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Contents
- 1 Physics in Euclidean Space and Flat Spacetime: Geometric Viewpoint
- 1.1 [N & R] Overview
- 1.2 Foundational Concepts
- 1.2.1 [N] Newtonian Concepts
- events, vectors, and spacetime diagrams 1.2.2 [R] Special Relativistic Concepts: Inertial frames, inertial coordinates,
- and its Invariance 1.2.3 [R] Special Relativistic Concepts: Principle of Relativity; the Interval
- 1.3 [N & R] Tensor Algebra Without a Coordinate System
- 1.4 Particle Kinetics and Lorentz Force Without a Reference Frame
- 1.4.1 [N] Newtonian Particle Kinetics
- and its Conservation, 4-Force 1.4.2 [R] Relativistic Particle Kinetics: World Lines, 4-Velocity, 4-Momentum
- 1.4.3 [R] Geometric Derivation of the Lorentz Force Law
- 1.5 Component Representation of Tensor Algebra
- 1.5.1 [N] Euclidean 3-space
- 1.5.2 [R] Minkowski Spacetime
- 1.5.3 [N & R] Slot-Naming Index Notation
- 1.6 [R] Particle Kinetics in Index Notation and in a Lorentz Frame
- 1.7 Orthogonal and Lorentz Transformations of Bases, and Spacetime Diagrams
- 1.7.1 [N] Euclidean 3-space: Orthogonal Transformations
- 1.7.2 [R] Minkowski Spacetime: Lorentz Transformations
- 1.7.3 [R] Spacetime Diagrams for Boosts
- 1.8 [R] Time Travel
- and Curl 1.9 [N & R] Directional Derivatives, Gradients, Levi-Civita Tensor, Cross Product
- 1.10 [R] Nature of Electric and Magnetic Fields; Maxwell’s Equations
- 1.11 Volumes, Integration, and Integral Conservation Laws
- 1.11.1 [N] Newtonian Volumes and Integration
- 1.11.2 [R] Spacetime Volumes and Integration
- 1.11.3 [R] Conservation of Charge in Spacetime
- 1.11.4 [R] Conservation of Particles, Baryons and Rest Mass
- 1.12 The Stress-Energy Tensor and Conservation of 4-Momentum
- 1.12.1 [N] Newtonian Stress Tensor and Momentum Conservation
ii
1.12.2 [R] Relativistic Stress-Energy Tensor.................. 58 1.12.3 [R] 4-Momentum Conservation...................... 60 1.12.4 [R] Stress-Energy Tensors for Perfect Fluid and Electromagnetic Field 61
quantities (locations in space or spacetime, momenta of particles, etc.) that are measured in some coordinate system or reference frame. For example, Newtonian vectorial quantities (momenta, electric fields, etc.) are triplets of numbers [e.g., (1, 9 , −4)] representing the vectors’ components on the axes of a spatial coordinate system, and relativistic 4-vectors are quadruplets of numbers representing components on the spacetime axes of some reference frame. By contrast, in this book, we shall express all physical quantities and laws in a geometric form, i.e. a form that is independent of any coordinate system or reference frame. For example, in Newtonian physics, momenta and electric fields will be vectors described as arrows that live in the 3-dimensional, flat Euclidean space of everyday experience. They require no coordinate system at all for their existence or description—though sometimes co- ordinates will be useful. We shall state physical laws, e.g. the Lorentz force law, as geomet- ric (i.e. coordinate-free) relationships between these geometric (i.e. coordinate-independent) quantities. By adopting this geometric viewpoint, we shall gain great conceptual power and often also computational power. For example, when we ignore experiment and simply ask what forms the laws of physics can possibly take (what forms are allowed by the requirement that the laws be geometric), we shall find remarkably little freedom. Coordinate independence strongly constrains the laws (see, e.g., Sec. 1.4 below). This power, together with the elegance of the geometric formulation, suggests that in some deep (ill-understood) sense, Nature’s physical laws are geometric and have nothing whatsoever to do with coordinates or reference frames. The mathematical foundation for our geometric viewpoint is differential geometry (also called “tensor analysis” by physicists). This differential geometry can be thought of as an extension of the vector analysis with which all readers should be familiar. There are three different frameworks for the classical physical laws that scientists use, and correspondingly three different geometric arenas for the laws; cf. Fig. 1.1. General relativity is the most accurate classical framework; it formulates the laws as geometric relationships in the arena of curved 4-dimensional spacetime. Special relativity is the limit of general relativity in the complete absence of gravity; its arena is flat, 4-dimensional Minkowski spacetime^1. Newtonian physics is the limit of general relativity when (i) gravity is