Landau and Lifshitz, Classical Field Theory.Chapter One, The Principle of Relativity.

In a separate document I have explained that section one of L&L’s text involves more than just the principle of relativity, but also the assumption that there exists a finitely-valued maximal velocity of signal propagation, denoted by c, and that the value of c is independent of the relative velocities between different inertial frames of reference.

In fact, the authors were describing in § 1 more than just the basic principle of relativity, that physical laws are the same in all inertial frames of reference, but also trying to give a justification for, without coming right out and saying it, all of the assumptions inherent in Einstein’s statement of Special Relativity.

This statement consists of three basic parts, the first part being that the principle of relativity holds. The second part of this statement is what L&L refer to as Einstein’s Principle of Relativity, that information cannot be propagated with infinite speed. Neither of these two parts requires particularly great insight to obtain them, as I tried to suggest these two statements are simply logical conclusions based on how we are to view, interpret, and describe our universe.

The same can be said concerning Newton’s Three Laws, that the first two laws really don’t say very much, they just give names for things we seem to observe around us. It has often been stated that “all of classical physics comes from Newton’s third law”, the one stating that the force exerted by one body on another is equal and opposite to the force exerted by the second body on the first.

Thus, we come to the third part of Einstein’s statement of Special Relativity, that not only is the maximal speed of information propagation the same in all frames of reference, but that this speed is actually the speed of light itself. That is, out of all the ways to explain how Newtonian mechanics and Maxwell’s equations can co-exist, making sense in a mutually consistent way, the simplest explanation is to assume the maximal speed of information propagation is actually the speed of light, a constant that also seems to magically appear in the midst of Maxwell’s equations when considering the units of electric and magnetic field intensities.

Einstein firmly believed that no physical theory could be presented as valid unless there existed experimental verification, or at least a way to prepare an experiment in order to test the theory. It would not be difficult to invent a theory in which signals could be propagated at a speed faster than that of light, and if any experimental observation could be explained by such a theory then so much the better. But no such experimental observation exists, in fact all observation indicates that indeed there isn’t anything that can move at a speed faster than that of light. If this makes the speed of light hold some kind of special meaning, then there must be some kind of experimental observation which justifies its special place in the universe. Lo and behold, between 1880 and 1900 there were many justifications of Maxwell’s equations, verifying that this constant speed of light does indeed seem to hold some special status in the universe. All Einstein was doing in finalizing his 1905 statement was to use Occam’s Razor, i.e. come up with the simplest explanation possible for everything that has been observed.

Sections 2-5 of L&L’s text are presented next.

An event is something that happens at a particular point in space at a particular time.

Another way to say this is that an event is a point in space-time.

Let space-time be described by a given coordina

1 1 1 1

te system .

Imagine event # 1 to be the propagation of a signal (perhaps a beam of light), originating

at some point , , in space at some time , and let event # 2 be the arrival of this

signal at

S

x y z t

2 2 2 2 1

2 1

some other point , , in space at some later time .

If denotes the signal velocity, then is the distance through which the

signal propagates in moving from one body to the other. B

x y z t t

c d c t t

2 2 222 1 2 1 2 1

ut also, the Pythagorean

Theorem gives the spatial distance between the two bodies according to the sum of

squares equation, . Setting these two expressions

for equal to each ot

d x x y y z z

d

2 2 2 22

her gives the fundamental space-time equation for signal

propagation, namely that 0.

Next, we imagine the observation of events # 1 and # 2 from the perspective of a different

space-tim

x y z c t

e coordinate system , which is either stationary with respect to or is moving

with respect to in a straight line with constant speed. If we measure distances and time

intervals in , obtaining v

S S

S

S

alues , , , and , we may find that none of these

quantities have the same numerical values as those measured in , but (assuming, and

again this is a big assumption, that the signal velocity

x y z t

S

c

2 2 2 22

is invariant under such a

coordinate transformation) we must obtain the same equation as before, i.e.

0.

Taking the limiting value of this relation to its differential form (i.e.

x y z c t

d

2 2 2 2 2 2

is the value assumed

in the limit as 0), we define the quantity , the space-time

arclength differential, and require this quantity to remain invariant whenever we change from

a

x

x ds c dt dx dy dz

given coordinate system (inertial reference frame) to any other equivalent coordinate

system (i.e. any other inertial reference frame).

Now, if we are in the system and we consider ourselves as at rS

22 2

est, i.e. not moving, so that

0, then the only part left to obtain the invariant arclength differential is

that given by . This means that, while we see sailing by even though

x y z

ds c dt S

observers in view themselves as not moving, the only way to keep the arclength

differential invariant is to assume (from our point of view) that observers in are

measuring time using a variable

S

S

t

1 which satisfies . This quantity , which

might be different from the variable used to measure time in the system , is called

the proper time for , but what it really represents is the v

cdt ds t

t S

S

ariable we need in in

order to explain how time is being measured in .

