241SPECIALRELATIVITYANDFLATSPACETIMEin three dimensions do we have an interesting map from two dual vectors to a third dualvector. If you wanted to you could define a map from n -1 one-forms to a single one-form,but I'm not sure it would be of any use.Electrodynamics provides an especially compelling example of the use of differentialforms. From the definition of the exterior derivative, it is clear that equation (1.78) canbe concisely expressed as closure of the two-form Fuw:dF=0.(1.91)Does this mean that F is also exact? Yes; as we've noted, Minkowski space is topologicallytrivial, so all closed forms are exact. There must therefore be a one-form Aμ such thatF=dA.(1.92)This one-form is the familiarvector potential of electromagnetism,with the 0 componentgiven by the scalar potential, Ao = @. If one starts from the view that the A, is thefundamental field of electromagnetism, then (1.91) follows as an identity (as opposed to adynamical law, an equation of motion).Gauge invariance is expressed by the observationthat the theory is invariant under A → A + d for some scalar (zero-form) , and this isalso immediate from the relation (1.92). The other one of Maxwell's equations, (1.77), canbe expressed as an equation between three-forms:(1.93)d(*F) = 4元(*J) ,where the current one-form J is just the current four-vector with index lowered.Filling inthe details is left for you to do.As an intriguing aside, Hodge duality is the basis for one of the hottest topics in theoreticalphysics today. It's hard not to notice that the equations (1.91) and (1.93) look very similar.Indeed, if we set J= O, the equations are invariant under the“duality transformations"F→*F,(1.94)*F-→-F.We therefore say that the vacuum Maxwell's equations are duality invariant, while the invari-ance is spoiled in the presence of charges. We might imagine that magnetic as well as electricmonopoles existed in nature; then we could add a magnetic current term 4(*Jm)to theright hand side of (1.91),and the equations would be invariant under duality transformationsplus the additional replacement J Jm. (Of course a nonzero right hand side to (1.91) isinconsistent with F =dA, so this idea only works if A, is not a fundamental variable.)Longago Dirac considered the idea of magnetic monopoles and showed that a necessary conditionfor their existence is that the fundamental monopole charge be inversely proportional to
1 SPECIAL RELATIVITY AND FLAT SPACETIME 24 in three dimensions do we have an interesting map from two dual vectors to a third dual vector. If you wanted to you could define a map from n − 1 one-forms to a single one-form, but I’m not sure it would be of any use. Electrodynamics provides an especially compelling example of the use of differential forms. From the definition of the exterior derivative, it is clear that equation (1.78) can be concisely expressed as closure of the two-form Fµν : dF = 0 . (1.91) Does this mean that F is also exact? Yes; as we’ve noted, Minkowski space is topologically trivial, so all closed forms are exact. There must therefore be a one-form Aµ such that F = dA . (1.92) This one-form is the familiar vector potential of electromagnetism, with the 0 component given by the scalar potential, A0 = φ. If one starts from the view that the Aµ is the fundamental field of electromagnetism, then (1.91) follows as an identity (as opposed to a dynamical law, an equation of motion). Gauge invariance is expressed by the observation that the theory is invariant under A → A + dλ for some scalar (zero-form) λ, and this is also immediate from the relation (1.92). The other one of Maxwell’s equations, (1.77), can be expressed as an equation between three-forms: d(∗F) = 4π(∗J) , (1.93) where the current one-form J is just the current four-vector with index lowered. Filling in the details is left for you to do. As an intriguing aside, Hodge duality is the basis for one of the hottest topics in theoretical physics today. It’s hard not to notice that the equations (1.91) and (1.93) look very similar. Indeed, if we set Jµ = 0, the equations are invariant under the “duality transformations” F → ∗F , ∗F → −F . (1.94) We therefore say that the vacuum Maxwell’s equations are duality invariant, while the invariance is spoiled in the presence of charges. We might imagine that magnetic as well as electric monopoles existed in nature; then we could add a magnetic current term 4π(∗JM) to the right hand side of (1.91), and the equations would be invariant under duality transformations plus the additional replacement J ↔ JM. (Of course a nonzero right hand side to (1.91) is inconsistent with F = dA, so this idea only works if Aµ is not a fundamental variable.) Long ago Dirac considered the idea of magnetic monopoles and showed that a necessary condition for their existence is that the fundamental monopole charge be inversely proportional to
