Lecture notes on special theory of relativity on an undergraduate level, Lecture notes of Electrodynamics

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Lecture Notes

Relativity - The Special Theory

S. Murugesh

Last update: September 14, 2010

Contents

Chapter 1

The Issue

In this chapter we review the relevant aspects of classical mechanics, and the issues that led to the Special Theory of Relativity.

1.1 Galelian transformations and the Galeian group

Consider two observers in two reference frames, S and S′, designated by coordinates (x, y, z) and time t, and (x′, y′, z′) and time t′, respectively. Let the two reference frames be in uniform relative motion with respect to each other with constant velocity V = {vx, vy, vz }. Also, let the two reference frames differ in their orientation (which remains a constant), and differ in their origin in space and time. Then the two space and time coordinates are in general related by    

x′ y′ z′ t′

vx R 3 × 3 vy vz 0 0 0 1

x y z t

x 0 y 0 z 0 t 0

.^ (1.1)

Here, R 3 × 3 is a constant 3-parameter rotation (orthogonal 3×3) matrix representing the orientation of the S′^ frame with respect to S frame, and (x 0 , y 0 , z 0 ) and t 0 are the difference in their spacial and temporal origins. In all, there are 10 parameters that relate the two space and time coordinates (3 for constant spacial rotations in R 3 × 3 , 3 boost parameters -(vx, vy, vz ), 3 spacial displacements -(x 0 , y 0 , z 0 ), and one temporal displacement-t 0 ). Eq. (1.1) constitutes the most general Galelian transformation connecting two coordinate sys- tems. Any non-accelerating reference frame is defined as an inertial reference frame. Any reference frame related to any other inertial reference frame by a Galelian transformation will also be inertial. Indeed, the significance of such transformations arises from the fact - Forces (or accelerations) seen in one inertial reference frame is the same in all inertial reference frames. More strictly, the form of the equations of mechanics are invariant with respect to such transformations. I.e.,

mi¨ri = mi¨r′ i. (1.2)

Further, mi¨ri = Fi([r], t) (1.3)

will go over to mi¨r′ i = Fi([r(r′)], t(t′)) = F′ i([r′], t′). (1.4) Some facts about the transformation in Eq. (1.1) can be immediately seen: i) Identity transfor- mation is just a particular case of Eq. (1.1). ii) Inverse of Eq. (1.1) relating S′^ to S is also of the same form. iii) If we consider two such relations relating S to S′^ and S′^ to S′′, respectively, say

   

x′′ y′′ z′′ t′′

v′ x R′ 3 × 3 v y′ v z′ 0 0 0 1

x′ y′ z′ t′

x′ 0 y 0 ′ z 0 ′ t′ 0

.^ (1.5)

It can be worked out that the relation between S and S′′^ is again of the same form. I.e.,

   

x′′ y′′ z′′ t′′

v x′′ R′′ 3 × 3 v y′′ v z′′ 0 0 0 1

x y z t

x′′ 0 y 0 ′′ z 0 ′′ t′′ 0

 ,^ (1.6)

for suitable constant R′′ 3 × 3 , (v′′ x, v y′′ , v′′ z ), (x′′ 0 , y′′ 0 , z 0 ′′ ) and t′′ 0. Thus, if one considers the set of all possible transformations of the type Eq. (1.1), it forms a group under composition - the 10- parameter Galelian group. Moral: Equations of mechanics preserve their form under Galelian transformations. Or simply, mechanics pocesses Galelian group symmetry, or is Galelian invariant. Practically this means that we can write and follow the same equations, using one’s own lab coordinates, irrespective of our reference frame.

1.2 Maxwell’s equations do not agree

The results of the previous section constitute what is referred to as Galelian/Newtonian/Classical relativity, and consequently a relativity principle - a belief that laws of physics must be same in all inertial reference frames, wherein the reference frames are related by transformations of the type Eq. (1.1). However, Maxwell’s equations of electrodynamics do not show this invariance. Under the trans- formation x′^ = x − v 0 t, y′^ = y, z′^ = z andes t′^ = t, we have

∂ ∂x

∂x′^

∂y

∂y′^

∂z

∂z′^

∂t

= −v 0

∂x′^

∂t′^

3. Maxwell’s equations are fine and a relativity principle exists. But, the expressions as in Eq. (1.1) are wrong.

The third possibility was the one investigated by Einstein, and the one that resolved all issues.

