SPCS Special and General Relativity
Lecture 3: Spacetime Diagrams, Spacetime, Geometry
Introducing Spacetime
Classically we have an absolute time, which can be viewed as a one-dimensional Euclidean space, R, plus
an absolute three-dimensional space, R3. Time and space are two separate spaces and the movement in one space
(time) is independent of movement through the other (space).
What the postulates of special relativity tell us is that movement through space is no longer independent of
movement through time. We conclude that space and time are no longer independent and that the rate of movement
through time depends on your movement through space (velocity).
The result of this is that space and time are linked; we can no longer treat them as separate or absolute. To
quote Minkowski, “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and
only a union of the two will preserve an independent reality”.
To mathematically manifest this idea we introduce a new entity –spacetime – where time becomes the
fourth dimension tacked on to three-dimensional space. What we want to do is incorporate this concept into our
understanding of SR and see if it is useful. To do this we need to describe the geometry of this new entity,
spacetime.
Before beginning an in depth analysis of the geometry of spacetime, a graphical method of representing
events in spacetime will be introduced. This representation brings all of the effects we have previously discussed
into one realm. These diagrams, it will be seen, are intimately related to the geometrical aspects of spacetime. We
begin with these to ease into the full geometrical, mathematical formalism.
Spacetime Diagrams
Consider the following example: two people, Alice and Bob, observe each other. Alice
is atop Hoover Tower at Stanford. Bob is in a jet that travels at constant velocity passing by
Hoover tower on towards Berkeley’s Sather Tower. Let’s plot Bob’s trajectory as seen by Alice
(in Alice’s reference frame). The spatial origin is at Hoover Tower and the x axis is defined as
the line joining Hoover Tower and Sather Tower. Plotting time on the vertical axis we see that
Bob’s trajectory is a sloping line crossing the x-axis when Bob passes Alice and intercepts the
line representing the spatial position of Sather Tower when Alice observes Bob passing it.
There is nothing special about this plot; you may have created such plots learning
classical mechanics. However now we interpret this plot in a different manner. First of all, we
convert the vertical axis to measure time in meters. This is done by changing the vertical
scale to ct instead of just t. Again, this could be done for classical physics but here is the
key difference; interpreting this plot classically, time and space are separate, unrelated,
spaces. This is termed a non-metric space since the length along a line segment has no
significance. Consider a plot of time versus money, these are two unrelated dimensions.
The length along a line in this plot has no significant meaning. Contrast this to a plot of the
two dimensional plane. Here the two dimensions are related and the length along a curve
in this space is the distance traveled by the object. So, even though the plot looks the same
classically, they are interpreted differently.
Again, Bob’s trajectory in spacetime as viewed by Alice
is a straight line between the two events of Bob passing Alice and
Bob passing Sather Tower, as viewed by Alice. Points in
spacetime will be called events since they represent something
happening at a particular place at a particular time (x,y,z,t). A line
plotted on this diagram is termed a worldline. A worldline is
simply the path an object traces out through this four dimensional
space. Note, in order to plot all four dimensions on a diagram would
not be possible, thus we often restrict ourselves to one and twodimensional (spatial) problems – here to the one dimensional line, x and time.
Another important trajectory to indicate on this plot is one due to a light pulse
emanating from Alice’s position at time ct = 0 m. If we measure time in seconds we see that
this worldline would lie very close to the x axis. (In one second of time the light pulse would
travel 3 x 108 m, most of the way to the Moon). On this plot most of the area is covered by
objects traveling much less than the speed of light; that is, representing particles traveling close
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SPCS Special and General Relativity
to the speed of light would involve plotting nearly horizontal worldlines. This is because we have chosen the
standard convention of measuring x in meters and t in seconds.
However, it is more natural to plot the perpendicular axes in the same units with the same scale – meters. In
this coordinate system, a distance on the x axis, say 1 meter, is an equivalent distance along the ct axis, 1 meter = ∆t
= 3.3 ns. With this natural choice of scale, a 45o line represents 1 m of time in 1 m of distance, that is 1 = c∆t/∆x.
Such a worldline represents an object that is traveling at c. Significant here is that in all frames wordlines for objects
traveling at c will be represented at +45o or -45o, this being a consequence of the constancy of the speed of light, [c].
In these notes we will continually write ct as the distance, in meters, covered in time. In most modern
treatments of relativity the lazy physicist principle is invoked and one simply sets c = 1. This is what is done in the
textbook “Spacetime Physics” but will not be followed here. (In addition, professional physicists will often also set
G = 1, the universal constant for gravitation as well as h/2π, Planck’s constant). The reason that c is kept in here is to
be continually reminded of the large magnitude of the speed of light, to be constantly aware of the regions where
you can and can not approximate what happens using classical physics.
