1.1 Lorentz Transformation
It is a common misconception that Einstein based his special relativity theory on the Michaelson-Morley experiment. In the days of Michaelson and Morley it was thought that electromagnetic waves propagated by a medium which was called the Luminiferous Aether. The earth rotates on its axis and rotates around the sun with circles the galaxy which wanders about the local cluster group etc so it was expected that if a device could be made to detect the its motion with respect to the aether that it would yield a significantly nonzero result. Michaelson and Morley constructed just such a device and to their astonishment it yielded a null result. This is enough to dispel the aether theory in most peoples minds, but at the time some people tried to explain why one would not be able to measure a speed based on the motion of light in the device even though they insisted the Earth was in motion with respect to the medium. Lorentz was one such person who empirically derived the Lorentz transformation equations by introducing length contractions and time dilations into a transformation which would leave the speed c invariant to frame so that one could not use it to determine a speed with respect to the medium. Einstein is given so much credit because what he did different was to come up with two powerful postulates from a simple idea and from those postulates was able to derive the Lorentz transformations from first principles and developed special relativistic physics from there.
The idea that led Einstein to his postulates [Einstein A., 1905] is depicted in the figure.
Start out placing a magnet stationary on a table perpendicular to the plane of a loop of wire and pointed at the center. Wire the loop in series with a resister and a current meter to calculate from the current and resistance the voltage induced in the loop. Move the loop at constant velocity and note the voltage function. Next fix the loop with respect to the table and move the magnet at the same velocity except for opposite in direction. Note the voltage function. Either way you do the experiment you get the same voltage function, the same physics. Therefor it won't matter whether we say the magnet source is moving and the loop receiver is stationary or visa-versa as you get the same physics either way. This led Einstein to his first postulate of special relativity. The electrons form a current in response to the changing magnetic flux, or the way the magnetic field changes across the loop over time. Either perspective yields the same physics so the information about the magnet as received by the electrons must travel at a speed independent of whether we say the source is stationary and the receiver is moving or visa-versa. This led Einstein to realize that there must be a speed that is invariant to frame at which the electromagnetic information transfers. This led to his second postulate of special relativity.
The first postulate of can be worded as
The laws of physics are invariant to inertial frame transformation.
The second postulate can be worded as
The invariant speed c, is finite and is the vacuum speed of light.
These postulates are phrased a little different in virtually every text, but the core idea behind them as depicted above is the same.
I phrase the second in this manner for the reason that time dilation and such effects really have nothing directly to do with light itself. These are instead due to spacetime being structured such that the invariant speed c is finite. If we surprisingly discovered that light had some small amount of mass and thus really traveled at speeds just short of c, it would have no ramification on the special relativistic physics whatsoever. It would merely mean that we are using bad terminology, for instance in calling c speed particles "light-like". What actually distinguishes Lorentz transformations and special relativistic physics from Galilean transformations and Newtonian physics is that in the Lorentz transformations the invariant speed c is finite. In the mathematical limit as c goes to infinity the Lorentz transformations become Galilean and physics reduces to Newtonian physics. The statement that this invariant speed c is the vacuum speed of light merely tells us where to look experimentally for what that speed it. And should we find that light travels at speeds just less than c then one may merely remove the "and is the vacuum speed of light" part and the fact that physics is relativistic according to the remainder of the two postulates would be unaffected.
The first postulate tells us that it does not matter what inertial frames we take to be in motion, or what we take to be stationary as the laws of physics do not depend on inertial frame. In a sense it is odd that the theory is called relativity at all because according to the first postulate physics is invariant. Thus relativity is really a theory of invariance. What is really relative in the theory are time and space coordinates and things defined in such a way that they depend on the coordinate frames, the elements of the tensors we will learn about, not a tensor equation as a whole. The equations used to model the "laws" of physics must be invariant equations if we are to be consistent with the first postulate. As we shall see tensor equations are invariant and so in relativistic theory we write the laws of physics as tensor equations.
As demonstrated in problem 1.1.1, these two postulates of imply that the coordinate transformation that correctly describes boosts between different inertial frames is the Lorentz coordinate transformation.
Lets say that one observer uses an inertial coordinate frame S with coordinates (ct,x,y,z). Another observer uses another inertial coordinate frame S' given by (ct',x',y',z'). They will be in motion with respect to each other so that the S frame observer observes the other to moving at speed v an the +x direction and the S' frame observer will observe the other to be moving at the same speed in the -x' direction. Lets say we know the location of an event according to one observer's coordinates and wish to determine the location according to the other coordinates. We transform the coordinates of the event from the one to the other by doing a Lorentz coordinate transformation. In this case the Lorentz coordinate transformation equations are
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where we make the definitions
A more compact form of the transformation that allows the boost to be in any direction rather then restricting it to a coordinate axis is
(To include acceleration see section 5.4)
Inverted equation 1.1.1 becomes
and in differential form equation 1.1.1 becomes
and the inverse differential form is
c is called the Lorentz invariant speed and according to Einstein's second postulate, it is the finite vacuum speed of light.
