chapter 4 lecture: lorentz...

Chapter 4

Lecture: Lorentz Covariance

To go beyond Newtonian gravitation we must consider,with Einstein, the intimate relationship between the cur-vature of space and the gravitational field.

• Mathematically, this extension is bound inextricablyto the geometry of spacetime, and in particular tothe aspect of geometry that permits quantitative mea-surement of distances.

• Let us first consider these ideas within the 4-dimensional spacetime termedMinkowski space.

As we shall see, requiring covariance withinMinkowski space will lead us to thespecialtheory of relativity.

67

68 CHAPTER 4. LECTURE: LORENTZ COVARIANCE

4.0.4 Minkowski Space

In a particular inertial frame, introduce unit vectorse0, e1, e2, ande3 that point along thet, x, y, andz axes. Any 4-vectorb may beexpressed in the form,

b = b0e0+b1e1+b2e2 +b3e3.

and the scalar product of 4-vectors is given by

a·b = b·a = (aµeµ)·(bνeν) = eµ ·eνaµbν .

Note that generally we shall use

• non-bold symbols to denote 4-vectors

• bold symbols for 3-vectors.

Where there is potential for confusion, we use a notationsuch asbµ to stand generically for all components of a4-vector.

Introducing the definition

ηµν ≡ eµ ·eν ,

the scalar product may be expressed as

a·b = ηµνaµbν .

69

and thus the line element becomes

ds2 = −c2dt2 +dx2+dy2+dz2 = ηµνdxµdxν ,

where themetric tensor of flat spacetimemay be expressed as

ηµν =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

= diag(−1,1,1,1).

That is, the line element corresponds to the matrix equation

ds2 = (cdt dx dy dz)

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

cdt

dx

dy

dz

,

whereds2 represents the spacetime interval betweenx andx+dxwith

x = (x0,x1,x2,x3) = (ct,x1,x2,x3).

The Minkowski metric is sometimes termed apseudo-euclidean metricto emphasize that it is euclidean-like ex-cept for the difference in sign between the time and spaceterms in the line element.


Example 4.1

Let us use the metric to determine the relationship between the timecoordinatet and the proper timeτ, with τ2 ≡−s2/c2. From

ds2 = −c2dt2 +dx2+dy2+dz2,

we may write

dτ2 =−ds2

c2 =1c2(c2dt2−dx2−dy2−dz2)

= dt2

1−1c2

[(dxdt

)2

+

(dydt

)2

+

(dzdt

)2]

︸︷︷︸

v2

=

(

1−v2

c2

)

dt2.

wherev is the magnitude of the velocity. Therefore, theproper timeτ that elapses betweencoordinate timest1 andt2 is

τ12 =∫ t2

t1

(

1−v2

c2

)1/2

dt.

The proper time intervalτ12 is shorter than the coordinate time inter-val t2− t1 because the square root is always less than one. This is thetime dilation effect of special relativity, stated in general form. If thevelocity is constant, this reduces to

∆τ =

(

1−v2

c2

)1/2

∆t,

which is the usual statement oftime dilation in special relativity.

4.1. TENSORS IN MINKOWSKI SPACE 71

Table 4.1: Rank 0, 1, and 2 tensor transformation laws

Tensor Transformation law

Scalar ϕ ′ = ϕ

Covariant vector A′µ =

∂xν

∂x′µAν

Contravariant vector A′µ =∂x′µ

∂xν Aν

Covariant rank-2 T ′µν =

∂xα

∂x′µ∂xβ

∂x′νTαβ

Contravariant rank-2 T ′µν =∂x′µ

∂xα∂x′ν

∂xβ Tαβ

Mixed rank-2 T ′νµ =

∂xα

∂x′µ∂x′ν

∂xβ Tβα

4.1 Tensors in Minkowski Space

In Minkowski space,transformations between coordinate systems areindependent of spacetime.Thus derivatives appearing in the generaldefinitions of Table 4.1 for tensors are constants and for flatspacetimewe have the simplified tensor transformation laws

ϕ ′ = ϕ Scalar

A′µ = ΛµνAν Contravariant vector

A′µ = Λ ν

µ Aν Covariant vector

T ′µν = ΛµγΛ ν

δ Tγδ Contravariant rank-2 tensor

T ′µν = Λ γ

µ ΛδνTγδ Covariant rank-2 tensor

T ′µν = Λµ

γΛδνTγ

δ Mixed rank-2 tensor

where the matrixΛµν does not depend on the spacetime coordinates.