Redrawing our plot with this new rescaling increases the slope of Bob’s
worldline and sets the worldline of our light pulse to be a 45 degree line. This is the
conventional way of representing spacetime. (N.B. that the same results for setting c = 1,
although the factor of c no longer appears and time is measured in meters).
Now let’s relate this plot to a similar plot but interpreted by
Bob in his jet. On Alice’s plot (which will be labeled by x
and t) Bob moves off along the slanted worldline. In Bob’s
reference frame he sees himself as standing still and Alice
moving at constant velocity in the –x’ direction (Bob’s
coordinates will be labeled by x’ and t’).
In Alice’s frame, her own worldline is vertical along the ct
axis, because she does not move and is always at x = 0 m, at
Hoover Tower. In Bob’s frame his own worldline is vertical
as well since he is always at x’ = 0 m – always at the same
spot within his jet. With this in mind, we can attempt to combine Alice’s and Bob’s
spacetime plot into one single one. N.B. the two diagrams represent the same 4-dimensional spacetime but plotted in
different coordinate systems, one which corresponds naturally to Alice’s reference frame and the other to Bob’s. The
goal is to represent both coordinate systems on the same plot, this will help visualizing the geometry of 4dimensional spacetime.
It is easy to find how to properly plot Bob’s time axis within Alice’s plot. Begin by plotting Alice’s
spacetime plot (with axes x and ct) and add Bob’s worldline but as the time coordinate ct’. This coordinate
represents the IRF with the same velocity as Bob’s jet. The question is how to add Bob’s spatial direction, x’? This
will require a little work. To do this we will need to return to the discussion of simultaneous events viewed in both
frames.
Fig 14.
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SPCS Special and General Relativity
Simultaneity again
Consider two systems that move with respect to each other (collinearly). In frame A, there are a large
number of observers spaced regularly apart in this frame (10 meters apart). There are also a large number in frame B
spaced regularly apart as well. In frame A, it is observed that the spacing between the observers in B is the same as
in A and at one instant in time, the array of observers in A and in B line up (1 is across from 1’, 2 across from 2’,
etc.). At this instance all of these events are simultaneous in frame A and all observers are instructed to set clocks to
one predetermined time (say 0 s or Noon). Thus at this instant as viewed in frame A all clocks in both frames read
the same value. After this event the clocks in frame B will be observed to run slower than in , but clocks in frame A
will be synchronized and those in frame B will be synchronized.
Now examine these events in the B frame. First, in order to have the observers in B to appear to have the
same spacing as observed in A, then in B’s frame, the B observers must be farther apart than the A observers are
seen in A’s frame. This is due to length contraction.
In order to give a concrete example, let’s say that the dilation factor is 2 (v/c = 0.87) and that the observers
in A (as observed by A) are 10 meters apart. Thus in B’s frame, the B observers must be 20 meters apart in order to
have them appear to be 10 meters apart in A.
Since the B frame observers see the A frame to travel at speed v, but in the opposite direction, they will be
closer together due to length contraction. Since they are 10 meters apart in A and the dilation factor is 2, they must
appear to be 5 meters apart in B’s frame.
This is exactly the scenario discussed in the last lecture but now the spacetime diagrams will gives us a clue
on how to represent various IRFs on the same diagram. In frame A we have the following diagram.
Observe that the frame B observers (1’, 2’, etc.) move to the right at velocity v and are measured to be 10 m
apart by the frame A observers.
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SPCS Special and General Relativity
As mentioned previously, due to length contraction, in order for the B observers to appear 10 m apart to A,
they must be 20 meters apart in their own frame. And, since they observe A to travel at the same speed, they must
appear 5 m apart from their perspective. The spacetime diagram is thus the following.
Notice that in this diagram the blue lines (A observers: 1,2, etc.) are closer together, this is the effect of
length contraction. It is important to remember that to determine length in any one frame a horizontal cross section
(a time slice) needs to be considered. Here, if a horizontal line is drawn and the distance between any two adjacent A
worldlines is measured it comes out to be the measured distance in this frame (5 meters in this case). In order to
measure the length of a moving object you, obviously, would not measure its ends at different times – the logical
way is to determine the location of the end points at any one time. This is how you observe length contraction with
spacetime diagrams.
An important feature in this discussion is the location of events considered simultaneous in A and B. The
first diagram singled out a series of events simultaneous in A (1-1’, 2-2’, etc.) in which all clocks were set to the
same value. Notice the location of these events in frame B, where 1 meets 1’, 2 meets 2’ etc. They do not occur at
the same time in frame B –this is lack of simultaneity discussed before. Now define a line of simultaneity, as a line
on the diagram indicating all events that appear simultaneous in a particular frame. In the A diagram the events 1-1’,
2-2’, etc. all lie along (and map out) a line of simultaneity in frame A. Notice how this line of simultaneity appears
in the frame B: by connecting the intersections of 1 and 1’, 2 and 2’ etc. we see that it slopes up and to the left. In
frame A it is a horizontal line (a line of equal time) but in frame B it is not.
