Space-time diagrams refer to an inertial frame. Let us
consider two inertial frames, A (Alice) and B (Bob). For simplicity, let us
only consider one spatial dimension. Assume Bob moves with constant velocity
v with respect to Alice.
Initially we are only concerned with how to draw a space-time diagram in Alice's
reference frame.
Alice's rules for drawing space-time diagrams:
Line of constant position
The two red dots represent two events with the same space but different time coordinates.
Two lines of constant position
The red dots
represent events with the same space coordinate and different time
coordinates.
The blue dots also represent events with the same space
coordinate and different time coordinates,
but the space coordinate of the
blue dots is different from the space coordinate of the red dots.
Two lines of constant time
The red dots represent events with the same
time and different space coordinates.
The blue dots also represent events
with the same time coordinate and different space coordinates,
but the time
coordinate of the blue dots is different from the time coordinate of the red
dots.
The blue and the red point are two events with the same space coordinate,
separated by one unit of ct.
The red and the green dot are two events with
the same time coordinate separated by a distance of one unit.
A particularly important
collection of events for an object is the set of all events at which the object
is present. The totality of all such events is represented by a continuous line
in the space-time diagram. This line, which represents the entire history of
the object, is called the world line of the object. An object which is
stationary in Alice's frame of reference throughout its entire history is
represented by the line of constant position. An object moving with constant
velocity in Alice's frame of reference is represented by a straight line that is
not parallel to any line of constant position, since the object is at different
positions at different times.
The world lines of objects moving with the same constant velocity are parallel
lines. An especially important world line is the space-time trajectory of a
photon.
The dotted lines represent the world lines of two photons traveling in
opposite directions.
For the angles we have α + β = 90^{o}.
Alice's space-time diagram
The above rules completely
determine the structure and orientation of the system of lines of constant time
and constant position that Alice uses to locate events in space and time except
for two choices still available to her:
(a) She is free to choose the scale factor λ.
(b) She is free to choose the angle 2α
that her lines of constant time make with her lines of constant position or,
equivalently, the angle a that both
families of lines make with the photon lines. Her choices of λ and
α fix μ.
Her choice of α depends on how she wants to use her diagram. If she is only using it to chart events in her own reference frame, then a symmetric choice is to take 2α to be 90^{o}, so that her lines of constant position are vertical and her lines of constant time are horizontal. If, however, she wishes to compare her description of events with the description of the same events provided by an observer in another inertial frame of reference, then taking 2α to be 90^{o} may not be the best choice.
How can Bob use Alice's space-time diagram and convert it to a space time diagram drawn in his inertial frame?
Bob's frame of reference has velocity v with respect to Alice's frame. Bob's lines of constant position are therefore parallel to the word line of an object that is moving with velocity v in Alice's frame. Bob's lines of constant position are parallel straight lines that are not parallel to Alice's lines of constant position. The faster Bob moves with respect to Alice, the more they tilt away from Alice's lines. Lines of constant time and lines of constant position in Alice's diagram, drawn through any two points on one of Bob's lines of constant position, define a parallelogram. The ratio of the lengths of the sides of this parallelogram is just the relative speed v of the two reference frames.
The brown line is a line of constant position in Bob's frame,
i.e. it is the
world line of an object at rest in Bob's frame.
The orange lines are lines of
constant position and the green lines are lines of constant time in Alice's
frame.
The velocity of Bob with respect to Alice is v/c = a/b.
In
Alice's frame the
position of the object changes by an amount μa
and the time by and amount μb/c.
What are Bob's lines of constant time?
Consider a set of events associated with a clock synchronization experiment, on a train that is stationary in Bob's frame of reference. The left end, right end, and middle of this a train are represented in Alice's space-time diagram by parallel world lines of objects moving with constant velocity v. The lines are equally spaced, since Alice and Bob agree on which point describes the middle of the train. Two photons emitted together in the middle of the train travel in opposite directions with the same speed c. The train is at rest in Bob's frame, and the photons arrive at the opposite ends of the train at the same time in Bob's reference frame. In Alice's frame we draw a pair of 45^{o} lines starting at a point on the world line of the middle of the train. These lines represent the trajectories of photons moving toward the front and rear. The points of intersection of the two photons with the two ends of the train represent simultaneous events in Bob's frame and therefore lie on one of his lines of constant time.
The diagram is drawn in Alice's frame.
The brown lines are the world
lines of three equally spaced objects which are stationary in Bob's frame.
The low black dot represents the event of two photons being emitted in opposite
directions from the middle object.
The upper black dots represent the two
events of the photons being detected by the other two objects.
In Bob's
frame of reference these are simultaneous events.
The green line joining
the upper two dots is a line of constant time in Bob's
frame.
Bob's lines of constant time in Alice's diagram must make the same angle with the photon trajectories as his lines of constant position. In addition, the spacing of Bob' lines of constant c*time one unit apart is the same as the spacing of Bob's lines of constant position one unit of distance apart in Alice's diagram.
How to construct Bob's space-time diagram given Alice's diagram
Draw Alice's diagram. Choose the orientation of the ct and x axes. Draw the world line of an object moving with velocity v in Alice's frame. That world line makes an angle θ with Alice's ct axis. It is Bob's ct' axis. Bob's x' axis makes an angle -θ with Alice's x axis.
The rules for the orientation of Alice's lines of constant time and constant position and the relationship between their scales impose restrictions on the lines of constant position and time that Bob must use, if he wishes to represent events with the same points that Alice uses in her diagram. Those restrictions have exactly the same form as the rules we originally imposed on Alice. It is therefore impossible for anybody else to tell which of them made the diagram first, following rules 1-13, and which of them subsequently imposed his or her own lines of constant time and position on the other's diagram. This symmetry is required by the principle of relativity.
Example:
Let Alice choose her lines of constant time and her lines of constant velocity perpendicular to each other. Let the lines all be spaced 5 units apart. Let Bob move with v = 0.2c i with respect to Alice, γ = 1.02.
Alice's lines of constant velocity are shown in the
diagram.
The lines are also spaced 5 units apart if Bob's scale factor
λ'
is related to Alice's scale factor
λ by
λ'
=
λ [(1 - v^{2}/c^{2})/(1 + v^{2}/c^{2})]^{½}.
[Assume the space and time coordinate of an event are 0 in
both Alice's and Bob's diagram. A second event has space coordinate zero
in Alice's diagram and time coordinate ct' = 5 units in Bobs diagram. Its
time coordinate t in Alice's diagram is given by ct' = γct,
t = t'/γ . The intersections of Bob's lines of
constant time spaced by cΔt' = 5 units intersect
Alice's lines of constant position every cΔt'= 5/γ
units.
Therefore
λ'/(λ/γ )
= cosθ, where tanθ =
Δx/Δct = v/c. Now
cosθ = 1/(tan^{2}θ
+1)^{½},
so
λ'
=
λ[(1 - v^{2}/c^{2})/(1 + v^{2}/c^{2})]^{½}.]
Bob's scale factor μ' is related to Alice's scale factor m by
μ'
=
μ[(1 + v^{2}/c^{2})/(1 - v^{2}/c^{2})]^{½}.
The rhombus bounded by lines of constant time and constant position associated with events one unit of ct and one unit of distance apart is a unit rhombus. The scale factors used by different frames of reference are related by the rule that unit rhombi used by different observers all have the same area. Since the height of a unit rhombus is the scale factor λ and its base is the scale factor μ, the analytical expression of this geometric rule is that for any two frames of reference λ_{A}μ_{A} = λ_{B}μ_{B}.
The unit rhombus for some frame of reference.