S

S

Now suppose we are in coordinate system with spatial coordinates , , and time

measured by the variable . Then, in this coordinate system, we can define velocity

components / , / ,x y z

S x y z

t

v dx dt v dy dt v dz

2 2 222

2 2 2 22

/ , and again use the Pythagorean Theorem

to give the equation . Multiplying this equation by

gives , which means the arclength differential

dt

dx dy dzv dt

dt dt dt

dx dy dz v dt

2 2 2 2 2 22 2 2

2 22

2 22 2 22 2

2 2 2

is given by

.

We now equate this with the proper time equation, , which gives

11 , or 1 .

This shows proper time (the tim

ds c dt dx dy dz c v dt

ds c dt

v vdt c v dt dt dt dt

c c c

e elapsing in the moving system as measured by the

fixed system ) is slower than the time as measured in . This pheomenon is called the

Lorentz contraction of time. Note that, in the limit as

S

S S

c

, we have , which

implies , and so absolute time (a universal clock) is the result of assuming the

speed of light is infinite (i.e. information transfer is instantaneous). Note also that, if

dt dt

t t

, then , which means in our everyday experience the Lorentz time dilation

effect is not noticable.

Here is another strange thing. If we are in the system , which we consider as stationary,

a

v c dt dt

S

nd we view moving past us, then we would go through exactly the same steps in

calculating the proper time for , which is really the variable we need to describe how

time is measured in relati

S

dt S

S

2

2

ve to . Since exactly the same calculations are used, we

obtain exactly the same answer, 1 . That is, observers in both systems see

time as moving slower in the other system. The reason

S

vdt dt

c

behind this seeming paradox is

again that these "proper" times don't really have physical meaning, they are just variables

used to measure time in systems other than our own. (????? L&L don't address this.)

Galilean Transformations.

Recalling the Galilean principle of relativity is that the speed of information propagation

is infinite, which implies time is measured the same in all coordinate systems. Thus, we

could define the transformation , given by , , , , as a

Galilean transformation. In this setting, is the relative (constant) velocity between

and , with our choice

S S x x vt y y z z t t

v

S S

of coordinate axes so that the motion is purely in the -direction.

(Note the sign of depends on which system we're in, so that .) What makes

this change of coordinates Galilean is the ass

x

v x x vt umption that . (Thus, .)t t x x vt

.

Note we can also assume the speed of light is finite, as long as we assume information

can be propagated at some speed faster than . Thus, the arclength differentia

Galilean Transformations continued

c

c

2

2 2 22 2 2 2 2 2 2

l can

still be calculated under a Galilean transformation. But is it invariant?

Taking differentials we have , , , , which gives

ds

dx dx vdt y dy dz dz dt dt

ds c dt dx dy dz c dt dx vdt dy d

2

2 2 2 22 2

2

2

unless 0.

Thus, the space-time arclength differential is not invariant under Galilean transformation,

and so this set of transformation equations becomes inv

z

c v dt dx vdx dt dy dz

ds v

alid under Einstein's principle of

special relativity. The reason why we want to use special relativity in choosing coordinate

transformations is because the correct choice will make not only Galileo's and Newton's

equations correct but also Maxwell's equations. Eventually we'll do the calculations which

show Maxwell's equations do not remain invariant under Galilean transformations, the main

reason why Einstein sought a different set of transformation equations.

To help determine what type of transformations we should be using, we can consider the

same thought experiment as in the preceding example, i.e. and moving relative to

each other with constant velocity (so that they are equivalent frames of reference), with

the spatial axes chosen so that the constant velocity vector is in the -di

S S

x

rection. This time,

however, we will assume , which makes the equation invalid.

Instead, we need to write the transformation of in terms of and (not ), and so we

need to determine

t t x x vt

x x t t

constants and such that . The reason why we can

assume and to be constants is because we are using inertial reference frames moving

with constant velocity in the absence of any extern

A B x Ax Bt

A B

al forces. Thus, anything involving

second derivatives must give zero, and so we assume linear relations between the space-time

variables.

Since we are assuming , we will also need constants determit t ning the transformation

equation for . However, to keep the units (distance) the same in both equations, we will

instead find constants and such that (again assuming ). We should

t

t

C D ct Cx Dct c c hen also go back and redefine and so that the spatial transformation equation becomes

We now need to determine what values of , , , will keep the space-time arclength

differential invari

A B

x Ax Bct

A B C D

ant.

2 2 2 2 2 2

2 22 2

2 2 2 22 2

2 2 2 22 2 2 2 2

.