In addition, for flat spacetime we may use a coordinatesystem for which the second term of

A′µ ,ν = Aα ,β

∂xβ

∂x′ν∂xα

∂x′µ︸︷︷︸

Tensor

+ Aα∂ 2xα

∂x′ν∂x′µ︸︷︷︸

Not a tensor

can be transformed away and in flat spacetimecovariantderivatives are equivalent to partial derivatives.

In the Minkowski transformation laws theΛµν are ele-

ments ofLorentz transformations,to which we now turnour attention.

4.2. LORENTZ TRANSFORMATIONS 73

4.2 Lorentz Transformations

In 3-dimensional euclidean space, rotations are a partic-ularly important class of transformations because theychange the direction for a 3-vector but preserve its length.

• We wish to generalize this idea to investigate abstractrotations in the 4-dimensional Minkowski space.

• Such rotations in Minkowski space are termedLorentz transformations.


e1

e2

φ

x

x1

x2

e1'

e2'

'x1x2'

φ

Consider a rotation of the coordinate system in euclidean space, asillustrated in the figure above.

• For the length of an arbitrary vectorxxx to be unchanged by thistransformation means thatxxx·xxx= xxx′·xxx′, which (sincexxx·xxx= gi j xix j)requires that the transformation matrixR implementing the rota-tion x′i = Ri

jxj act on the metric tensorgi j in the following way

Rgi j RT = gi j ,

whereRT denotes the transpose ofR.

• For euclidean space the metric tensor is just the unit matrix sothe above requirement reduces toRRT = 1, which is the conditionthatRbe anorthogonal matrix.

• Thus, we obtain by this somewhat pedantic route the well-knownresult thatrotations in euclidean space are implemented by or-thogonal matrices.

• But the requirementRgi j RT = gi j for rotations isvalid generally,not just for euclidean spaces. Therefore, let us use it as guidanceto constructinggeneralized rotations in Minkowski space.


• By analogy with the above discussion of rotations in euclideanspace, we seek a set of transformations that leave the lengthof a4-vector invariant in the Minkowski space.

• We write the coordinate transformation in matrix form,

dx′µ = Λµνdxν ,

where we expect the transformation matrixΛµν to satisfy the

analog ofRgi j RT = gi j for the Minkowski metricηµν ,

ΛηµνΛT = ηµν ,

Or explicitly in terms of components,Λ ρµ Λσ

νηρσ = ηµν .

• Let us now use this property to construct the elements of thetransformation matrixΛµ

ν These will include

– rotations about the spatial axes (corresponding to rotationswithin inertial systems) and

– transformations between inertial systems moving at differ-ent constant velocities that are termedLorentz boosts.

We consider first the simple case of rotations about thezaxis.


4.2.1 Rotations

For rotations about thez axis The transformation may we written inmatrix notation as

(

x′1

x′2

)

= R

(

x1

x2

)

=

(

a b

c d

)(

x1

x2

)

.

wherea, b, c, andd parameterize the transformation matrix.

• Rotations about a single axis correspond to a 2-dimensional prob-lem with euclidean metric, so the conditionRgi j RT = gi j is

(

a b

c d

)

︸︷︷︸

R

(

1 0

0 1

)

︸︷︷︸gi j

(

a c

b d

)

︸︷︷︸

RT

=

(

1 0

0 1

)

︸︷︷︸gi j

,

• Carrying out the matrix multiplications on the left side gives(

a2+b2 ac+bd

ac+bd c2+d2

)

=

(

1 0

0 1

)

,

and comparison of the two sides of the equation implies that

a2+b2 = 1 c2+d2 = 1 ac+bd= 0.