Chapter 2

The Special Theory

We shall deliberately ignore the chronological development of the theory 1 , and start directly from the answer.

2.1 The Postulates

The resolution came in the form of two (just two!!) postulates.

PI. The principle of relativity: Laws of physics must be the same in all inertial reference frames.

Though this assertion may sound nothing new, it has to be appreciated that, first of all, this is a postulate. Besides, the change is in its privilege, now as an apriori assertion. The second postulate brings in some fundamental changes in our notion of space and time. While the following sections in this chapter are devoted to a more detailed discussion on these aspects, we shall briefly define the bare minimum first, just enough material to state the postulate. The special theory forces us to look upon space and time not independently, but as a space-time continuum. Just as we speak of a point in 3-space given by 3-coordinates, we have Events(noun) designated by 4 coordinates - 3 spacial and 1 temporal. Thus we have a 4-dimensional space-time, and every space-time point is defined as an ’Event’. For a start it may be convenent to think of these Events as usual events(verb). Consider two Events in space-time, say (ti; xi, yi, zi), i = 1, 2 (say two firecrackers bursting in the sky at two different points at different times). Contrary to our usual notion that time intervals ∆t = (t 2 − t 1 ), and lengths ∆l^2 = (x 2 − x 1 )^2 + (y 2 − y 1 )^2 + (z 2 − z 1 )^2 , are independently invariant in any reference frame, we have

PII. The space time interval between two Events, defined as

∆s^2 = c^2 ∆t^2 − ∆l^2 , (2.1) (^1) Just as we do not care anymore about the chronological development of Newton’s laws, which after all were not written overnight.

x x

y

x

ct

x

y ct

x

θ

v

θ c

tan = v_

a b

c d

Figure 2.1: Two simple situations and their world line view: a) an object at rest, and c) a object in uniform motion along x direction in space, and their respective world lines c) and d). World lines of objects in uniform motion are straight lines. y and z directions are not shown in b) and d).

Let S′^ be another reference frame moving with respect to S with some velocity such that the same two events in S′^ are (ct′ 1 ; x′ 1 ) and (ct′ 2 ; x′ 2 ), and the velocity of the object v′. From PII we have

∆s^2 = (c^2 − v^2 )(t 2 − t 1 )^2 = (c^2 − v′^2 )(t′ 2 − t′ 1 )^2 = ∆s′ 2

. (2.4)

When velocities v 1 and v 2 are small compared to c, Eq. (2.4) reduces to

(t 2 − t 1 )^2 = (t′ 2 − t′ 1 )^2 , (2.5)

or simply, one gets back Galelian relativity. Indeed, practically relativistic effects are significant only at velocities of the order 0. 5 c and more.

2.3.2 Objects with velocity c are special

Suppose the object in the previous section is moving at speed v = c in S. Then, evidently

∆s^2 = 0. (2.6)

ct

x

ct

x

a

b

Figure 2.2: a) The world line of a accelerating object, moving in the x direction. The curvature of the world line is a measure of its acceleration. The other two spacial directions are suppressed for simplicity. b) The world line of a planet orbiting in the x − y plane. The world line for the planet is a helix (i.e, if the orbit is circular).

Consequently, ∆s′^2 = 0 and v′^ = c. I.e., if an object is moving at speed c in some reference then it moves with the same speed c in all reference frames. Objects traveling with speed c are treated by space-time with a little added privilege. Hence, the velocity of light in vacuum, being c, will be the same as seen from any reference frame, just as several experiments had suggested^2. Indeed, from Eq. (2.4), it follows that an object moving with a speed less than (greater than) c in one reference will be seen to be moving with a speed less than (greater than) c in any reference frame. The invariance of the space-time intervals allows us to classify two events into three types:

2.3.3 Time like, space like and light like intervals: Causality

Two events are said to be time like, space like or light like separated if their space-time intervals are such that ∆s^2 > 0, ∆s^2 < 0 or ∆s^2 = 0 respectively. The primary significance being that they maintain this classification in any reference frame. We wish to see how the spacetime separation between events look like in different reference frames. Of two events A and B, let A coincide with the origin in two reference frames S and S′^ (i.e, space-time origin of the two reference frames coincide, and also coincide with the event A). Given B = (ct; x, y, z) in frame S, we wish to see where B = (ct′, x′, y′, z′) can possibly be in S′. Since the

(^2) ’The speed of light in vacuum is same in all reference frame’ is in fact the version of PII originally given by Einstein, and is equivalent to the version we have used. Einstein’s version is deliberately avoided for the following reasons: a) the postulate PII as we have stated is functionally more useful, b) it shows the precise difference between Galelian and Special relativity and c) on a naive reading Einstein’s version is likely to give an impression that we are talking about some special property of light. Whereas, it should be realized that what we have encountered is a property of space-time. After all, it is not just light, but any object moving with speed c that has the same speed in all reference frames.

past

light cone

time like

space like

light like

x y

future

ct

Figure 2.3: The light cone attached to an event in space time (origin). The events in the future fall inside the cone in the +ve t direction. The events outside the cone are space like seperated from the origin and are not accessible. If a future event falls on the paraboloid inside the cone, then in any other reference frame (with the same origin) the event will be some other point in the same paraboloid. The actual structure is 4 dimensional. z direction is suppressed for simplicity.

= c^2 (t′ 2 − t′ 1 )^2 − v^2 (t′ 2 − t′ 1 )^2. (2.11)

Since the two intervals are same by PII, it follows that

(t 2 − t 1 ) =

v^2 c^2

) · (t′ 2 − t′ 1 ) (2.12)

or (t′ 2 − t′ 1 ). = γ(t 2 − t 1 ), (2.13)

where γ = 1/

(1 − v^2 /c^2 ). So, the times shown by a moving clock will be different from the one it shows when it is at rest. It can be seen that 1 ≤ γ ≤ ∞. (t′ 2 − t′ 1 ) is the time interval between two ’ticks’ in a frame in which the clock is moving. From Eq. (2.13) it follows that (t′ 2 − t′ 1 ) ≥ (t 2 − t 1 ). Moving clocks run slower by a factor γ.

x y

ct

A

B

B’

C

C’

x

x

x

x

Figure 2.4: Event B might transform to B′^ in some other reference frame S′. However, the time ordering between events A and B is preserved (A′^ = A). This is true for any two events that are time like separated. But the same is not true for events C and A, which are space like separated. C transforms to C′^ in S′. Since the time ordering is reversed, A precedes C in S, but is reversed in S′.

Definition: The Proper time between two events is defined as the time interval between the events in a frame in which the two events happen at the same place. The time shown by a clock in a reference frame in which the clock is at rest is the Proper time shown by the clock. Evidently, proper time is not defined between two events that are space like separated.

2.5 Length contraction: Proper length

Having seen that time interval measurements in two reference frames are different, it is natural to expect the same about length measurements too. The definition of proper length goes along the same line as that of proper time. Definition: The Proper length of an object is its length measured in a frame in which the

Chapter 3

Four Vectors and the ∗ product

As we saw in the last chapter, special relativity identifies space and time not as independent entities but as two facets of a space-time continuum. Consequently, a point in space time is identified 4 coordinates, one time and three space. Thus the position vector of an object in space time is given by^1 r = (ct; x, y, z) = (ct, r). (3.1)

3.1 The four-velocity vector

Every observer makes measurements in her own lab frame in her own clock - her own proper time. Evidently, thats one time on which everybody will agree. It seems natural therefore to define the four velocity of an object as the proper time derivative of its position 4-vector. However, between two observers in relative motion we know they do not measure the same time interval between two given events. Indeed, the proper time interval ∆τ is related to the time interval observed in any other frame ∆t by

∆τ =

∆t γ

The time derivatives in the two reference frames are related by

d dτ

= γ

d dt

The four velocity of a moving object is defined thus:

v =

dr dτ

= γ

dr dt

= γ(c; v). (3.4)