(Exact by definition)
c = 299792458m/s
If c were to be infinite then the Lorentz transformation equations would be the Galilean transformations
When speeds are large enough so that c can no longer be taken to be infinite then the Galilean transformations can no longer be used and the special relativistic phenomena such as time dilation and length contraction are observed
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Problem 1.1.1. Consider a signal sent along an x, x' axis sent at t = 0, where S' moves relative to the S in the + x direction with speed v. If the speed of the signal is the same speed c according to either frame, and assuming the coordinate transformation between the two takes the form
t= at' + bx'
x = ex' + ft'
y = y'
z = z'
derive the Lorentz coordinate transformation in the form of Eqn. 1.1.1 and the expression for g .
Invert the Lorentz transformations starting in the form of Eqn 1.1.1 to arrive at Eqn 1.1.4
Notice that this is the same result as if we had just switched the primes on the coordinates and the sign on b . Considering Einstein's first postulate, why must this be so?
Write the inverse of equation 1.1.3.
Consider S' to be a boost from S in the <1,1,1> direction. Use equation 1.1.3 to write out the transformations for the ct, x, y and z coordinates.
Problem 1.1.5 Show that in a limit where c is large compared to v, Eqn.1.1.1 reduces to Eqn. 1.1.8.
Relative Space and Time
The differential form of one of the Lorentz coordinate transformation equations Eqn 1.1.5 is
Thus extended to a finite interval this becomes.
1.2 Relative Space and Time 5
Given the interval in time and space between two events according to S' , this equation gives the interval in time between the events according to S. Now consider the case that the events happen at the same location according to S' , for instance the ticks on the S' observer's watch. In this case Dx' = 0 and we have
To distinguish that the time interval is for events at constant location according to the proper frame we often write the proper time interval as and call it proper time.
From this equation we see that the time interval between the events according to S is longer than the time interval between the events according to S'. This phenomenon is called time dilation. Time intervals between events are not absolute, but depend on whose coordinates you use.
We can extend this phenomenon to the case in which one of the observers is accelerated. Though special relativity is really only directly concerned with inertial frames, we can consider an accelerated state to be a state of transitions or boosts between different inertial frames. We will next let the S' observer enter a state of acceleration. For small time intervals we can say the S' observer is an inertial frame observer. Thus the equation
holds valid for describing how much time goes by according to the S observer given how much the S' observer has aged even if the S' observer is accelerating.
Now consider the case that the two events happen at the same time according to S'. Then Dct' = 0 and Eqn 1.2.1 becomes
From this we see that if the events have a displacement in space along the x' direction, then they happen at different times according to S. Thus the very notion of simultaneity is relative. Events simultaneous according to one coordinate frame are not all simultaneous according to another.
Next, consider the following differential Lorentz coordinate transformation equation from Eqn.1.1.6
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Extend over a finite interval to arrive at the corresponding equation for two displaced events
Lets say that the S' observer puts a fire cracker on each end of a measuring stick of length L0 oriented along the x' direction and sets them off timed so that the events occur at the same time according to the S frame. Then the length of the stick according to the S frame L will be given by the spatial displacement between the events. Then we have and . Inserting these three inputs into the above equation results in
This is called length contraction.
A train moves at a constant speed v along a straight track. There is an on board experimenter who sets up a laser pulse generator on the floor facing straight up at a mirror on the ceiling at a height h. The pulse travels up to the mirror and straight back down in a round-trip time according to this frame given by . (Denote the on board observer's frame with primes.) An external observer is stationary with respect to the ground and observes the experiment. According to him the light pulse makes a triangular path in a round trip time of . If the light pulse travels at the same speed of c according to either observer, show that the times are related by Eqn. 1.2.2
According to the S frame observer of problem 1.2.1, the track has a total length of L0 much larger than the length of the train, and so finds that the train takes a time t = L0/v to travel from one end to the other. The S ' frame observer finds that it takes a time of t' from the time the he's inside the train station at one end of the track until he's inside the train station at the other end of the track and so this observer uses L = vt' measure the length of the track. Use this information with Eqn 1.2.2 to arrive at Eqn. 1.2.5.
Epsilon Eridani is a K2V type star, which is much like our sun, only orange red. It also has at
1.2 Relative Space and Time 7
least one confirmed planet and likely has more. It is also very close at 3 parsecs away.
(1pc = 3.26 ly) So, let's say radio contact becomes established with an alien civilization there. Ambassadors and scientists are eventually sent there in a starship. It travels at a near constant speed there and back at v = (4/5)c.
How long does the trip take according to the crew?
Lets say that the event that the earth sends off a signal of goodbye to the outgoing ship and the event that the aliens send off a welcome signal to the ship are simultaneous according to the earth based coordinate frame. These signals are sent according to the earth frame at the same time that the ship is half way along its journey. Which signal does the ship receive first? How far apart in time were the sending of the signals according to the ship based coordinate frame?
If the ship sent a pod on a trip back to the earth again at (4/5)c according to the Earth based coordinate frame, what happens to the time ordering of the events according to an observer on the pod?
A starship leaves Earth on a round trip to another star varying its acceleration so that
where , c = 1 ly/y, and T ' is the time in years for the round trip according to the crew of the starship.
