Editing Special relativity (section)

== Beyond the basics ==
=== Rapidity ===
{{Main|Rapidity}}
{{multiple image
 | direction         = horizontal
 | width1            = 135
 | image1            = Trig functions (sine and cosine).svg
 | caption1          = Figure 7-1a.  A ray through the [[unit circle]] {{nowrap|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = 1}} in the point {{nowrap|1=(cos ''a'', sin ''a'')}}, where ''a'' is twice the area between the ray, the circle, and the ''x''-axis.
 | width2            = 190
 | image2            = Hyperbolic functions-2.svg
 | caption2          = Figure 7-1b. A ray through the [[unit hyperbola]] {{nowrap|1=''x''<sup>2</sup> − ''y''<sup>2</sup> = 1}} in the point {{nowrap|1=(cosh ''a'', sinh ''a'')}}, where ''a'' is twice the area between the ray, the hyperbola, and the ''x''-axis.
}}
[[File:Sinh+cosh+tanh.svg|thumb|180px|Figure 7–2. Plot of the three basic [[Hyperbolic function]]s: hyperbolic sine ([[:File:Hyperbolic Sine.svg|sinh]]), hyperbolic cosine ([[:File:Hyperbolic Cosine.svg|cosh]]) and hyperbolic tangent ([[:File:Hyperbolic Tangent.svg|tanh]]). Sinh is red, cosh is blue and tanh is green.]]
Lorentz transformations relate coordinates of events in one reference frame to those of another frame. Relativistic composition of velocities is used to add two velocities together. The formulas to perform the latter computations are nonlinear, making them more complex than the corresponding Galilean formulas.

This nonlinearity is an artifact of our choice of parameters.<ref name="Taylor1992" />{{rp|47–59}} We have previously noted that in an {{nowrap|1=''x''–''ct''}} spacetime diagram, the points at some constant spacetime interval from the origin form an invariant hyperbola. We have also noted that the coordinate systems of two spacetime reference frames in standard configuration are hyperbolically rotated with respect to each other.

The natural functions for expressing these relationships are the [[Hyperbolic functions|hyperbolic analogs of the trigonometric functions]]. Fig.&nbsp;7-1a shows a [[unit circle]] with sin(''a'') and cos(''a''), the only difference between this diagram and the familiar unit circle of elementary trigonometry being that ''a'' is interpreted, not as the angle between the ray and the {{nowrap|1=''x''-axis}}, but as twice the area of the sector swept out by the ray from the {{nowrap|1=''x''-axis}}. Numerically, the angle and {{nowrap|1=2 × area}} measures for the unit circle are identical. Fig.&nbsp;7-1b shows a [[unit hyperbola]] with sinh(''a'') and cosh(''a''), where ''a'' is likewise interpreted as twice the tinted area.<ref>{{cite book|last1=Thomas|first1=George B.|last2=Weir|first2=Maurice D.|last3=Hass|first3=Joel|last4=Giordano|first4=Frank R.|title=Thomas' Calculus: Early Transcendentals|date=2008|publisher=Pearson Education, Inc.|location=Boston|isbn=978-0-321-49575-4|page=533|edition=Eleventh}}</ref> Fig.&nbsp;7-2 presents plots of the sinh, cosh, and tanh functions.

For the unit circle, the slope of the ray is given by
: <math>\text{slope} = \tan a  = \frac{\sin a }{\cos a }.</math>

In the Cartesian plane, rotation of point {{nowrap|1=(''x'', ''y'')}} into point {{nowrap|1=(''x''{{'}}, ''y''{{'}})}} by angle ''&theta;'' is given by
: <math>
\begin{pmatrix}
x' \\
y' \\
\end{pmatrix} = \begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \\
\end{pmatrix}\begin{pmatrix}
x \\
y \\
\end{pmatrix}.</math>

In a spacetime diagram, the velocity parameter <math>\beta</math> is the analog of slope. The ''rapidity'', ''&phi;'', is defined by<ref name="Morin2007" />{{rp|96–99}}
: <math>\beta \equiv \tanh \phi \equiv \frac{v}{c},</math>
where
: <math>\tanh \phi = \frac{\sinh \phi}{\cosh \phi} = \frac{e^\phi-e^{-\phi}}{e^\phi+e^{-\phi}}.</math>

The rapidity defined above is very useful in special relativity because many expressions take on a considerably simpler form when expressed in terms of it. For example, rapidity is simply additive in the collinear velocity-addition formula;<ref name="Taylor1992" />{{rp|47–59}}
: <math>\beta = \frac{\beta_1 + \beta_2}{1 + \beta_1 \beta_2} = </math> <math>\frac{\tanh \phi_1 + \tanh \phi_2}{1 + \tanh \phi_1 \tanh \phi_2} =</math> <math>\tanh(\phi_1 + \phi_2),</math>
or in other words, {{tmath|1= \phi = \phi_1 + \phi_2 }}.