weak but not necessarily absent, (ii) relative speeds of particles and materials are small compared to the speed of light c, and (iii) all stresses (pressures) are small compared to the total density of mass-energy; its arena is flat, 3-dimensional Euclidean space with time separated off and made universal (by contrast with the frame-dependent time of relativity). In Parts I–V of this book (statistical physics, optics, elasticity theory, fluid mechanics, plasma physics) we shall confine ourselves to the Newtonian and special relativistic formula- tions of the laws, and accordingly our arenas will be flat Euclidean space and flat Minkowski spacetime. In Part VI we shall extend many of the laws we have studied into the domain of strong gravity (general relativity), i.e., the arena of curved spacetime. In Parts I and II (statistical physics and optics), in addition to confining ourselves to flat space or flat spacetime, we shall avoid any sophisticated use of curvilinear coordinates; i.e., when using coordinates in nontrivial ways, we shall confine ourselves to Cartesian coordinates
(^1) so-called because it was Hermann Minkowski (1908) who identified the special relativistic invariant interval as defining a metric in spacetime, and who elucidated the resulting geometry of flat spacetime.
Special Relativity Classical Physics in the absence of gravity Arena: Flat, Minkowski spacetime
vanishing gravity
General Relativity The most accurate framework for Classical Physics Arena: Curved spacetime
weak gravity small speeds small stresses
Newtonian Physics Approximation to relativistic physics Arena: Flat, Euclidean 3-space, plus universal time
low speeds small stresses add weak gravity
Fig. 1.1: The three frameworks and arenas for the classical laws of physics, and their relationship to each other.
in Euclidean space, and Lorentz coordinates in Minkowski spacetime. This chapter is an introduction to all the differential geometric tools that we shall need in these limited arenas. In Parts III, IV, and V, when studying elasticity theory, fluid mechanics, and plasma physics, we will use curvilinear coordinates in nontrivial ways. As a foundation for them, at the beginning of Part III we will extend our flat-space differential geometric tools to curvilinear coordinate systems (e.g. cylindrical and spherical coordinates). Finally, at the beginning of Part VI, we shall extend our geometric tools to the arena of curved spacetime. In this chapter we shall alternate back and forth, one section after another, between the laws of physics and flat-space differential geometry, using each to illustrate and illuminate the other. We begin in Sec. 1.2 by recalling the foundational concepts of Newtonian physics and of special relativity. Then in Sec. 1.3 we develop our first set of differential geometric tools: the tools of coordinate-free tensor algebra. In Sec. 1.4 we illustrate our tensor-algebra tools by using them to describe—without any coordinate system or reference frame whatsoever—the kinematics of point particles that move through the Euclidean space of Newtonian physics and through relativity’s Minkowski spacetime; the particles are allowed to collide with each other and be accelerated by an electromagnetic field. In Sec. 1.5, we extend the tools of tensor algebra to the domain of Cartesian and Lorentz coordinate systems, and then in Sec. 1. we use these extended tensorial tools to restudy the motions, collisions, and electromagnetic accelerations of particles. In Sec. 1.7 we discuss rotations in Euclidean space and Lorentz transformations in Minkowski spacetime, and we develop relativistic spacetime diagrams in some depth and use them to study such relativistic phenomena as length contraction, time dilation, and simultaneity breakdown. In Sec. 1.8 we illustrate the tools we have developed by asking whether the laws of relativity permit a highly advanced civilization to build time machines for traveling backward in time as well as forward. In Sec. 1.9 we develop additional differential geometric tools: directional derivatives, gradients, and the Levi-Civita tensor, and in Sec. 1.10 we use these tools to discuss Maxwell’s equations and the geometric nature of electric and magnetic fields. In Sec. 1.11 we develop our final set of geometric tools: volume elements and the integration of tensors over spacetime, and in Sec. 1.12 we use these tools to define the stress tensor of Newtonian physics and relativity’s stress-energy tensor, and to formulate very general versions of the conservation of 4-momentum.
