equations A5 and A6 must be equivalent:
⇒ 𝛾2
=
1
1 −
𝑉2
𝑐2
Appendix B – Composition of
Velocities
The equation governing the conservation of
velocities between two inertial reference frames
moving with some relative speed 𝑉 may be
derived straightforwardly by considering the
differential forms of the rotation-free Lorentz
Transformations [19]
i.e. equations 2.4.3 and
2.4.5:
𝑑𝑥′
= 𝛾(𝑑𝑥 − 𝑉𝑑𝑡)
𝑑𝑡′
= 𝛾 (𝑑𝑡 −
𝑉
𝑐2
𝑑𝑥)
No other equations are required, as the relative
motion between reference frames is assumed to
be constant and rectilinear. By dividing
equation 2.4.3 by 2.4.5:
𝑑𝑥′
𝑑𝑡′
=
𝛾(𝑑𝑥 − 𝑉𝑑𝑡)
𝛾 (𝑑𝑡 −
𝑉
𝑐2 𝑑𝑥)
=
𝑑𝑥 − 𝑉𝑑𝑡
𝑑𝑡 (1 −
𝑉
𝑐2
𝑑𝑥
𝑑𝑡
)
⇒ 𝑣′ =
𝑣 − 𝑉
1 −
𝑉
𝑐2 𝑣
Assuming that 𝑉 is some fraction of the speed
of light, the equation above simplifies to the
following:
𝑣′ =
𝑣 − 𝑉
1 − 𝑉𝑣

34
10. Literature Review
Literature Review: Resolving an example of the Triplet Paradox in Special
Relativity
Llyr Humphries
PHM5860 Major Project
Institute of Mathematics, Physics and Computer Sciences, Aberystwyth University
November 2015
Abstract
The purpose of this report is to outline the key research, necessary equations and logical
thinking pertaining to the project mentioned above. The study will involve an investigation of the
Special Theory of Relativity and its consequences on a group of triplets moving at relativistic (near-
light) speeds relative to each other. An analysis of a scenario in which a set of twins are moving at
relativistic speeds (relative to each other) will serve as a starting point before eventually moving on to
a scenario involving triplets, and possibly (depending on time restraints) more complex versions of the
twin and triplet paradoxes. An important aspect of the report will be to understand how a paradox seems
to arise in a relativistic-twin scenario. Incidentally, it will be of greater importance to demonstrate how
the paradox arises from an inadequate understanding of Special Relativity, and to demonstrate how the
paradox can be resolved. Likewise, the same demonstrations will be performed for a triplet scenario;
triplet A remains at Earth, while triplets B and C move in opposite directions away from Earth to
equidistant stars with equal but opposite speeds and accelerations.

35
Contents
Introduction.........................................................................................................................................36
The Theory of Special Relativity .......................................................................................................36
The Luminiferous Aether and the Michaelson-Morley Experiment..............................................37
Time Dilation.......................................................................................................................................39
The Lorentz Transformations............................................................................................................40
The Twin Paradox...............................................................................................................................40
The Triplet Paradox ...........................................................................................................................41
References............................................................................................................................................42

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