The Lorentz transformations take a simple form when expressed in terms of rapidity. The ''&gamma;'' factor can be written as
: <math>\gamma = \frac{1}{\sqrt{1 - \beta^2}} = \frac{1}{\sqrt{1 - \tanh^2 \phi}}</math> <math>= \cosh \phi,</math>
: <math>\gamma \beta = \frac{\beta}{\sqrt{1 - \beta^2}} = \frac{\tanh \phi}{\sqrt{1 - \tanh^2 \phi}}</math> <math>= \sinh \phi.</math>

Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called ''boosts''.

Substituting ''&gamma;'' and ''&gamma;β'' into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in the {{nowrap|1=''x''-direction}} may be written as
: <math>
  \begin{pmatrix}
    c t' \\
    x'
  \end{pmatrix}
  =
  \begin{pmatrix}
    \cosh \phi & -\sinh \phi \\
    -\sinh \phi & \cosh \phi
  \end{pmatrix}
  \begin{pmatrix}
    ct \\
    x
  \end{pmatrix},</math>
and the inverse Lorentz boost in the {{nowrap|1=''x''-direction}} may be written as
: <math>
  \begin{pmatrix}
    c t \\
    x
  \end{pmatrix}
  =
  \begin{pmatrix}
    \cosh \phi & \sinh \phi \\
    \sinh \phi & \cosh \phi
  \end{pmatrix}
  \begin{pmatrix}
    c t' \\
    x'
  \end{pmatrix}.</math>

In other words, Lorentz boosts represent [[hyperbolic rotation]]s in Minkowski spacetime.<ref name="Morin2007" />{{rp|96–99}}

The advantages of using hyperbolic functions are such that some textbooks such as the classic ones by Taylor and Wheeler introduce their use at a very early stage.<ref name="Taylor1992">{{cite book|url=https://archive.org/details/spacetime_physics/|title=Spacetime Physics|last1=Taylor|first1=Edwin F.|last2=Wheeler|first2=John Archibald|date=1992|publisher=W. H. Freeman|isbn=0-7167-2327-1|edition=2nd}}</ref><ref group=note>Rapidity arises naturally as a coordinates on the pure [[Representation theory of the Lorentz group#Conventions and Lie algebra bases|boost generators]] inside the [[Lie algebra]] algebra of the Lorentz group. Likewise, rotation angles arise naturally as coordinates (modulo {{nowrap|2{{pi}}}}) on the pure [[Representation theory of the Lorentz group#Conventions and Lie algebra bases|rotation generators]] in the Lie algebra. (Together they coordinatize the whole Lie algebra.) A notable difference is that the resulting rotations are periodic in the rotation angle, while the resulting boosts are not periodic in rapidity (but rather one-to-one). The similarity between boosts and rotations is formal resemblance.</ref>
{{anchor|4‑vectors}}

=== 4‑vectors ===
{{Main|Four-vector}}
Four‑vectors have been mentioned above in context of the energy–momentum {{nowrap|1=4‑vector}}, but without any great emphasis. Indeed, none of the elementary derivations of special relativity require them. But once understood, {{nowrap|1=4‑vectors}}, and more generally [[tensors]], greatly simplify the mathematics and conceptual understanding of special relativity. Working exclusively with such objects leads to formulas that are ''manifestly'' relativistically invariant, which is a considerable advantage in non-trivial contexts. For instance, demonstrating relativistic invariance of [[Maxwell's equations]] in their usual form is not trivial, while it is merely a routine calculation, really no more than an observation, using the [[field strength tensor]] formulation.<ref name=Post_1962>{{cite book|title=Formal Structure of Electromagnetics: General Covariance and Electromagnetics|date=1962|publisher=Dover Publications Inc.|isbn=978-0-486-65427-0|author=E. J. Post}}</ref>

On the other hand, general relativity, from the outset, relies heavily on {{nowrap|1=4‑vectors}}, and more generally tensors, representing physically relevant entities. Relating these via equations that do not rely on specific coordinates requires tensors, capable of connecting such {{nowrap|1=4‑vectors}} even within a ''curved'' spacetime, and not just within a ''flat'' one as in special relativity. The study of tensors is outside the scope of this article, which provides only a basic discussion of spacetime.

