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PHY3101 Detweiler
January 30, 2008
Relativity
Notes on lecture 10
The interval
The interval between two events is the same when measured in any frame of reference:
±interval2 = −c2 ∆t2 + ∆x2 + ∆y 2 + ∆z 2
The value of the interval is independent of which coordinate system is in use.
Definition: Timelike interval
We say that the interval between two events is timelike if the right hand side of the interval
squared is negative. If the interval between two events is timelike then there is some frame of
reference where the events occur at the same place, and there is no frame of reference where
the events occur at the same time. And the event that occurs first in one frame of reference,
occurs first in all frames of reference. The value of a timelike interval determines the proper
time between the two events—that is the the time between the events as measured in a
frame of reference where they occur at the same position.
Example: Given two timelike-related events find a frame of reference in which the two
events occur at the same place.
We are looking for the speed v of the boosted frame of reference in which ∆x0 = 0. But
∆x − v∆t
∆x0 = p
1 − v 2 /c2
so we require ∆x − v∆t/c2 = 0 ⇒ v = ∆x/∆t.
Definition: Spacelike interval
We say that the interval between two events is spacelike if the right hand side of the interval
squared is positive. If the interval between two events is spacelike then there is some frame of
reference where the events occur at the same time, and there is no frame of reference where
the events occur at the same place. The value of a spacelike interval is actually the proper
distance between the two events—that is the distance as measured in a frame of reference
where the two events occur at the same time.
Example: Given two spacelike-related events find a frame of reference in which the two
events occur at the same time.
We are looking for the speed v of the boosted frame of reference in which ∆t0 = 0. But
∆t − v∆x/c2
∆t0 = p
1 − v 2 /c2
so we require ∆t − v∆x/c2 = 0 ⇒ v = c2 ∆t/∆x.
Definition: Light-like interval
We say that the interval is light-like or null if the interval is zero. If the interval is light-like
(or null) then a ray of light can go from one event to the other. If the interval between two
events is light-like then there is no frame of reference where the events occur at either the
same time or at the same place. And the event that occurs first in one frame of reference,
occurs first in all frames of reference.
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“Pythagorean Theorem” intuition does not work in a space-time diagram
ct #3
light
*
x’
ct"
#2a
#2
*
*
ct’
*
#1
x
Twin paradox
The traveling twin leaves the Earth at event #1, arrives at a distant planet and instantaneously turns around at event #2 and finally arrives back on Earth at event #3. While
traveling from #1 to #2 the line ct0 marks the worldline of the traveling twin, and the proper
time between events #1 and #2 is ∆t0(2,1) = t02 −t01 . In the Earth’s frame of reference, the time
p
interval between #1 and #2 is “time dilated” to be ∆t(2,1) = t2 − t1 = ∆t0(2,1) / 1 − v 2 /c2 .
This is also the time interval between events #1 and #2a because event #2a is simultaneous
with #2 in the Earth’s frame of reference.
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Faster than light allows time travel
In the first figure, notice that t = 0 on the x-axis and that x = 0 on the ct-axis. The line
labeled ct0 is the world line of an object moving with a position given by x = vt where
v = 54 c, and this is the ct0 axis (where x0 = 0) for the “moving” frame of reference. Also
the x0 -axis is the line where t0 = 0 and also where t = vx/c2 , which is consistent with the
Lorentz transformations.
x=0
ct
x’=0
ct’
light
x’
ct’=0
*#2
#1
*
*#−1
x
ct=0
Look at the location in spacetime of events #1 and #2. It is clear that x1 = 0 and t1 = 0
and also that x2 > 0 and that t2 > 0. We deduce that event #1 occurs before event #2
with this coordinate system.
In this diagram it is more difficult to understand the primed coordinates, but it should
be clear that the x0 axis is the line where ct0 = 0. The Lorentz transformations then imply
that this same line is described by t = vx/c2 in the unprimed coordinates. Also, it should
be clear that the t0 axis is the line where x0 = 0. The Lorentz transformations then imply
that this same line is described by x = vt in the unprimed coordinates.