• These requirements are satisfied by the choices

a = cosϕ b = sinϕ c = −sinϕ d = cosϕ,

and we obtain the expected result for an ordinary rotation,(

x′1

x′2

)

= R

(

x1

x2

)

=

(

cosϕ sinϕ−sinϕ cosϕ

)(

x1

x2

)

.


v

x' x

Figure 4.1:A Lorentz boost along the positivex axis.

Now, let’s apply this same technique to determine the ele-ments of aLorentz boost transformation.

4.2.2 Lorentz Boosts

Consider a boost from one inertial system to a 2nd one moving in thepositive direction at uniform velocity along thex axis (Fig. 4.1).

• They andzcoordinates are unaffected by this boost, so this alsois effectively a 2-dimensional transformation ont andx,

(

cdt′

dx′

)

=

(

a b

c d

)(

cdt

dx

)

• We can write the conditionΛηµνΛT = ηµν out explicitly as(

a b

c d

)

︸︷︷︸

Λ

(

−1 0

0 1

)

︸︷︷︸ηµν

(

a c

b d

)

︸︷︷︸

ΛT

=

(

−1 0

0 1

)

︸︷︷︸ηµν

,

(identical to the rotation case, except for theindefinite metric).


• Multiplying the matrices on the left side and comparing with thematrix on the right side in

(

a b

c d

) (

−1 0

0 1

) (

a c

b d

)

=

(

−1 0

0 1

)

,

gives the conditions

a2−b2 = 1 −c2+d2 = 1 −ac+bd= 0,

• These are satisfied if we choose

a = coshξ b = sinhξ c = sinhξ d = coshξ ,

whereξ is a hyperbolic variable taking the values−∞ ≤ ξ ≤ ∞.

• Therefore, the boost transformation may be written as(

cdt′

dx′

)

=

(

coshξ sinhξsinhξ coshξ

)(

cdt

dx

)

.

Which may be compared with the rotational result

(

x′1

x′2

)

=

(

cosϕ sinϕ−sinϕ cosϕ

)(

x1

x2

)

.


Box 4.1 Minkowski Rotations

The respective derivations make clear that the appearance of hy-perbolic functions in the boosts, rather than trigonometric func-tions as in rotations, traces to the role of the indefinite metricgµν = diag(−1,1) in the boosts.

• The hyperbolic functions suggest that the boost transfor-mations are “rotations”in Minkowski space.

• But these rotations

– mix space and time, and

– will have unusual properties since they correspond torotations through imaginary angles (see Exercise 4.6).

• These unusual properties follow from the metric:

– The invariant interval that is being conserved is notthe length of vectors in space or the length of timeintervals separately.

– Rather it is the specificmixture of time and space in-tervals implied by the Minkowski line element withindefinite metric:

ds2 = (cdt dx dy dz)

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

cdt

dx

dy

dz

,


v/c

+1-1

= - tanh−1 v

c

ξ

Figure 4.2:Dependence of the Lorentz parameterξ on β = v/c.

We can put the Lorentz boost transformation into a more familiarform by relating the boost parameterξ to the boost velocity.

• The velocity of the boosted system isv = x/t. From(

cdt′

dx′

)

=

(


)(

cdt

dx

)

,

the origin (x′ = 0) of the boosted system is

x′ = ctsinhξ +xcoshξ = 0.

• Therefore,x/t = −csinhξ/coshξ , from which we conclude that

β ≡vc

=xct

= −sinhξcoshξ

= − tanhξ .

• This relationship betweenξ andβ is plotted inFig. 4.2.


• Utilizing

β ≡vc

=xct

= −sinhξcoshξ

= − tanhξ .

the identitycosh2ξ −sinh2ξ = 1, and the definition

γ ≡

(

1−v2

c2

)−1/2

of the Lorentzγ factor, we may write

coshξ =1

√

1−sinh2ξ/cosh2ξ=

1√

1−v2/c2= γ,

• From this result and

β = −sinhξcoshξ

we obtainsinhξ = −β coshξ = −β γ.