==== Definition of 4-vectors ====
A 4-tuple, {{tmath|1=A=\left(A_{0}, A_{1}, A_{2}, A_{3}\right)}} is a "4-vector" if its component ''A<sub>i</sub>'' transform between frames according to the Lorentz transformation.

If using {{tmath|1=(ct, x, y, z)}} coordinates, ''A'' is a {{nowrap|1=4–vector}} if it transforms (in the {{nowrap|1=''x''-direction}}) according to
: <math>\begin{align}
  A_0' &= \gamma \left( A_0 - (v/c) A_1 \right)  \\ 
  A_1' &= \gamma \left( A_1 - (v/c) A_0 \right)\\
  A_2' &= A_2 \\ 
  A_3' &= A_3
\end{align} ,</math>
which comes from simply replacing ''ct'' with ''A''<sub>0</sub> and ''x'' with ''A''<sub>1</sub> in the earlier presentation of the [[#Lorentz transformations|'''Lorentz transformation.''']]

As usual, when we write ''x'', ''t'', etc. we generally mean Δ''x'', Δ''t'' etc.

The last three components of a {{nowrap|1=4–vector}} must be a standard vector in three-dimensional space. Therefore, a {{nowrap|1=4–vector}} must transform like {{tmath|1=(c \Delta t, \Delta x, \Delta y, \Delta z)}} under Lorentz transformations as well as rotations.<ref name="Schutz1985">{{cite book| last1=Schutz |first1= Bernard F. |title=A first course in general relativity|date=1985|publisher=Cambridge University Press|location=Cambridge, UK|isbn=0521277035|page=26}}</ref>{{rp|36–59}}

==== Properties of 4-vectors ====
* '''Closure under linear combination:''' If ''A'' and ''B'' are {{nowrap|1=4-vectors}}, then {{tmath|1=C = aA + aB}} is also a {{nowrap|1=4-vector}}.
* '''Inner-product invariance:''' If ''A'' and ''B'' are {{nowrap|1=4-vectors}}, then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a {{nowrap|1=3-vector}}. In the following, <math>\vec{A}</math> and <math>\vec{B}</math> are {{nowrap|1=3-vectors}}:
*: <math>A \cdot B \equiv </math> <math>A_0 B_0 - A_1 B_1 - A_2 B_2 - A_3 B_3 \equiv </math> <math>A_0 B_0 - \vec{A} \cdot \vec{B}</math>
: In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in {{nowrap|1=3-space}}.
: Two vectors are said to be ''orthogonal'' if {{tmath|1= A \cdot B = 0 }}. Unlike the case with {{nowrap|1=3-vectors}}, orthogonal {{nowrap|1=4-vectors}} are not necessarily at right angles to each other. The rule is that two {{nowrap|1=4-vectors}} are orthogonal if they are offset by equal and opposite angles from the 45° line, which is the world line of a light ray. This implies that a lightlike {{nowrap|1=4-vector}} is orthogonal to ''itself''.
* '''Invariance of the magnitude of a vector:''' The magnitude of a vector is the inner product of a {{nowrap|1=4-vector}} with itself, and is a frame-independent property. As with intervals, the magnitude may be positive, negative or zero, so that the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one for which {{tmath|1= A \cdot A = 0 }}, while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval <math>c^2 t^2 - x^2</math> and the invariant length of the relativistic momentum vector {{tmath|1= E^2 - p^2 c^2 }}.<ref name="Morin2007" />{{rp|178–181}}<ref name="Schutz1985" />{{rp|36–59}}