,

2

2

2 .

Comparing

Galilean Transformations continued

dx Adx Bcdt cdt Cdx Dcdt

ds c dt dx dy dz

C dx CDdx cdt D cdt

A dx ABdx cdt B cdt dy dz

D B c dt A C dx dy dz

c CD AB dx dt

2 2 2 2 22

2

this with , i.e. that the form of the

space-time arclength differential is invariant, gives 0 (or / / ,

which is easiest satisfied by taking and . We also

ds c dt dx dy dz

ds CD AB C B A D

C B D A

2 2 2 2

2 2

2

have 1,

which with gives 1, and this is satisfied by defining cosh , sinh

for some (to be determined) parameter .

Thus, in our example, the most general way to keep invari

D B A C

C B A B A B

ds

ant is to assume the equations

cosh sinh and sinh cosh . Geometrically, represents

hyperbolic rotation in the -plane, but in a more general setting might mean something

a b

x x ct ct x ct

xt

it more abstract. However is to be interpreted, it's pretty clear that its value must be

independent of the origin of the coordinate system of . (Otherwise we just choose an

equivalent frame of re

S

ference placing the origin wherever we want.)

Therefore, without loss of generality, we can assume 0, and whatever value of we

obtain must be the value even without this assumption. Thus, we now a

x

1

ssume the equations

sinhsinh , cosh , which gives tanh , which means must

cosh

have the value tanh / . This can be simplified a bit more using the assumed

relative motion b

x ctx ct ct ct

ct ct

x ct

etween and , namely . (That is, this Galilean equation is

still valid, it just can't be used as one of the equations in the space-time coordinate

transformation.) Taking 0 gives

S S x x vt

x x vt

1

2 22

2 2 2

/ / /

tanh / . (Note that is a dimensionless quantity.)

1Using the hyperbolic trig identity 1 tanh sech , we have

cosh

1 1 /cosh , and sinh cosh tan

1 tanh 1 / 1

x ct vt ct v c

v c

v c

v c v

2 2

2

2 2

./

Thus, now that has been determined, we can generalize back to 0, and so our

general transformation equations cosh sinh , sinh cosh

/take the form ,

1 /

c

x

x x ct ct x ct

t v cx vtx t

v c

2 2, equations for a Lorentz transformation.

1 /

x

v c

2 2

2 2

The astute reader will recognize the quantity 1 / , which is typically denoted by ,

as the time-dilation scale factor obtained earlier, 1 / . When 0,

but more importantly when / is

v c

dt v c dt dt v

v c

2 2 small enough to consider / as being negligible,

then 1 and we obtain with , i.e. this case includes phenomena described

by Galilean mechanics. Note this also occurs when considering

v c

x x vt t t the limit .

Otherwise, is a quantity which is always smaller than 1, with 0 as .

This same scale factor appears in the denominators of both transformation equations

just derived (the Loren

c

v c

tz transformation), and this scale factor is referred to as the

Lorentz contraction. Thus, observers in view events in with not only time appearing

to have been slowed down, but also lengths have

S S

been contracted. Again, observers in

come to the same conclusion, so that both systems think the other system has been contracted

in both time and space. The only way to keep explaining why this se

S

emingly paradoxical

result is obtained is to just keep chanting, hey, everything is relative, man.

Another result of relativity is that velocity vectors in different coordinate systems are not

added by s

1 2 1 2

imply adding their components, as would be true in Galilean mechanics. Instead,

if and are velocities in two systems and , and we wish to find the resultant velocity

as measured from some

v v S S

v 1 2

1 21 2

1 22

1 2 0 1 2 02 20

given system , the fact that and are measured differently

means we do not get simply , but .1

2This also means when both and , and when .

1 /

S t t

v vv v v v

v vc

v c v c v c v v v v vv c

This relativistic velocity-addition equation negates an argument Isaac Asimov once used

to prove our universe must be finite. Since we observe the universe is expanding, and

since there is no absolute frame of reference, this means all objects are moving away

from each other at about the same rate, independent of where they are in the universe.

If we were to imagine an infinitely long line of stars, all initially separated by the same

distance, and then use this condition of uniform expansion independent of location, then

if we start with any star at random, the ones next to it move away with ve 0

0

locity , which

means the ones on either side of them move away with velocity 2 , and so on. Thus,

if there were infinitely many stars, eventually we would see stars moving away from us

with some velo

v

v

0city , which we have decided is not possible. Thus, Asmiov

concluded that the theory of relativity necessarily implies a bounded universe.

The fact that velocities are not added in this simplistic

Nv c

t way shows this argument to be

invalid, and there is nothing in relativity theory disallowing the existence of a universe

which is unbounded in both time and space.