difference of two velocity measurements at times separated by dt and multiplying by 1/dt generates the acceleration a = dv/dt. Multiplying by the fluid element’s (scalar) mass m gives the force F = ma that produced the acceleration; dividing an electrically produced force by the fluid element’s charge q gives another vector, the electric field E = F/q, and so on. We can define inner products [Eq. (1.9a) below] of pairs of vectors at a point (e.g., force and displacement) to obtain a new scalar (e.g., work), and cross products [Eq. (1.60a)] of vectors to obtain a new vector (e.g., torque). By examining how a differentiable scalar field changes from point to point, we can define its gradient [Eq. (1.54b)]. In this fashion, which should be familiar to the reader and will be elucidated and generalized below, we can construct all of the standard scalars and vectors of Newtonian physics. What is important is that these physical quantities require no coordinate system for their definition. They are geometric (coordinate-independent) objects residing in Euclidean 3-space at a particular time. It is a fundamental (though often ignored) principle of physics that the Newtonian physical laws are all expressible as geometric relationships between these types of geometric objects, and these relationships do not depend upon any coordinate system or orientation of axes, nor on any reference frame (on any purported velocity of the Euclidean space in which the measurements are made).^2 We shall return to this principle throughout this book. It is the Newtonian analog of Einstein’s Principle of Relativity (Sec. 1.2.3 below).
1.2.2 [R] Special Relativistic Concepts: Inertial frames, inertial
coordinates, events, vectors, and spacetime diagrams
Because the nature and geometry of Minkowski spacetime are far less obvious intuitively than those of Euclidean 3-space, we shall need a crutch in our development of the Minkowski foundational concepts. That crutch will be inertial reference frames. We shall use them to develop in turn the following frame-independent Minkowski-spacetime concepts: events, 4- vectors, the principle of relativity, geometrized units, the interval and its invariance, and spacetime diagrams. An inertial reference frame is a (conceptual) three-dimensional latticework of measuring rods and clocks (Fig. 1.3) with the following properties: (i) The latticework moves freely through spacetime (i.e., no forces act on it), and is attached to gyroscopes so it does not rotate with respect to distant, celestial objects. (ii) The measuring rods form an orthogonal lattice and the length intervals marked on them are uniform when compared to, e.g., the wavelength of light emitted by some standard type of atom or molecule; and therefore the rods form an orthonormal, Cartesian coordinate system with the coordinate x measured along one axis, y along another, and z along the third. (iii) The clocks are densely packed throughout the latticework so that, ideally, there is a separate clock at every lattice point. (iv ) The clocks tick uniformly when compared, e.g., to the period of the light emitted by some standard type of atom or molecule; i.e., they are ideal clocks. (v ) The clocks are synchronized by the Einstein synchronization process: If a pulse of light, emitted by one of the clocks, bounces off a mirror attached to another and then returns, the time of bounce tb
(^2) By changing the velocity of Euclidean space, one adds a constant velocity to all particles, but this leaves
the laws, e.g. Newton’s F = ma, unchanged.
Fig. 1.3: An inertial reference frame. From Taylor and Wheeler (1992).
as measured by the clock that does the bouncing is the average of the times of emission and reception as measured by the emitting and receiving clock: tb = 12 (te + tr).^3 Our second fundamental relativistic concept is the event. An event is a precise location in space at a precise moment of time; i.e., a precise location (or “point”) in 4-dimensional spacetime. We sometimes will denote events by capital script letters such as P and Q — the same notation as for points in Euclidean 3-space; there need be no confusion, since we will avoid dealing with 3-space points and Minkowski-spacetime points simultaneously. A 4-vector (also often referred to as a vector in spacetime) is a straight arrow ∆~x reaching from one event P to another Q; equivalently, ∆~x is a direction in spacetime at the event P where it lives, together with a number that tell us its length. We often will deal with 4-vectors and ordinary (3-space) vectors simultaneously, so we shall need different notations for them: bold-face Roman font for 3-vectors, ∆x, and arrowed italic font for 4-vectors, ∆~x. Sometimes we shall identify an event P in spacetime by its vectorial separation ~xP from some arbitrarily chosen event in spacetime, the “origin” O. An inertial reference frame provides us with a coordinate system for spacetime. The coordinates (x^0 , x^1 , x^2 , x^3 ) = (t, x, y, z) which it associates with an event P are P’s location (x, y, z) in the frame’s latticework of measuring rods, and the time t of P as measured by the clock that sits in the lattice at the event’s location. (Many apparent paradoxes in special relativity result from failing to remember that the time t of an event is always measured by a clock that resides at the event, and never by clocks that reside elsewhere in spacetime.) It is useful to depict events on spacetime diagrams, in which the time coordinate t = x^0 of some inertial frame is plotted upward, and two of the frame’s three spatial coordinates, x = x^1 and y = x^2 , are plotted horizontally. Figure 1.4 is an example. Two events P and Q are shown there, along with their vectorial separations ~xP and ~xQ from the origin and the vector ∆~x = ~xQ − ~xP that separates them from each other. The coordinates of P and Q,
(^3) For a deeper discussion of the nature of ideal clocks and ideal measuring rods see, e.g., pp. 23–29 and 395–399 of Misner, Thorne, and Wheeler (1973).