• Thus, insertingcoshξ = γ and sinhξ = −β γ in the Lorentztransformation gives

(

cdt′

dx′

)

=

(


)(

cdt

dx

)

=

(

γ −γβ−γβ γ

)(

cdt

dx

)

.

= γ

(

1 −β−β 1

)(

cdt

dx

)

.


• Writing the matrix expression(

cdt′

dx′

)

= γ

(

1 −β−β 1

)(

cdt

dx

)

.

out explicitly for finite intervals gives the Lorentz boost equa-tions (for the specific case of a positive boost along thex axis) instandard textbook form,

t ′ = γ(

t −vxc2

)

x′ = γ(x−vt)

y′ = y z′ = z,

• The inverse transformation corresponds to the replacement v→−v.

• Clearly these reduce to the Galilean boost equations

xxx′ = xxx′(xxx,t) = xxx−vvvt t′ = t ′(xxx,t) = t.

in the limit thatv/c vanishes, as we would expect.

• It is easily verified (Exercise 4.1) that the Lorentz transforma-tions leave invariant the spacetime intervalds2.


Figure 4.3:The light cone diagram for two space and one time dimensions.

4.2.3 The Light Cone Structure of Minkowski Spacetime

By virtue of the line element (which defines a cone)

ds2 = −c2dt2 +dx2+dy2+dz2,

the Minkowski spacetime may be classified according to the lightcone diagram exhibited in Fig. 4.3.

The light cone is a 3-dimensional surface in the 4-dimensional spacetime and events in spacetime may becharacterized according to whether they are inside of, out-side of, or on the light cone.


The standard terminology (assuming our(−1,1,1,1) metric signa-ture):

• If ds2 < 0 the interval is termedtimelike. In that caseds/c is thetime measured by a clock moving freely fromx to x+ dx (theproper time;see below).

• If ds2 > 0 the interval is termedspacelike. Then|ds2|1/2 may beinterpreted as the length of a ruler with ends atx andx+dx, asmeasured by an observer at rest with respect to the ruler.

• If ds2 = 0 the interval is calledlightlike (or sometimesnull).Then the pointsx andx+dxare connected by signals moving atlight speed.


The light cone classification clarifies the distinction be-tween Minkowski spacetime and a 4D euclidean space inthat two points in the Minkowski spacetime may be sepa-rated by a distance whose square could be

• positive,

• negative, or

• zero

which embodies impossibilities for a euclidean space.

In particular, lightlike particles have world-lines confined to the light cone and the squareof the separation of any two points on a light-like worldline iszero.


Example 4.2

The Minkowski line element in one space and one time dimension[often termed(1+ 1) dimensions] isds2 = −c2dt2 + dx2. Thus, ifds2 = 0

−c2dt2+dx2 = 0 −→

(dxdt

)2

= c2 −→ v = ±c.

We can generalize this result easily to the full space and we concludethat

• Events in Minkowski space separated by a null interval (ds2 = 0)are connected by signals moving at light velocity,v = c.

• If the time (ct) and space axes have the same scales, this meansthat the worldline of a freely propagating photon (or any mass-less particle moving at light velocity) always make±45◦ anglesin the lightcone diagram.

• By similar arguments, events at timelike separations (inside thelightcone) are connected by signals withv < c, and

• Those with spacelike separations (outside the lightcone)couldbe connected only by signals withv > c (which would violatecausality).


ct

y

x

Figure 4.4:We may imagine a lightcone attached to every point of spacetime.

We have placed the lightcone in the earlier illustration atthe origin of our coordinate system, but in general we mayimagine a lightcone attached to every point in the space-time, as illustrated in Fig. 4.4.


World line of

massive

particleWorld line

of photon

ct

y

x

ct

y

x

Figure 4.5:Worldlines for massive particles and massless particles like photons.