251SPECIALRELATIVITYANDFLATSPACETIMEthe fundamental electric charge. Now, the fundamental electric charge is a small number;electrodynamics is "weakly coupled",which is why perturbation theory is so remarkablysuccessful in quantum electrodynamics (QED).But Dirac's condition on magnetic chargesimplies that a dualitytransformation takes a theory of weakly coupled electric chargestoatheory of strongly coupled magnetic monopoles (and vice-versa). Unfortunately monopolesdon't exist (as far as we know), so these ideas aren't directly applicable to electromagnetismbut there are some theories (such as supersymmetric non-abelian gauge theories)for whichit has been long conjectured that some sort of duality symmetry may exist. If it did, wewouldhavetheopportunitytoanalyzeatheorywhichlooked strongly coupled (andthereforehardto solve)bylooking attheweakly coupled dual version.Recentlyworkby SeibergandWitten and others has provided very strong evidence that this is exactly what happens incertain theories. The hope is that these techniques will allow us to explore various phenom-enawhich weknowexist instrongly coupled quantum field theories, such as confinementofquarksin hadrons.We've now gone over essentially everything there is to know about the care and feeding oftensors. In the next section we will look more carefully at the rigorous definitions of manifoldsand tensors, but the basic mechanics have been pretty well covered.Before jumping to moreabstract mathematics, let's review how physics works in Minkowski spacetime.Start with the worldline of a single particle. This is specified by a map R M, whereM is the manifold representing spacetime; we usually think of the path as a parameterizedcurve r"(). As mentioned earlier, the tangent vector to this path is dr/d> (note that itdepends on the parameterization). An object of primary interest is the norm of the tangentvector,whichservestocharacterizethepath;ifthetangentvectoristimelike/null/spacelikeat some parameter value,we say that the path is timelike/null/spacelike at that point.Thisexplains why the same words are used to classify vectors in the tangent space and intervalsbetween two points-because a straight line connecting, say, two timelike separated pointswill itself be timelike at every point along the path.Nevertheless, it's important to be aware of the sleight of hand which is being pulled here.The metric, as a (o,2) tensor, is a machine which acts on two vectors (or two copies of thesame vector) to produce a number. It is therefore very natural to classify tangent vectorsaccording to the sign of their norm. But the interval between two points isn't somethingquite so natural; it depends on a specific choice of path (a “straight line") which connectsthe points, and this choice in turn depends on the fact that spacetime is fat (which allowsa unique choice of straight line between the points). A more natural object is the lineelement, or infinitesimal interval:ds?= nwdec'dr".(1.95)From this definition it is tempting to take the square root and integrate along a path toobtain a finite interval.But since ds? need not be positive, we define different procedures
1 SPECIAL RELATIVITY AND FLAT SPACETIME 25 the fundamental electric charge. Now, the fundamental electric charge is a small number; electrodynamics is “weakly coupled”, which is why perturbation theory is so remarkably successful in quantum electrodynamics (QED). But Dirac’s condition on magnetic charges implies that a duality transformation takes a theory of weakly coupled electric charges to a theory of strongly coupled magnetic monopoles (and vice-versa). Unfortunately monopoles don’t exist (as far as we know), so these ideas aren’t directly applicable to electromagnetism; but there are some theories (such as supersymmetric non-abelian gauge theories) for which it has been long conjectured that some sort of duality symmetry may exist. If it did, we would have the opportunity to analyze a theory which looked strongly coupled (and therefore hard to solve) by looking at the weakly coupled dual version. Recently work by Seiberg and Witten and others has provided very strong evidence that this is exactly what happens in certain theories. The hope is that these techniques will allow us to explore various phenomena which we know exist in strongly coupled quantum field theories, such as confinement of quarks in hadrons. We’ve now gone over essentially everything there is to know about the care and feeding of tensors. In the next section we will look more carefully at the rigorous definitions of manifolds and tensors, but the basic mechanics have been pretty well covered. Before jumping to more abstract mathematics, let’s review how physics works in Minkowski spacetime. Start with the worldline of a single particle. This is specified by a map R → M, where M is the manifold representing spacetime; we usually think of the path as a parameterized curve x µ (λ). As mentioned earlier, the tangent vector to this path is dxµ /dλ (note that it depends on the parameterization). An object of primary interest is the norm of the tangent vector, which