spacetime from P to Q. Different observers in different inertial frames will attribute different coordinates to each birth and different components to the births’ vectorial separation; but all observers can agree that they are talking about the same events P and Q in spacetime and the same separation vector ∆~x. In this sense, P, Q, and ∆~x are frame-independent, geometric objects (points and arrows) that reside in spacetime.
1.2.3 [R] Special Relativistic Concepts: Principle of Relativity;
the Interval and its Invariance
The principle of relativity states that Every (special relativistic) law of physics must be ex- pressible as a geometric, frame-independent relationship between geometric, frame-independent objects, i.e. objects such as points in spacetime and vectors and tensors, which represent physical quantities such as events and particle momenta and the electromagnetic field. Since the laws are all geometric (i.e., unrelated to any reference frame or coordinate system), there is no way that they can distinguish one inertial reference frame from any other. This leads to an alternative form of the principle of relativity (one commonly used in elementary textbooks and equivalent to the above): All the (special relativistic) laws of physics are the same in every inertial reference frame, everywhere in spacetime. A more operational version of this principle is the following: Give identical instructions for a specific physics experiment to two different observers in two different inertial reference frames at the same or different locations in Minkowski (i.e., gravity-free) spacetime. The experiment must be self-contained, i.e., it must not involve observations of the external universe’s properties (the “environment”), though it might utilize carefully calibrated tools derived from the external universe. For example, an unacceptable experiment would be a measurement of the anisotropy of the Universe’s cosmic microwave radiation and a computation therefrom of the observer’s velocity relative to the radiation’s mean rest frame; such an experiment studies the Universal environment, not the fundamental laws of physics. An acceptable experiment would be a measurement of the speed of light using the rods and clocks of the observer’s own frame, or a measurement of cross sections for elementary particle reactions using cosmic-ray particles whose incoming energies and compositions are measured as initial conditions for the experiment. The principle of relativity says that in these or any other similarly self-contained experiments, the two observers in their two different inertial frames must obtain identically the same experimental results—to within the accuracy of their experimental techniques. Since the experimental results are governed by the (nongravitational) laws of physics, this is equivalent to the statement that all physical laws are the same in the two inertial frames. Perhaps the most central of special relativistic laws is the one stating that the speed of light c in vacuum is frame-independent, i.e., is a constant, independent of the inertial reference frame in which it is measured. In other words, there is no aether that supports light’s vibrations and in the process influences its speed — a remarkable fact that came as a great experimental surprise to physicists at the end of the nineteenth century. The constancy of the speed of light is built into Maxwell’s equations. In order for the Maxwell equations to be frame independent, the speed of light, which appears in them, must also be frame independent. In this sense, the constancy of the speed of light follows from the Principle of Relativity; it is not an independent postulate. This is illustrated in Box 1.2.