==== Examples of 4-vectors ====
* '''Displacement 4-vector:''' Otherwise known as the ''spacetime separation'', this is {{nowrap|1=(''&Delta;t, &Delta;x, &Delta;y, &Delta;z''),}} or for infinitesimal separations, {{nowrap|1=(''dt'', ''dx'', ''dy'', ''dz'')}}.
*: <math>dS \equiv (dt, dx, dy, dz)</math>
* '''Velocity 4-vector:''' This results when the displacement {{nowrap|1=4-vector}} is divided by <math>d \tau</math>, where <math>d \tau</math> is the proper time between the two events that yield ''dt'', ''dx'', ''dy'', and ''dz''.
*: <math>V \equiv \frac{dS}{d \tau} = \frac{(dt, dx, dy, dz)}{dt/\gamma} = </math> <math>\gamma \left(1, \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \right) = </math> <math>(\gamma, \gamma \vec{v} ) </math>
{{multiple image
 | direction         = horizontal
 | width1            = 150
 | image1            = Momentarily Comoving Reference Frame.gif
 | caption1          = Figure 7-3a. The momentarily comoving reference frames of an accelerating particle as observed from a stationary frame.
 | width2            = 150
 | image2            = Lorentz transform of world line.gif
 | caption2          = Figure 7-3b. The momentarily comoving reference frames along the trajectory of an accelerating observer (center).
}}
: The {{nowrap|1=4-velocity}} is tangent to the world line of a particle, and has a length equal to one unit of time in the frame of the particle.
: An accelerated particle does not have an inertial frame in which it is always at rest. However, an inertial frame can always be found that is momentarily comoving with the particle. This frame, the ''momentarily comoving reference frame'' (MCRF), enables application of special relativity to the analysis of accelerated particles.
: Since photons move on null lines, <math>d \tau = 0</math> for a photon, and a {{nowrap|1=4-velocity}} cannot be defined. There is no frame in which a photon is at rest, and no MCRF can be established along a photon's path.
* '''Energy–momentum 4-vector:'''
*: <math>P \equiv (E/c, \vec{p}) = (E/c, p_x, p_y, p_z)</math>
: As indicated before, there are varying treatments for the energy–momentum {{nowrap|1=4-vector}} so that one may also see it expressed as <math>(E, \vec{p})</math> or {{tmath|1= (E, \vec{p}c) }}. The first component is the total energy (including mass) of the particle (or system of particles) in a given frame, while the remaining components are its spatial momentum. The energy–momentum {{nowrap|1=4-vector}} is a conserved quantity.
* '''Acceleration 4-vector:''' This results from taking the derivative of the velocity {{nowrap|1=4-vector}} with respect to {{tmath|1= \tau }}.
*: <math>A \equiv \frac{dV}{d \tau} = </math> <math>\frac{d}{d \tau} (\gamma, \gamma \vec{v}) = </math> <math>\gamma \left( \frac{d \gamma}{dt}, \frac{d(\gamma \vec{v})}{dt} \right)</math>
* '''Force 4-vector:''' This is the derivative of the momentum {{nowrap|1=4-vector}} with respect to <math>\tau .</math>
*: <math>F \equiv \frac{dP}{d \tau} = </math> <math>\gamma \left(\frac{dE}{dt}, \frac{d \vec{p}}{dt} \right) = </math> <math> \gamma \left( \frac{dE}{dt},\vec{f} \right) </math>

As expected, the final components of the above {{nowrap|1=4-vectors}} are all standard {{nowrap|1=3-vectors}} corresponding to spatial {{nowrap|1=3-momentum}}, {{nowrap|1=3-force}} etc.<ref name="Morin2007" />{{rp|178–181}}<ref name="Schutz1985" />{{rp|36–59}}

==== 4-vectors and physical law ====
The first postulate of special relativity declares the equivalency of all inertial frames. A physical law holding in one frame must apply in all frames, since otherwise it would be possible to differentiate between frames. Newtonian momenta fail to behave properly under Lorentzian transformation, and Einstein preferred to change the definition of momentum to one involving {{nowrap|1=4-vectors}} rather than give up on conservation of momentum.

Physical laws must be based on constructs that are frame independent. This means that physical laws may take the form of equations connecting scalars, which are always frame independent. However, equations involving {{nowrap|1=4-vectors}} require the use of tensors with appropriate rank, which themselves can be thought of as being built up from {{nowrap|1=4-vectors}}.<ref name="Morin2007" />{{rp|186}}
{{anchor|Acceleration}}