The key here is that the events indicated all lie along the x axis in frame A. In other words, the line of
simultaneity just introduced is equivalent to the x axis. Thus we have found how to represent the x axis in frame B
(it slopes up and to the left).
The original question posed was to find the x’ axis in the diagram of frame A. To find out, just reverse the
discussion. Find a series of events in frame B that all lie along the x’ axis and see where they appear in frame A’s
diagram. Examining the diagram in B we see we can use events 3’-0 (when observer 0 in A passes by observer 3’ in
B), 4’-4, and 5’-8. There are more but this is sufficient to map out the x’ axis in A. Look back at the frame A
diagram and see where these worldlines intersect – along a line that slopes up and to the right. Thus we can now
include the x’ axis to our plot (also noting that the worldline of observer 4’ maps out the ct’ axis, i.e. is at x’=0).
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SPCS Special and General Relativity
Notice that a light ray emitted at the origin passes halfway in between
the ct’ and x’ axis in this diagram – exactly what we expect since a 45o line
bisects two perpendicular lines (which these are in their own frame). The
following diagram considers two frames with some arbitrary relative velocity.
There are several features that need to be pointed out in this diagram. Notice that
the transformation from Alice’s to Bob’s frame is performed by ‘scrunching’ the
axis together about the lightcone line. The lightcone lies halfway between the
time and position axis for both frames – that is at 45o. It will be left as an
exercise to show that the angle between ct and ct’ is the same as that between x
and x’. Notice that for reference frames which move very fast the ct’ and x’ axes
approach each and would actually meet for frames moving at c.
In the following
figure we display several different inertial reference frames
as observed in Alice’s frame. Notice that the lightcone will
always appear at 45o in each of these frames, that is in each
frame light is observed to propagate at c.
How is Alice’s frame represented in Bob’s frame?
To display that, simply ‘unscrunch’ Bob’s axes so that they
are perpendicular (he is at rest in his own frame). In so doing,
Alice’s axes scrunch towards the left moving light cone (Alice appears to be traveling to the left in Alice’s frame).
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SPCS Special and General Relativity
Observing the kinematical special relativistic effects in diagrams.
In constructing the diagrams it has already been pointed out how the lack of simultaneity [Sim] is
represented in spacetime diagrams. Length contraction is also directly observable in diagrams as discussed above. In
the last two diagrams, the dotted lines represent the ends of a stick at rest in frame B. It is clear from the diagram
that the observed length in A, found by making a time slice in the A diagram, is shorter than that observed in the B
frame.
The other key effect is time dilation. In order to determine this from the diagrams it must be noted that the
non-orthogonal frame forms an oblique coordinate system when viewed in the other frame. Thus it is instructive to
add lines of equal x’ and ct’ within the A diagram.
The way to read this diagram is that the time between the two events (1 and 2) is c∆t’ in frame B and c∆t in
frame A. The later found by drawing a horizontal line across from the event 2 to the ct axis. To compare the time
intervals one needs to find the intersection of equal ct’ time of event 2 to the ct axis. This relates the scale of the two
axes. It is clear that the line c∆t’ (blue line along the ct axis) is shorter than the line c∆t. Thus ∆t > ∆t’. This is time
dilation, frame A observes time between ticks of the B clock to be longer (clock B runs slower).
If it is not clear how the above works consider the following. The event 1 appears at the origin of both
frames. Event 2 occurs at the point indicated in the diagram. The sloping line passing through event 2, parallel to the
x’ axis, is the locus of events that occur at the same time. To relate the passage of time in frame B to A not that the
intersection of this line with the ct axis (the top end of the blue line) indicates an event in B occurring at the same
time as event 2. Thus the time difference between 1 and 2 is easily read off as the vertical distance between 1 and
the intersection point (represented by the blue line). This ought to clear up difficulties of relating time measurements
in the two frames.
Notice that the slope of the light line, as calculated in S’ (oblique coordinates) is still the same. The change
∆x’ equals the change c∆t’ so that ∆x’/c∆t’ = 1  ∆x’/∆t’ = c = vlight. The transformation instills the constancy of c.
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SPCS Special and General Relativity
What remains is to demonstrate addition of velocities [AV] and the limitation of the speed of light [Noc] in
terms of diagrams.