Box 1. Measuring the Speed of Light Without Light
r
q, μ
Q
a e
r a m
q, μ (^) v
Q
In some inertial reference frame we perform two experiments using two particles, one with a large charge Q; the other, a test particle, with a much smaller charge q and mass μ. In the first experiment we place the two particles at rest, separated by a distance |∆x| ≡ r and measure the electrical repulsive acceleration ae of q (left diagram). In Gaussian cgs units (where the speed of light shows up explicitly instead of via ǫoμo = 1/c^2 ), the acceleration is ae = qQ/r^2 μ. In the second experiment, we connect Q to ground by a long wire, and we place q at the distance |∆x| = r from the wire and set it moving at speed v parallel to the wire. The charge Q flows down the wire with an e-folding time τ so the current is I = dQ/dτ = (Q/τ )e−t/τ^. At early times 0 < t ≪ τ , this current I = Q/τ produces a solenoidal magnetic field at q with field strength B = (2/cr)(Q/τ ), and this field exerts a magnetic force on q, giving it an acceleration am = q(v/c)B/μ = 2 vqQ/c^2 τ r/μ. The ratio of the electric acceleration in the first experiment to the magnetic acceleration in the second experiment is ae/am = c^2 τ / 2 rv. Therefore, we can measure the speed of light c in our chosen inertial frame by performing this pair of experiments, carefully measuring the separation r, speed v, current Q/τ , and accelerations, and then simply computing c =
(2rv/τ )(ae/am). The principle of relativity insists that the result of this pair of experiments should be independent of the inertial frame in which they are performed. Therefore, the speed of light c which appears in Maxwell’s equations must be frame-independent. In this sense, the constancy of the speed of light follows from the Principle of Relativity as applied to Maxwell’s equations.
The constancy of the speed of light was verified with very high precision in an era when the units of length (centimeters) and the units of time (seconds) were defined independently. By 1983, the constancy had become so universally accepted that it was used to redefine the meter (which is hard to measure precisely) in terms of the second (which is much easier to measure with modern technology^4 ): The meter is now related to the second in such a way
(^4) The second is defined as the duration of 9,192,631,770 periods of the radiation produced by a certain
hyperfine transition in the ground state of a 133 Cs atom that is at rest in empty space. Today (2008) all fundamental physical units except mass units (e.g. the kilogram) are defined similarly in terms of fundamental constants of nature.
Fig. 1.5: Geometry for proving the invariance of the interval.
Choose the spatial coordinate systems of the primed and unprimed frames in such a way that (i) their relative motion (with speed β that will not enter into our analysis) is along the x direction and the x′^ direction, (ii) event P lies on the x and x′^ axes, and (iii) event Q lies in the x-y plane and in the x′-y′^ plane, as shown in Fig. 1.5. Then evaluate the interval between P and Q in the unprimed frame by the following construction: Place a mirror parallel to the x-z plane at precisely the height h that permits a photon, emitted from P, to travel along the dashed line of Fig. 1.5 to the mirror, then reflect off the mirror and continue along the dashed path, arriving at event Q. If the mirror were placed lower, the photon would arrive at the spatial location of Q sooner than the time of Q; if placed higher, it would arrive later. Then the distance the photon travels (the length of the two-segment dashed line) is equal to c∆t = ∆t, where ∆t is the time between events P and Q as measured in the unprimed frame. If the mirror had not been present, the photon would have arrived at event R after time ∆t, so c∆t is the distance between P and R. From the diagram it is easy to see that the height of R above the x axis is 2h − ∆y, and the Pythagorean theorem then implies that
(∆s)^2 = −(∆t)^2 + (∆x)^2 + (∆y)^2 = −(2h − ∆y)^2 + (∆y)^2. (1.6a)
The same construction in the primed frame must give the same formula, but with primes
(∆s′)^2 = −(∆t′)^2 + (∆x′)^2 + (∆y′)^2 = −(2h′^ − ∆y′)^2 + (∆y′)^2. (1.6b)
The proof that (∆s′)^2 = (∆s)^2 then reduces to showing that the principle of relativity requires that distances perpendicular to the direction of relative motion of two frames be the same as measured in the two frames, h′^ = h, ∆y′^ = ∆y. We leave it to the reader to develop a careful argument for this [Ex. 1.2]. Because of its frame invariance, the interval (∆s)^2 can be regarded as a geometric property of the vector ∆~x that reaches from P to Q; we shall call it the squared length (∆~x)^2 of ∆~x:
(∆~x)^2 ≡ (∆s)^2. (1.7)
Note that this squared length, despite its name, can be negative (for timelike ∆~x) or zero (for null ∆~x) as well as positive (for spacelike ∆~x).