=== Acceleration ===
{{Further|Acceleration (special relativity)}}

Special relativity does accommodate [[acceleration (special relativity)|accelerations]] as well as [[Rindler coordinates|accelerating frames of reference]].<ref>{{cite book |title=Relativity Made Relatively Easy |edition=illustrated |first1=Andrew M. |last1=Steane |publisher=OUP Oxford |year=2012 |isbn=978-0-19-966286-9 |page=226 |url=https://books.google.com/books?id=75rCErZkh7EC}} [https://books.google.com/books?id=75rCErZkh7EC&pg=PA226 Extract of page 226]</ref>
It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames.<ref>{{cite book |title=Explorations in Mathematical Physics: The Concepts Behind an Elegant Language |edition=illustrated |first1=Don |last1=Koks |publisher=Springer Science & Business Media |year=2006 |isbn=978-0-387-32793-8 |page=234 |url=https://books.google.com/books?id=ObMb7l9-9loC}} [https://books.google.com/books?id=ObMb7l9-9loC&pg=PA234 Extract of page 234]</ref> It is only when gravitation is significant that general relativity is required.<ref name="PhysicsFAQ">{{cite web|last1=Gibbs|first1=Philip|title=Can Special Relativity Handle Acceleration?|url=http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html|website=The Physics and Relativity FAQ|publisher=math.ucr.edu|access-date=28 May 2017|archive-date=7 June 2017|archive-url=https://web.archive.org/web/20170607102331/http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html|url-status=live}}</ref>

Properly handling accelerating frames does require some care, however. The difference between special and general relativity is that (1) In special relativity, all velocities are relative, but acceleration is absolute. (2) In general relativity, all motion is relative, whether inertial, accelerating, or rotating. To accommodate this difference, general relativity uses curved spacetime.<ref name="PhysicsFAQ" />

In this section, we analyze several scenarios involving accelerated reference frames.

{{anchor|Dewan–Beran–Bell spaceship paradox}}

==== Dewan–Beran–Bell spaceship paradox ====
{{Main|Bell's spaceship paradox}}

The Dewan–Beran–Bell spaceship paradox ([[Bell's spaceship paradox]]) is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues.

[[File:Bell's spaceship paradox - two spaceships - initial setup.png|thumb|Figure 7–4. Dewan–Beran–Bell spaceship paradox]]
In Fig.&nbsp;7-4, two identical spaceships float in space and are at rest relative to each other. They are connected by a string that is capable of only a limited amount of stretching before breaking. At a given instant in our frame, the observer frame, both spaceships accelerate in the same direction along the line between them with the same constant proper acceleration.<ref group=note>In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured.</ref> Will the string break?

When the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution. Two lines of reasoning lead to opposite conclusions. Both arguments, which are presented below, are flawed even though one of them yields the correct answer.<ref name="Morin2007" />{{rp|106,120–122}}
# To observers in the rest frame, the spaceships start a distance ''L'' apart and remain the same distance apart during acceleration. During acceleration, ''L'' is a length contracted distance of the distance {{nowrap|1=''L{{'}} = &gamma;L''}} in the frame of the accelerating spaceships. After a sufficiently long time, ''&gamma;'' will increase to a sufficiently large factor that the string must break.
# Let ''A'' and ''B'' be the rear and front spaceships. In the frame of the spaceships, each spaceship sees the other spaceship doing the same thing that it is doing. ''A'' says that ''B'' has the same acceleration that he has, and ''B'' sees that ''A'' matches her every move. So the spaceships stay the same distance apart, and the string does not break.<ref name="Morin2007" />{{rp|106,120–122}}

The problem with the first argument is that there is no "frame of the spaceships." There cannot be, because the two spaceships measure a growing distance between the two. Because there is no common frame of the spaceships, the length of the string is ill-defined. Nevertheless, the conclusion is correct, and the argument is mostly right. The second argument, however, completely ignores the relativity of simultaneity.<ref name="Morin2007" />{{rp|106,120–122}}

[[File:Bell spaceship paradox.svg|thumb|Figure 7–5. The curved lines represent the world lines of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A' and B', the observers stop accelerating. The dashed lines are lines of simultaneity for either observer before acceleration begins and after acceleration stops.]]
A spacetime diagram (Fig.&nbsp;7-5) makes the correct solution to this paradox almost immediately evident. Two observers in Minkowski spacetime accelerate with constant magnitude <math>k</math> acceleration for proper time <math>\sigma</math> (acceleration and elapsed time measured by the observers themselves, not some inertial observer). They are comoving and inertial before and after this phase. In Minkowski geometry, the length along the line of simultaneity <math>A'B''</math> turns out to be greater than the length along the line of simultaneity {{tmath|1= AB }}.