In order to see these a few transformations between frames will be illustrated. Consider, as above, two
events 1 and 2 however now we want to label these as occurring in the frame 0 (given by axes ct and x). The two
events may represent an object at rest in this frame. In another frame 1 (with axes ct’ and x’) the object is observed
to move to the right at some speed. The first diagram below indicates the spacetime diagram in frame 1. Here the
object is observed to be moving at some speed v’. Now consider yet another frame (Frame 2 with ct” and x”) in
which the frame 1 is observed to be moving at the same speed v’. The question is, what is the speed of the object,
v”, in this frame? Classically it would be v’ + v’. however if we iterate this process by introducing more frames,
each moving at speed v’ with respect to the next one, we see that the speed of the original object (represented by the
worldline between 1 and 2) approaches but never equals c.
This iterative process is meant to demonstrate how an object observed to move at less than c in one frame
will be observed to be traveling at less than c in all frames. If you do not feel confident in that conclusion, play
around with other examples, observing the rules developed here. This discussion explains how both addition of
velocities [AV] and the limitation of the speed of light [Noc] can be understood from spacetime diagrams.
Continually accelerating object
To elaborate upon how the limitation of the speed of light is observed in diagrams another
gedankenexperiment is proposed. A spaceship undergoes constant acceleration for
a very long period of time. Forego any discussion of how this might occur and
assume that the thrust produced by the ship is constant and its mass remains the
same. Thus in any inertial reference frame that is momentarily at rest with respect
to the ship it accelerates from rest to greater speeds. The goal here is to analyze its
motion from a series of such instantaneous comoving frames (ICF), taken to be
inertial frames.
Begin with one particular reference frame, S, where the ship starts from
rest. This could be its starting point where it is launched from rest, it need not be
just as long as it is momentarily at rest at time t = 0. In this frame the ship
accelerates in the positive x direction. When the speed of the ship is observed to be
exactly c/2 in S it passes a space station that is traveling at c/2 as well. This space
station is an ICF call it S’, for the instant (t’=0) that the ship passes it is at rest with
respect to the station. Afterwards (t’ > 0) it accelerates to greater speeds. Thus the
motion of the space ship in S’ is exactly the same as it was in S as it speeds up to
c/2.
When the space ship reaches c/2 as measured in S’ it passes another space
station that is also traveling at c/2. This third ICF we can label S”. Again, as
viewed from S” the ship starts from rest (at x”=0 and t”=0) and accelerates up to
c/2 where we introduce yet another ICF, S”’. Repeat this process indefinitely.
Examining a spacetime diagram of any two neighboring ICFs in the chain the worldline of the spaceship is
a gently curving line (this is an indication that it is accelerating. In the figure the view is from the frame S’i (that is,
S with i primes) and the next ICF (S with i+1 primes) is introduced when the ship reaches ß = 1/2. It is important to
note that the specific neighboring frames we choose does not matter -the diagram is the same. For example if i=0
then S is the first frame and S’ is the first space station.
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SPCS Special and General Relativity
Returning to the original ICF, S, and including all
of the other ICFs (only up to S”” shown) it is seen
that the object continually accelerates yet reaches
an asymptotic speed of c. It never reaches exactly
c from the perspective of S but gets
infinitesimally close. Each neighboring ICF
observes each other to travel at a speed of c/2.
This is the same scenario introduced before with
the beltways, each traveling at c/2 with respect to
its neighbors.
The important point here is that, though
this diagram is for the frame S, it could represent
any frame (like S’i). All frames see this same
scenario of the ship accelerating away from rest
and approaching c asymptotically. This begins to
highlight the fact that the geometry of spacetime
is not our familiar old Euclidean geometry but
something strange. If you keep doubling your
speed every hour, say, a friend who watches you
will never see you reach c. An analogy would be a
strange room where you can see the far wall a
short distance away yet when you walk towards it
you never reach it. Someone watching you would
see each step you take be half of the previous,
although to you each step is the same. This is
reminiscent of Zeno’s paradox yet what is
important to take away from this is that the
geometry is no longer Euclidean.
Viewing this from the space ship’s frame
(not an inertial one but a diagram can still be
constructed) at any one time the same diagram to
the left would be valid but as time went on each
further ICF would approach the origin as the ship
passes them. The ship always remains at the
origin (this is what is meant by its frame). Though
difficult to visualize (and an animation is
available) it appears like one is unraveling layers
of an infinite onion. The layers being each ICF
yet the ship never reaches the end (at c).
There is still much to discuss about spacetime diagrams, in particular regions of spacetime that are allowed
to a material object and how to properly measure events in spacetime. Before covering those areas the geometry of
spacetime needs to be explored further. What comes next is a formal way of describing the geometry of any space.
The method is more abstract and mathematical but extremely powerful in deriving the results discussed thus far and
going further to introduce dynamics. What will come to be clear is that special relativity is nothing but geometry, a
point that will be reiterated many times.
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