How does compare to the acceleration of Earth's surface gravity in the same units?
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With this value for , it would take a time of T ' = 7.21 y to do a round trip past Alpha Centauri which is the closest star at 4.23 ly away. This is less than 8.46 y. What is the time for this round trip according to an observer on Earth? (Hint - Integrate Eqn. 1.2.3)
The point of introducing pseudo paradoxes in introduction to special relativistic physics, is to demonstrate to the student that it is easy to leave some aspect of the special theory out of a relativistic scenario you may consider and come up with something that looks like a paradox that really isn't. So when one finds what appears to be a paradox in relativistic physics one needs to go looking for what aspect was missed. For example one might consider the consequences of time travel associated with moving faster than light, however Lorentz transformation imbedded into the tensor laws such as how momentum and energy relate between frames through a momentum 4-vector prevents one from doing this with special relativistic physics so that causality is preserved [Zeeman, E. C., 1964].
Consider a space vehicle that travels to another star at a constant velocity, then immediately whips around the star for the return trip at the same speed. We on Earth observe that the clocks on board the ship run slow due to time dilation throughout the entire trip both outgoing and incoming. On the way out, according to the ship frame it is the Earth that is moving away and so it is the Earth clocks that run slow due to time dilation. Also on the way back, according to the ship frame it is the Earth that is approaching the ship and so again the Earth clocks run slow due to time dilation. This presents a problem called the twin paradox. The problem is how to answer how the clocks read when the ship and earth arrive together again and what causes the difference. The total time dilation can be calculated by 1.2.2 or 1.1.3, but only if the accelerated frame is taken as the primed frame. The question would be why this round trip case is not symmetrical. If the two remained in inertial states then each would observe the other as aging slower in a symmetric fashion. But in this round trip the end result couldn't work symmetrically because that would lead to a true paradox. One can explain this from various perspectives just as there are various frames that can be used to describe the situation. The simplest explanation is that the accelerated frame of the ship is actually a piecewise construction of two different inertial frames for which there is a lack of simultaneity. Because of the piecewise construction of the accelerated frame, the ship observer reckons that clocks in the direction of the acceleration undergo an advance during the acceleration by an amount that depends on how far away they are. Primes ( ' ) will indicate the ship frame. T ' will be the time it takes the ship to reach the star according to the ship frame and as a function of proper time is.
The coordinate transformation from the accelerated ship coordinates to the inertial Earth coordinates is
This set of transformation equations results in Lorentz transformation on the way out as well as on the way in and gives the solution that the ship clocks read less time upon their arrival back at Earth. Symmetric time dilation only occurs during the portions of the trip where both observers maintain inertial states. During acceleration the symmetry is broken and both observers will always agree on how much they should age differently in a round trip.
1.3 Paradoxes 9
Consider a train whose proper length is greater than the proper length of a tunnel. The train moves at near c speeds so that it is extremely length contracted according to the frame of the tunnel. Let's say it is so length contracted that it fits inside the tunnel. Gates at the ends of the tunnel are set so that they each close once it's entirely inside.
According to the train observer the train is still while the tunnel moves the other direction. Therefor according to his coordinates it is the tunnel whose length is contracted. The proper length of the tunnel is itself shorter than the length of the train and so the tunnel is length contracted beyond this so that the train never fits to be enclosed by the gates.
This leads to a superficially apparent contradiction called the length contraction paradox. As with all the so-called paradoxes of the theory this only superficially seems to be a contradiction and is easily shown not to be a true contradiction. The solution is the realization that we did not account for relative simultaneity in this mind experiment. According to the tunnel frame the events that the two gates close are simultaneous. According to the train observer's coordinate system the two events still occur, but they are not simultaneous. According to the train observer's coordinate system the event that the gate closes behind the train happens after the front end of the train has smashed through the front gate which had already been closed.
To demonstrate this simply consider equation (1.1.6a-b)
Considering the events of the doors closing at the same time according to the unprimed tunnel frame we find
According to 1.3.2b the train fits between these events as long as Dx between the events is restricted by
Since length contraction is given by this means that the train will always fit between the events as long as between the events is restricted by
According to 1.3.2a this means that the events of the doors closing just barely at the ends of the train will happen at times differing by
This says that according to the primed frame the doors just shut at different times.
It is also relevant to note that since these events are spacelike in their separation, they can not be connected by signals restricted to the speed of light. This means that should the door at the end of the tunnel jam smashing the train, the back of the train would still fully enter the tunnel before it feels the compression wave.
No matter what happens there is no paradox.
A ship travels as in the discussion leading to Eqn 1.3.1. Just prior to t' = T ', the event that the ship begins acceleration back toward Earth, and the event that someone back on Earth blows out the candles on a 21st birthday cake are simultaneous. If the speed is , and T ' = (13/2) y, how much older is the birthday boy according to the accelerated ship frame just after the acceleration?
Einstein A., 1905, "Zur Elektrodynamik bewegter Korper," Ann. Phys. 17, 891-921
Zeeman, E. C., 1964, "Causality implies the Lorentz group," J. Math. Phys. 5, 190-493