This invariant interval between two events is as fundamental to Minkowski spacetime as the Euclidean distance between two points is to flat 3-space. Just as the Euclidean distance gives rise to the geometry of 3-space, as embodied, e.g., in Euclid’s axioms, so the interval gives rise to the geometry of spacetime, which we shall be exploring. If this spacetime geometry were as intuitively obvious to humans as is Euclidean geometry, we would not need the crutch of inertial reference frames to arrive at it. Nature (presumably) has no need for such a crutch. To Nature (it seems evident), the geometry of Minkowski spacetime, as embodied in the invariant interval, is among the most fundamental aspects of physical law. Before we leave this central idea, we should emphasize that vacuum electromagnetic radiation is not the only type of wave in nature. In this course, we shall encounter dispersive media, like optical fibers or plasmas, where electromagnetic signals travel slower than c, and we shall analyze sound waves and seismic waves where the governing laws do not involve electromagnetism at all. How do these fit into our special relativistic framework? The answer is simple. Each of these waves requires an underlying medium that is at rest in one particular frame (not necessarily inertial) and the velocity of the wave, specifically the group velocity, is most simply calculated in this frame from the waves’ and medium’s fundamental laws. We can then use the kinematic rules of Lorentz transformations to compute the velocity in another frame. However, if we had chosen to compute the wave speed in the second frame directly, using the same fundamental laws, we would have gotten the same answer, albeit perhaps with greater effort. All waves are in full compliance with the principle of relativity. What is special about vacuum electromagnetic waves and, by extension, photons, is that no medium (or “ether” as it used to be called) is needed for them to propagate. Their speed is therefore the same in all frames. This raises an interesting question. What about other waves that do not require an underlying medium? What about electron de Broglie waves? Here the fundamental wave equation, Schr¨odinger’s or Dirac’s, is mathematically different from Maxwell’s and contains an important parameter, the electron rest mass. This allows the fundamental laws of rela- tivistic quantum mechanics to be written in a form that is the same in all inertial reference frames and that allows an electron, considered as either a wave or a particle, to travel at a different speed when measured in a different frame. What about non-electromagnetic waves whose quanta have vanishing rest mass? For a long while, we thought that neutrinos provided a good example, but we now know from experiment that their rest masses are non-zero. However, there are other particles that have not yet been detected, including photinos (the hypothesized, supersymmetric partners to photons) and gravitons (and their associated gravitational waves which we shall discuss in Chapter 26), that are believed to exist without a rest mass (or an ether!), just like photons. Must these travel at the same speed as photons? The answer to this question, according to the principle of relativity, is “yes”. The reason is simple. Suppose there were two such waves (or particles) whose governing laws led to different speeds, c and c′^ < c, each the same in all reference frames. If we then move with speed c′^ in the direction of propagation of the second wave, we would bring it to rest, in conflict with our hypothesis that its speed is frame-independent. Therefore all signals, whose governing laws require them to travel with a speed that has no governing parameters (no rest mass and no underlying medium with physical properties) must travel with a unique speed which we call “c”. The speed of light
1.3 [N & R] Tensor Algebra Without a Coordinate Sys-
tem
We now pause in our development of the geometric view of physical laws, to introduce, in a coordinate-free way, some fundamental concepts of differential geometry: tensors, the inner product, the metric tensor, the tensor product, and contraction of tensors. In this section we shall allow the space in which the concepts live to be either 4-dimensional Minkowski spacetime, or 3-dimensional Euclidean space; we shall denote its dimensionality by N; and we shall use spacetime’s arrowed notation A~ for vectors even though the space might be Euclidean 3-space. We have already defined a vector A~ as a straight arrow from one point, say P, in our space to another, say Q. Because our space is flat, there is a unique and obvious way to transport such an arrow from one location to another, keeping its length and direction unchanged.^5 Accordingly, we shall regard vectors as unchanged by such transport. This enables us to ignore the issue of where in space a vector actually resides; it is completely determined by its direction and its length.
7.95 (^) T
Fig. 1.6: A rank-3 tensor T.