The length increase can be calculated with the help of the Lorentz transformation. If, as illustrated in Fig.&nbsp;7-5, the acceleration is finished, the ships will remain at a constant offset in some frame {{tmath|1= S' }}. If <math>x_{A}</math> and <math>x_{B}=x_{A}+L</math> are the ships' positions in {{tmath|1= S }}, the positions in frame <math>S'</math> are:<ref name="Franklin">{{cite journal |author=Franklin, Jerrold |title=Lorentz contraction, Bell's spaceships, and rigid body motion in special relativity |journal=European Journal of Physics |volume=31 |year=2010 |pages=291–298 |doi=10.1088/0143-0807/31/2/006 |bibcode = 2010EJPh...31..291F |issue=2 |arxiv = 0906.1919|s2cid=18059490 }}</ref>
: <math>\begin{align}
x'_{A}& = \gamma\left(x_{A}-vt\right)\\
x'_{B}& = \gamma\left(x_{A}+L-vt\right)\\
L'& = x'_{B}-x'_{A} =\gamma L
\end{align}</math>

The "paradox", as it were, comes from the way that Bell constructed his example. In the usual discussion of Lorentz contraction, the rest length is fixed and the moving length shortens as measured in frame {{tmath|1= S }}. As shown in Fig.&nbsp;7-5, Bell's example asserts the moving lengths <math>AB</math> and <math>A'B'</math> measured in frame <math>S</math> to be fixed, thereby forcing the rest frame length <math>A'B''</math> in frame <math>S'</math> to increase.

{{anchor|Accelerated observer with horizon}}

==== Accelerated observer with horizon ====
{{Main|Event horizon#Apparent horizon of an accelerated particle|Rindler coordinates}}

Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as [[event horizons]]. In the text accompanying [[Spacetime#Invariant hyperbola|Section "Invariant hyperbola" of the article Spacetime]], the magenta hyperbolae represented actual paths that are tracked by a constantly accelerating traveler in spacetime. During periods of positive acceleration, the traveler's velocity just ''approaches'' the speed of light, while, measured in our frame, the traveler's acceleration constantly decreases.

[[File:Accelerated relativistic observer with horizon.png|thumb|Figure 7–6. Accelerated relativistic observer with horizon. Another well-drawn illustration of the same topic may be viewed [[:File:ConstantAcceleration02.jpg|'''here''']]. ]]
Fig.&nbsp;7-6 details various features of the traveler's motions with more specificity. At any given moment, her space axis is formed by a line passing through the origin and her current position on the hyperbola, while her time axis is the tangent to the hyperbola at her position. The velocity parameter <math>\beta</math> approaches a limit of one as <math>ct</math> increases. Likewise, <math>\gamma</math> approaches infinity.

The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows:
# We remember that {{tmath|1= \beta = ct/x }}.
# Since {{tmath|1= c^2 t^2 - x^2 = s^2 }}, we conclude that {{tmath|1= \beta (ct) = ct/ \sqrt{c^2 t^2 - s^2} }}.
# <math>\gamma = 1/\sqrt{1 - \beta ^2} = </math> <math>\sqrt{c^2 t^2 - s^2}/s</math>
# From the relativistic force law, <math>F = dp/dt = </math>{{tmath|1= dpc/d(ct) = d(\beta \gamma m c^2)/d(ct) }}.
# Substituting <math>\beta(ct)</math> from step 2 and the expression for <math>\gamma</math> from step 3 yields {{tmath|1= F = mc^2 / s }}, which is a constant expression.<ref name="Bais">{{cite book|last1=Bais|first1=Sander|title=Very Special Relativity: An Illustrated Guide|url=https://archive.org/details/veryspecialrelat0000bais|url-access=registration|date=2007|publisher=Harvard University Press|location=Cambridge, Massachusetts|isbn=978-0-674-02611-7}}</ref>{{rp|110–113}}

Fig.&nbsp;7-6 illustrates a specific calculated scenario. Terence (A) and Stella (B) initially stand together 100 light&nbsp;hours from the origin. Stella lifts off at time 0, her spacecraft accelerating at 0.01&nbsp;''c'' per hour. Every twenty hours, Terence radios updates to Stella about the situation at home (solid green lines). Stella receives these regular transmissions, but the increasing distance (offset in part by time dilation) causes her to receive Terence's communications later and later as measured on her clock, and she ''never'' receives any communications from Terence after 100 hours on his clock (dashed green lines).<ref name="Bais" />{{rp|110–113}}

After 100 hours according to Terence's clock, Stella enters a dark region. She has traveled outside Terence's timelike future. On the other hand, Terence can continue to '''receive''' Stella's messages to him indefinitely. He just has to wait long enough. Spacetime has been divided into distinct regions separated by an ''apparent'' event horizon. So long as Stella continues to accelerate, she can never know what takes place behind this horizon.<ref name="Bais" />{{rp|110–113}}