serves to characterize the path; if the tangent vector is timelike/null/spacelike at some parameter value λ, we say that the path is timelike/null/spacelike at that point. This explains why the same words are used to classify vectors in the tangent space and intervals between two points — because a straight line connecting, say, two timelike separated points will itself be timelike at every point along the path. Nevertheless, it’s important to be aware of the sleight of hand which is being pulled here. The metric, as a (0, 2) tensor, is a machine which acts on two vectors (or two copies of the same vector) to produce a number. It is therefore very natural to classify tangent vectors according to the sign of their norm. But the interval between two points isn’t something quite so natural; it depends on a specific choice of path (a “straight line”) which connects the points, and this choice in turn depends on the fact that spacetime is flat (which allows a unique choice of straight line between the points). A more natural object is the line element, or infinitesimal interval: ds2 = ηµν dxµ dxν . (1.95) From this definition it is tempting to take the square root and integrate along a path to obtain a finite interval. But since ds2 need not be positive, we define different procedures
261 SPECIALRELATIVITYANDFLATSPACETIMExha)timelikenulldxtdaspacelikefordifferentcases.ForspacelikepathswedefinethepathlengthdcudavAs =(1.96)d入,nxdwhere the integral is taken over the path. For null paths the interval is zero, so no extraformula is required.Fortimelikepaths wedefine theproper timedcudrv△T=(1.97)d入ddxwhich will be positive. Of course we may consider paths that are timelike in some places andspacelike in others, but fortunately it is seldom necessary since the paths of physical particlesneverchangetheir character(massiveparticlesmoveontimelikepaths,masslessparticlesmove on null paths).Furthermore, thephrase“proper time"is especiallyappropriate, sinceT actually measures the time elapsed on a physical clock carried along the path.This point ofview makes the“twin paradox"and similar puzzles very clear; two worldlines,not necessarilystraight,which intersect at two different events in spacetime will have proper times measuredby the integral (1.97) along the appropriate paths, and these two numbers will in general bedifferent even if the people travelling along them wereborn at the same time.Let'smovefrom theconsiderationofpathsingeneralto thepathsof massiveparticles(which will always be timelike).Since the proper time is measured by a clock travelling ona timelike worldline, it is convenient to use as the parameter along the path. That is, weuse (1.97) to compute r(), which (if > is a good parameter in the first place) we can invertto obtain (t),after which we can think of the path as r(). The tangent vector in this
1 SPECIAL RELATIVITY AND FLAT SPACETIME 26 t x spacelike null timelike dx - d x ( ) λ µ µ λ for different cases. For spacelike paths we define the path length ∆s = Z s ηµν dxµ dλ dxν dλ dλ , (1.96) where the integral is taken over the path. For null paths the interval is zero, so no extra formula is required. For timelike paths we define the proper time ∆τ = Z s −ηµν dxµ dλ dxν dλ dλ , (1.97) which will be positive. Of course we may consider paths that are timelike in some places and spacelike in others, but fortunately it is seldom necessary since the paths of physical particles never change their character (massive particles move on timelike paths, massless particles move on null paths). Furthermore, the phrase “proper time” is especially appropriate, since τ actually measures the time elapsed on a physical clock carried along the path. This point of view makes the “twin paradox” and similar puzzles very clear; two worldlines, not necessarily straight, which intersect at two different events in spacetime will have proper times measured by the integral (1.97) along the appropriate paths, and these two numbers will in general be different even if the people travelling along them were born at the same time. Let’s move from the consideration of paths in general to the paths of massive particles (which will always be timelike). Since the proper time is measured by a clock travelling on a timelike worldline, it is convenient to use τ as the parameter along the path. That is, we use (1.97) to compute τ (λ), which (if λ is a good parameter in the first place) we can invert to obtain λ(τ ), after which we can think of the path as x µ (τ ). The tangent vector in this