A tangent to the worldline of any particle defines the localvelocity of the particle at that point and constant veloc-ity implies straight worldlines. Therefore, as illustrated inFig. 4.5,

• Light must always travel in straight lines (inMinkowski space;not in curved space), and alwayson the lightcone, sincev = c = constant. Thus pho-tons have constant local velocities.

• Worldlines for any massive particle lie inside the lo-cal lightcone sincev ≤ c (timelike trajectory, sincealways within the lightcone).

• Worldline for the massive particle in this example iscurved(acceleration).

• For non-accelerated massive particles the worldlinewould be straight, but alwayswithin the lightcone.


Figure 4.6:The light cone diagram for two space and one time dimensions.

4.2.4 Causality and Spacetime

The causal properties of Minkowski spacetime are encoded inits lightcone structure, which requires thatv≤ c for all signals.

• Each point in spacetime may be viewed as lying at the apex of alight cone (“Now” ).

• An event at the origin of a light cone may influence any event inits forward light cone (the “Future”).

• The event at the origin of the light cone may be influenced byevents in its backward light cone (the “Past”).

• Events at spacelike separations are causally disconnected fromthe event at the origin.

• Events on the light cone are connected by signals that travel ex-actly atc.


The light cone is a surface separating the knowable fromthe unknowable for an observer at the apex of the lightcone.

This light cone structure of spacetime ensures that all ve-locities obey locally the constraintv≤ c. Since velocitiesare defined and measured locally, covariant field theoriesin either flat or curved space are guaranteed to respect thespeed limitv≤ c, irrespective of whether globally veloci-ties appear to exceedc.

EXAMPLE: In the Hubble expansion of theUniverse, galaxies beyond a certain distance(the horizon) would recede from us at veloc-ities in excess ofc. However, all local mea-surements in that expanding, possibly curved,space would determine the velocity of light tobec.

4.3. LORENTZ TRANSFORMATIONS IN SPACETIME DIAGRAMS 91

x

ct

x'

ct'β =

1

ct = x βct =

xβ

-1

φ = tan-1(v/c)

−φ

Figure 4.7:Lorentz boost transformation in a spacetime diagram.

4.3 Lorentz Transformations in Spacetime Diagrams

It is instructive to look at the action of Lorentz transfor-mations in the spacetime (lightcone) diagram. If we con-sider boosts only in thex direction, the relevant part of thespacetime diagram in some inertial frame corresponds toa plot with axesct andx, as illustrated in the figure above.


x

ct

x'

ct'β =

1

ct = x β

ct =

xβ

-1

φ = tan-1(v/c)

−φ

Let us now ask what happens to these axes under the Lorentz boost

ct′ = cγ(

t −vxc2

)

x′ = γ(x−vt).

• The t ′ axis corresponds tox′ = 0. From the 2nd equationx =

vt → x/c = (v/c)t = β t so thatct = xβ−1, whereβ = v/c.

• Likewise, thex′ axis corresponds tot ′ = 0, which implies fromthe 1st equation thatct = (v/c)x = xβ .

• Thus, the equations of thex′ andt ′ axes (in the(x,ct) coordinatesystem) arect = xβ andct = xβ−1, respectively.

• Thex′ = 0 andt ′ axes for the boosted system are also shown inthe figure for a boost corresponding to a positive value ofβ .

• Time and space axes are rotated by same angle, but inoppositedirectionsby the boost (due to theindefinite Minkowski metric).

• Rotation angle related to boost velocity throughtanϕ = v/c.


Ordinary rotations (the two axes rotate by the same angle in thesamedirection):

e1

e2

φ

x

x1

x2

e1'

e2'

'x1x2'

φ

Lorentz boost “rotations” (the two axes rotate by the same angle butin opposite directions):

x

ct

x'

ct'β =

1

ct = x β

ct =

xβ

-1

φ = tan-1(v/c)

−φ


x

ct

x'

ct'

tB

tA

Constant t '

A

BConstant t

Consta

nt x'

Co

nsta

nt x

C

D

xC xD

Figure 4.8:Comparison of events in boosted and unboosted reference frames.