Let’s look at each of the quantities in Eq. (3.4) carefully. The situation is the following: In a given frame (say, the rest frame) an object, whose position vector is r, is observed to be moving with velocity v. τ is the time as measured in the object’s own reference frame - its proper time. The four

(^1) The semicolon separating time and space coordinates need not be taken seriously. Henceforth 3-vectors will be

denoted by bold faces, and 4-vectors by underlined bold faces.

velocity of the object in the object’s own reference frame will be (and so will be the four velocity of anyone or anything in its own reference frame)

v = (c; 0), (3.5)

just as its four position will be r = (cτ ; 0). (3.6)

Notice that the four velocities in Eq. (3.4) and Eq. (3.5) are one and the same, but just observed from two different reference frames.

3.2 The ∗ product

Analogous to the · product of three-vectors, we define the ∗ product of two four-vectors A = (at; ax, ay, az ) and B = (bt; bx, by, bz ) as

A ∗ B = atbt − axbx − ayby − az bz. (3.7)

With this definition, we notice v ∗ v = c^2. (3.8) A four vector connects two space time points. Consequently, PII can be rewritten as

• PII:The ∗ product of any two four-vectors defined in space time is an invariant in any reference frame.

How did this happen? Notice that the four-position in equations (3.1) and (3.6) are just our dr′ and dr from the previous chapter, if the respective origins were to be the first event, and assuming the two origins to coincide. Then, dr′^ ∗ dr′^ = dr ∗ dr, by PII. We just divide by dτ 2 on either side to get Eq. (3.8). The invariance implies that the ∗ product of any two four velocities must be c^2.

3.3 The four-accelaration

The four-accelaration is defined as

a =

dv dτ

= γ

d dt

(γc; γ~v). (3.9)

Differentiating Eq. (3.8), we get v ∗ a = 0. (3.10)

Thus, a and v are always orthogonal (in the sense of the ∗ product).

The velocity as seen from the ground frame is given by

dx dt

dx/dτ dt/dτ

γ

dx dτ

From the expression for v we find γ = cosh(aτ /c) and dx/dτ = c sinh(aτ /c). Thus the velocity as seen by the ground observer is c tanh(aτ /c). So, inspite of continued uniform acceleration, the object never reaches c in finite time. It is important to realize here that when we say uniform acceleration, we are referring to the acceleration as seen by the accelerating object itself, in its own reference frame.

Chapter 4

Velocity addition and Lorentz

transformations

We have noted already that in going from one reference frame to another the simple velocity addition rule does not hold. In this chapter we shall find the correct relativistic expression for adding velocities, the relations connecting space-time coordinates in two reference frames in relative motion - the Lorentz transformation relations.

4.1 Velocity addition

Let two observers 1 and 2 in relative motion observe an object 3 in motion. Say 1 is the ground reference frame, 2 moving with respect to 1 along their common x direction with velocity v 21 , and let 3 also move in the x direction with speed v 32 with respect to 2. Evidently the velocity of 1 with respect to 2 is v 12 = −v 21. The task is to find v 31 , the velocity of 3 with respect to 1. Let us write the corresponding four velocities:

v 21 = γ 21 (c; v 21 , 0 , 0), (4.1)

v 32 = γ 32 (c; v 32 , 0 , 0), (4.2) v 12 = γ 12 (c; v 12 , 0 , 0) = γ 21 (c; −v 21 , 0 , 0) (4.3) v 31 = γ 31 (c; v 31 , 0 , 0) =? (4.4)

where, as usual, γij = 1/

1 − v ij^2 /c^2. To determine v 31 we shall use the fact that the ∗ product of two four-vectors is invariant in any reference frame, and that the four-velocity of any object in its own reference frame is (c; 0), since in its own reference frame it is at rest. Thus, the speed of 1 in its own reference frame is zero, and consequently the four-vector v 11 = (c; 0). (4.5)

From the invariance of the ∗ product it follows that

v 32 ∗ v 12 = v 31 ∗ v 11. (4.6)