271 SPECIALRELATIVITYANDFLATSPACETIMEparameterization is known as the four-velocity,U:UM=deu(1.98)drSince d+? = -nudrdr", the four-velocity is automatically normalized:nμUU"= -1 .(1.99)(It will always be negative, since we are only defining it for timelike trajectories. You coulddefine an analogous vectorforspacelike paths as well; null pathsgive some extra problemssince the norm is zero.) In the rest frame of a particle, its four-velocity has componentsU= (1, 0,0,0).A related vector is the energy-momentum four-vector, defined byp"=mUu,(1.100)where m is the mass of the particle. The mass is a fixed quantity independent of inertialframe; what you may be used to thinking of as the "rest mass." It turns out to be muchmore convenient to take this as the mass once and for all, rather than thinking of mass asdepending on velocity.The energy of a particle is simply po, the timelike component of itsenergy-momentum vector. Since it's only one component of a four-vector, it is not invariantunderLorentztransformations;that's tobeexpected,however,sincetheenergy ofaparticleat rest is not the same as that of the same particle in motion.In the particle's rest frame wehave po = m; recalling that we have set c = 1, we find that we have found the equation thatmade Einstein a celebrity, E = mc2. (The field equations of general relativity are actuallymuch more important than this one, but "Rμw-Rgμw = 8πGTμ" doesn't elicit the visceralreaction that you get from “E = mc2".) In a moving frame we can find the components ofpby performing a Lorentz transformation; for a particle moving with (three-) velocity along the axiswe have(1.101)p = (m, vm, 0, 0) ,where = 1/V1-u2. For small u, this gives po = m + mu? (what we usually think ofas rest energy plus kinetic energy) and pl = mu (what we usually think of as [Newtonian]momentum). So the energy-momentum vector lives up to its name.The centerpiece of pre-relativity physics is Newton's 2nd Law, or f =ma =dp/dt.Ananalogous equation should hold in SR, and the requirement that it be tensorial leads usdirectly to introduce a force four-vector f satisfyingd2d(1.102)fH=mp"(T) .ndr2a(t) =dtThe simplest example of a force in Newtonian physics is the force due to gravity. In relativity,however, gravity is not described by a force, but rather by the curvature of spacetime itself
1 SPECIAL RELATIVITY AND FLAT SPACETIME 27 parameterization is known as the four-velocity, U µ : U µ = dxµ dτ . (1.98) Since dτ 2 = −ηµν dxµdxν , the four-velocity is automatically normalized: ηµνU µU ν = −1 . (1.99) (It will always be negative, since we are only defining it for timelike trajectories. You could define an analogous vector for spacelike paths as well; null paths give some extra problems since the norm is zero.) In the rest frame of a particle, its four-velocity has components U µ = (1, 0, 0, 0). A related vector is the energy-momentum four-vector, defined by p µ = mUµ , (1.100) where m is the mass of the particle. The mass is a fixed quantity independent of inertial frame; what you may be used to thinking of as the “rest mass.” It turns out to be much more convenient to take this as the mass once and for all, rather than thinking of mass as depending on velocity. The energy of a particle is simply p 0 , the timelike component of its energy-momentum vector. Since it’s only one component of a four-vector, it is not invariant under Lorentz transformations; that’s to be expected, however, since the energy of a particle at rest is not the same as that of the same particle in motion. In the particle’s rest frame we have p 0 = m; recalling that we have set c = 1, we find that we have found the equation that made Einstein a celebrity, E = mc2 . (The field equations of general relativity are actually much more important than this one, but “Rµν − 1 2Rgµν = 8πGTµν ” doesn’t elicit the visceral reaction that you get from “E = mc2”.) In a moving frame we can find the components of p µ by performing a Lorentz transformation; for a particle moving with (three-) velocity v along the x axis we have p µ = (γm, vγm, 0, 0) , (1.101) where γ = 1/ √ 1 − v 2 . For small v, this gives p 0 = m + 1 2mv2 (what we usually think of as rest energy plus kinetic energy) and p 1 = mv (what we usually think of as [Newtonian] momentum). So the energy-momentum vector lives up to its name. The centerpiece of pre-relativity physics is Newton’s 2nd Law, or f = ma = dp/dt. An analogous equation should hold in SR, and the requirement that it be tensorial leads us directly to introduce a force four-vector f µ satisfying f µ = m d 2 dτ 2 x µ (τ ) = d dτ p µ (τ ) . (1.102) The simplest example of a force in Newtonian physics is the force due to gravity. In relativity, however, gravity is not described by a force, but rather by the curvature of spacetime itself