Most of special relativity follows directly from this figure.

For example, relativity of simultaneity follows directly,as illustratedin Fig. 4.8.

• Points A and B lie on the samet ′ line, so they aresimultaneousin the boosted frame.

• But from the dashed projections on thect axis,event A occursbefore event B in the unboosted frame.

• Likewise, points C and D lie at the same value ofx′ in theboosted frame and so arespatially congruent, but in the un-boosted framexC 6= xD.

Relativistic time dilation and space contrac-tion effects follow rather directly from theseobservations.


Example 4.3

The time registered by a clock moving between two points in space-time depends on the path followed, as suggested by

dτ2 =

(

1−v2

c2

)

dt2.

The proper timeτ is the time registered by a clock carriedby an observer on a spacetime path.

That this is true even if the path returns to the initial spatial positionis the source of thetwin paradoxof special relativity.

• Twins are initially at rest in the same inertial frame. Twin2travels atv ∼ c to a distant star and then returns at the samespeed to the starting point; twin 1 remains at the starting point.

• The corresponding spacetime paths are:

x

t

Twin 2

Twin 1Distant

star

x0

t2

t2

• The elapsed time on the clock carried by Twin 2 is always smallerbecause of the square root factor in the above equation.


x

t

Twin 2

Twin 1Distant

star

x0

t2

t2

• The (seeming) paradox arises if one describes things from thepoint of view of Twin 2, who sees Twin 1 move away and thenback. This seems to be symmetric with the case of Twin 1 watch-ing Twin 2 move away and then back.

• But it isn’t: the twins travel different worldlines, and differentdistances along these worldlines.

• Their clocks record the proper time on their respective world-lines and thus differ when they are rejoined, indicating unam-biguously that Twin 2 is younger at the end of the journey.


Space Contraction

Consider the following schematic spacetime diagram, wherea rod ofproper lengthL0, as measured in its own rest frame(t,x), is orientedalong thex axis.

x

ct

x'

ct'

c∆tConstant t

L0

L'

The adjective “proper” in relativity generally denotesaquantity measured in the rest frame of the object.

Fundamental measurement issues:

• Distances must be measured betweenspacetime pointsat the same time.

• Elapsed times must be measured atspacetime pointsat the same place.

Example:For an arrow in flight its length is not generallygiven by the difference between the location of its tip atone time and its tail at a different time.


x

ct

x'

ct'

c∆tConstant t

L0

L'

x

ct

x'

ct'

Constant t '

Constant t '

Consta

nt x'

Consta

nt x'

• The frame(t ′,x′) is boosted by a velocityv along the positivexaxis relative to the(t,x) frame. Therefore, in the primed framethe rod will have a velocityv in the negativex′ direction.

• Determining the lengthL observed in the primed frame requiresthat the positions of the ends of the rod be measuredsimultane-ously in that frame.The axis labeledx′ corresponds to constantt ′ (see bottom figure above), so the distance marked asL is thelength in the primed frame.


x

ct

x'

ct'

c∆tConstant t

L0

L'

Mercator

(preserves angles,

distorts sizes)

Map Projections

Source: http://www.culturaldetective.com/worldmaps.html

• This distanceL seemslonger thanL0, but this is deceiving be-causewe are looking at a slice of Minkowski spacetime repre-sented on a piece of euclidean paper(the printer was fresh outof Minkowski-space paper :).


x

ct

x'

ct'

c∆tConstant t

L0

L'

• Much as for a Mercator projection of the globe onto a euclideansheet of paper (which gives misleading distance information—Greenland isn’t really larger than Brazil, and Africa has 14timesthe area of Greenland),we must trust the metric to determine thecorrect distance in a space.

• From the Minkowski indefinite-metric line element

ds2 = −c2dt2+dx2.

and the triangle in the figure above(Pythagorean theorem inMinkowski space),

L2 = L20− (c∆t)2.