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light
ct
x’=0
ct’
x
ct=0
#1
*
*
#−1
x’
ct’=0
*#2
A spacetime diagram for the same events but for the primed coordinates is shown in the
second figure. Event #1 is at the origin and
has coordinates x0 = 0 and t0p= 0. The
p
coordinates for event #2 are x02 = (x2 − vt2 )/ 1 − v 2 /c2 and t02 = (t2 − vx2 /c2 )/ 1 − v 2 /c2 .
The first figure shows qualitatively that t02 < 0, because event #2 lies below the line t0 = 0.
In the primed coordinate system, notice that the unprimed x axis is the line where t = 0,
which corresponds to the line t0 = vx0 /c2 . and that the unprimed t axis is the line where
x0 = 0, which corresponds to the line x0 = vt0 .
Now imagine that it is possible to travel faster than light. In the first diagram, you could
then start at event #1 and travel via warp-drive to #2.
Having arrived at #2 you could quickly change to the primed coordinate system of the
second figure, and again using warp-drive then return to an event at the same place, in the
unprimed coordinates, but just a little before #1 actually occurs. The net effect of this
operation is to put you back in your starting place, but a little before you started — you
would have gone back in time.
If tachyons actually exist, and we if can control them in some way, then we could send
information back into the past.
If we can travel back into the past, or send information back into the past, then we
might be able to create a logical paradox, by changing an event that we thought had already
occurred. This might be a good reason to claim that “The laws of physics forbid sending
information faster than the speed of light.”
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All evidence implies that this statement is true. However, it might be the case that
information might be able to travel faster than the speed of light, but that we do not have
free will (or at least enough free will) to go back and change the past.
Most physicists have thought about such things when they were younger. But, found
such ideas not a very fruitful way to increase their understanding.
Automobile and garage paradox
ct
ct’
x=0
back front
x’=0
x’
ct’=0
*
*
Doors are closed
x
ct=0
Above, is a spacetime diagram that helps “explain” problem #1 of homework set #4 —
that was the problem with the 11 ft long automobile fitting into the 10 ft garage. The figure
is in the frame of reference of the garage. The two light vertical lines are the locations of the
doors at either end of the garage. The two red lines are the front of the automobile (on the
right) and the back of the automobile on the left. The front and the back of the automobile
are both moving to the right at the same speed, so the red lines are parallel. The small blue
star on the left marks where the back of the automobile enters the garage, and the door on
the left can be closed. The small blue star on the right marks the event where the front of
the automobile leaves the garage and the door on the right must be opened.
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First look at the situation from the garage’s point of view:
Notice that the x-axis and the light grey horizontal line are both lines of “constant time” t
in the garage’s frame of reference. Also note that on the light grey line both the front and
the back of the automobile are inside the garage at the same time t.
Now, look at the situation from the automobile’s point of view:
Notice that the x0 -axis and the light brown lines sloping up and to the right are lines of
“constant time” t0 in the automobile’s frame of reference. Look at the brown line with
the blue star on the right that shows the front of the automobile leaving the garage. Now
look at the brown line with the blue star on the left that shows the back of the automobile
entering the garage. Here is the surprise: The front leaves before the back enters! — from
the automobile’s point of view. How did I come to that conclusion? The x0 -axis is the light
blue line where t0 = 0. The light brown lines are lines of constant t0 . Imagine moving the
blue line up while keeping it tilted. It reaches the blue star on the right before it reaches
the blue star on the left. So, from the point of view of the automobile, the event marked by
the blue star o the right occurs before the event marked by the blue star on the left.
Worm and rake paradox
We watch a 1.1 cm (proper length) long worm crawl East at a speed of 0.8c. And we watch
a very long rake, lined up East-West, slowly move South as shown on the blackboard. The
proper distance between the prongs of the rake is 1 cm. From our knowledge of the Lorentz
contraction,
p we know that the worm will measure the distance between the prongs to be
1.1 cm × 1 − v 2 /c2 = 0.66 cm and it fears that it will be cut in two. But we measure the
length of the worm to be only 0.6 cm and it appears that the worm is likely to fit between
the prongs and escape harm. Which of the following statements is true?
1. The rake necessarily cuts the worm in two.
2. The rake does not necessarily cut the worm in two.
3. The worm is necessarily cut in two from its point of view, but not from the rake’s
point of view.
4. None of the other statements is true.
5. None of these five statements is true (including this one).