281SPECIALRELATIVITYANDFLATSPACETIMEInstead, let us consider electromagnetism.The three-dimensional Lorentz force is givenby f = g(E +v× B), where g is the charge on the particle.We would like a tensorialgeneralization of this equation. There turns out to be a unique answer:fH=qUAFyH.(1.103)You can check for yourself that this reduces to the Newtonian version in the limit of smallvelocities. Notice how the requirement that the equation be tensorial, which is one way ofguaranteeing Lorentz invariance, severely restricted the possible expressions we could get.This is an example of a very general phenomenon, in which a small number of an apparentlyendless variety of possible physical laws are picked out by the demands of symmetry.Although pprovides a complete description of the energy and momentum of a particle,for extended systems it is necessary to go further and define the energy-momentum tensor(sometimes called the stress-energy tensor), T. This is a symmetric (2, 0) tensor which tellsus all we need to know about the energy-like aspects of a system: energy density, pressure,stress, and so forth. A general definition of Tw is "the flux of four-momentum p across asurface of constant rv",Tomakethis more concrete, let's considertheverygeneral categoryof matter which may be characterized as a fluid a continuum of matter described bymacroscopic quantities such as temperature, pressure, entropy, viscosity, etc. In fact thisdefinition is so general that it is of little use.In general relativity essentially all interestingtypesofmattercanbethoughtofasperfectfluids,fromstarstoelectromagneticfieldstothe entire universe. Schutz defines a perfect fluid to be one with no heat conduction and noviscosity,while Weinberg defines it as a fluid which looks isotropic in its rest frame; thesetwo viewpoints turn out to be equivalent. Operationally, you should think of a perfect fluidas one which may be completely characterized by its pressure and density.To understand perfect fuids, let's start with the even simpler example of dust. Dustis defined as a collection of particles at rest with respect to each other, or alternativelyas a perfect fluid with zero pressure. Since the particles all have an equal velocity in anyfixed inertial frame, we can imagine a “four-velocity field" U(r) defined all over spacetime.(Indeed, its components are the same at each point.) Define the number-flux four-vectorto beNH=nUH,(1.104)where n is the number density of the particles as measured in their rest frame. Then Nois the number density of particles as measured in any other frame, while Ni is the flux ofparticles in the ri direction. Let's now imagine that each of the particles have the same massm. Then in the rest frame the energy density of the dust is given by(1.105)p=nm
1 SPECIAL RELATIVITY AND FLAT SPACETIME 28 Instead, let us consider electromagnetism. The three-dimensional Lorentz force is given by f = q(E + v × B), where q is the charge on the particle. We would like a tensorial generalization of this equation. There turns out to be a unique answer: f µ = qUλFλ µ . (1.103) You can check for yourself that this reduces to the Newtonian version in the limit of small velocities. Notice how the requirement that the equation be tensorial, which is one way of guaranteeing Lorentz invariance, severely restricted the possible expressions we could get. This is an example of a very general phenomenon, in which a small number of an apparently endless variety of possible physical laws are picked out by the demands of symmetry. Although p µ provides a complete description of the energy and momentum of a particle, for extended systems it is necessary to go further and define the energy-momentum tensor (sometimes called the stress-energy tensor), T µν. This is a symmetric (2, 0) tensor which tells us all we need to know about the energy-like aspects of a system: energy density, pressure, stress, and so forth. A general definition of T µν is “the flux of four-momentum p µ across a surface of constant x ν ”. To make this more concrete, let’s consider the very general category of matter which may be characterized as a fluid — a continuum of matter described by macroscopic quantities such as temperature, pressure, entropy, viscosity, etc. In fact this definition is so general that it is of little use. In general relativity essentially all interesting types of matter can be thought of as perfect fluids, from stars to electromagnetic fields to the entire universe. Schutz defines a perfect fluid to be one with no heat conduction and no viscosity, while Weinberg defines it as a fluid which looks isotropic in its rest frame; these two viewpoints turn out to be equivalent. Operationally, you should think of a perfect fluid as one which may be completely characterized by its pressure and density. To understand perfect fluids, let’s start with the even simpler example of dust. Dust is defined as a collection of particles at rest with respect to each other, or alternatively as a perfect fluid with zero pressure. Since the particles all have an equal velocity in any fixed inertial frame, we can imagine a “four-velocity field” U µ (x) defined all over spacetime. (Indeed, its components are the same at each point.) Define the number-flux four-vector to be N µ = nUµ , (1.104) where n is the number density of the particles as measured in their rest frame. Then N0 is the number density of particles as measured in any other frame, while Ni is the flux of particles in the x i direction. Let’s now imagine that each of the particles have the same mass m. Then in the rest frame the energy density of the dust is given by ρ = nm . (1.105)