But the equation for thex′ axis givesc∆t = (v/c)L0, from which

L = (L20− (c∆t)2)1/2 =

(

L20−(v

cL0

)2)1/2

= L0(1−v2/c2)1/2,

which is thelength-contraction formula of special relativity: Lis shorterthan the proper lengthL0, even though it appears to belonger in the figure.


Mercator

(preserves angles,

distorts sizes)

Peters

(preserves sizes,

distorts angles)

Population

(preserves populations,

no distance information)

Map Projections

Source: http://www.culturaldetective.com/worldmaps.html

Figure 4.9:Map projections.


Invariance and Simultaneity

• In Galilean relativity, an event picks out a hyperplaneof simultaneity in the spacetime diagram consistingof all events occurring at the same time as the event.

• All observers agree on what constitutes this set of si-multaneous events becauseGalilean relativity of si-multaneity is independent of the observer.

• In Einstein’s relativity, simultaneity depends on theobserver and hyperplanes of constant coordinatetime have no invariant meaning.

• However, all observers agree on the position inspacetime of the lightcones associated with events,becausethe speed of light is invariant for all ob-servers.

The local lightcones define an invariantspacetime structure that may be used to clas-sify events.


(a) (b)

A

B

CD

Spacelike

Tim

elik

e

Ligh

tlike

A

C

ctct'

x

x'

φ1

Const

ant x'

φ1v =

c

(c)

A

ct ct'

x

x'

φ2

Constant t'

v =

c

Bφ2

Figure 4.10:(a) Timelike, lightlike (null), and spacelike separations. (b) Lorentztransformation that brings the timelike separated points Aand C of (a) into spatialcongruence (they lie along a line of constantx′ in the primed system). (c) Lorentztransformation that brings the spacelike separated pointsA and B of (a) into coin-cidence in time (they lie along a line of constantt ′ in the primed system.

• The spacetime separation between any two events (spacetimeinterval) may be classified in a relativistically invariantway as

1. timelike,

2. lightlike,

3. spacelike

by constructing the lightcone at one of the points, as illustratedin Fig. 4.10(a).


(a) (b)

A

B

CD

Spacelike

Tim

elik

e

Ligh

tlike

A

C

ctct'

x

x'

φ1

Const

ant x'

φ1v =

c

(c)

A

ct ct'

x

x'

φ2

Constant t'

v =

c

Bφ2

x

ct

x'

ct'

Constant t '

Constant t '

Consta

nt x'

Consta

nt x'

The geometry of the above two figures suggests another importantdistinction between points at spacelike separations [the line AB inFig. (a)] and timelike separations [the line AC in Fig. (a)]:

• If two events have atimelike separation, a Lorentz transforma-tion exists that can bring them into spatial congruence.Figure(b) illustrates geometrically a coordinate system(ct′,x′), relatedto the original system by anx-axis Lorentz boost ofv/c= tanϕ1,in which A and C have the same coordinatex′.

• If two events have aspacelike separation, a Lorentz transfor-mation exists that can synchronize the two points. Figure (c)illustrates anx-axis Lorentz boost byv/c= tanϕ2 to a system inwhich A and B have the same timet ′.


(a) (b)

A

B

CD

Spacelike

Tim

elik

e

Ligh

tlike

A

C

ctct'

x

x'

φ1

Const

ant x'

φ1v =

c

(c)

A

ct ct'

x

x'

φ2

Constant t'

v =

c

Bφ2

• Notice that the maximum values ofϕ1 andϕ2 are limited by thev = c line.

• Thus, the Lorentz transformation to bring point A into spatialcongruence with point C exists only if point C lies to the leftofthe v = c line and thus is separated by a timelike interval frompoint A.

• Likewise, the Lorentz transformation to synchronize point Awith point B exists only if B lies to the right of thev = c line,meaning that it is separated by a spacelike interval from A.

chapter 4 lecture: lorentz...

Documents