A rank-n tensor T is, by definition, a real-valued, linear function of n vectors. Pictorially we shall regard T as a box (Fig. 1.6) with n slots in its top, into which are inserted n vectors, and one slot in its end, out of which rolls computer paper with a single real number printed on it: the value that the tensor T has when evaluated as a function of the n inserted vectors. Notationally we shall denote the tensor by a bold-face sans-serif character T
T( , , , ︸ ︷︷ ︸ ). (1.8a)
տ n slots in which to put the vectors
If T is a rank-3 tensor (has 3 slots) as in Fig. 1.6, then its value on the vectors A, ~ B, ~ C~ will
be denoted T( A, ~B, ~ C~). Linearity of this function can be expressed as
T(e E~ + f F , ~ B, ~ C~) = eT( E, ~ B, ~ C~) + f T( F , ~ B, ~ C~) , (1.8b)
where e and f are real numbers, and similarly for the second and third slots. We have already defined the squared length ( A~)^2 ≡ A~^2 of a vector A~ as the squared distance (in 3-space) or interval (in spacetime) between the points at its tail and its tip. The inner product A~ · B~ of two vectors is defined in terms of the squared length by
A^ ~ · B~ ≡ 1 4
[
( A~ + B~)^2 − ( A~ − B~)^2
]
. (1.9a)
(^5) This is not so in curved spaces, as we shall see in Sec. 24.7.
In Euclidean space this is the standard inner product, familiar from elementary geometry. Because the inner product A~ · B~ is a linear function of each of its vectors, we can regard it as a tensor of rank 2. When so regarded, the inner product is denoted g( , ) and is called the metric tensor. In other words, the metric tensor g is that linear function of two vectors whose value is given by g( A, ~ B~) ≡ A~ · B .~ (1.9b)
Notice that, because A~ · B~ = B~ · A~, the metric tensor is symmetric in its two slots; i.e., one gets the same real number independently of the order in which one inserts the two vectors into the slots: g( A, ~ B~) = g( B, ~ A~) (1.9c)
With the aid of the inner product, we can regard any vector A~ as a tensor of rank one: The real number that is produced when an arbitrary vector C~ is inserted into A~’s slot is
A^ ~( C~) ≡ A~ · C .~ (1.9d)
Second-rank tensors appear frequently in the laws of physics—often in roles where one sticks a single vector into the second slot and leaves the first slot empty thereby producing a single-slotted entity, a vector. A familiar example is a rigid body’s (Newtonian) moment- of-inertia tensor I( , ). Insert the body’s angular velocity vector Ω into the second slot, and you get the body’s angular momentum vector J( ) = I( , Ω). Other examples are the stress tensor of a solid, a fluid, a plasma or a field (Sec. 1.12 below) and the electromagnetic field tensor (Secs. 1.4.3 and 1.10 below). From three (or any number of) vectors A~, B~, C~ we can construct a tensor, their tensor
product (also called outer product in contradistinction to the inner product A~ · B~), defined as follows:
A^ ~ ⊗ B~ ⊗ C~( E, ~ F , ~ G~) ≡ A~( E~) B~( F~ ) C~( G~) = ( A~ · E~)( B~ · F~ )( C~ · G~). (1.10a)
Here the first expression is the notation for the value of the new tensor, A~ ⊗ B~ ⊗ C~ evaluated on the three vectors E~, F~ , G~; the middle expression is the ordinary product of three real numbers, the value of A~ on E~, the value of B~ on F~ , and the value of C~ on G~; and the third expression is that same product with the three numbers rewritten as scalar products. Similar definitions can be given (and should be obvious) for the tensor product of any two or more tensors of any rank; for example, if T has rank 2 and S has rank 3, then
T ⊗ S( E, ~ F , ~ G, ~ H, ~ J~) ≡ T( E, ~ F~ )S( G, ~ H, ~ J~). (1.10b)
One last geometric (i.e. frame-independent) concept we shall need is contraction. We shall illustrate this concept first by a simple example, then give the general definition. From two vectors A~ and B~ we can construct the tensor product A~ ⊗ B~ (a second-rank tensor), and
we can also construct the scalar product A~ · B~ (a real number, i.e. a scalar, i.e. a rank- tensor ). The process of contraction is the construction of A~ · B~ from A~ ⊗ B~
contraction( A~ ⊗ B~) ≡ A~ · B .~ (1.11a)
(here we have used the vector cross product, which will not be introduced formally un- til Sec. 1.9 below). Obviously, these laws of motion are geometric relationships between geometric objects.
1.4.2 [R] Relativistic Particle Kinetics: World Lines, 4-Velocity,
4-Momentum and its Conservation, 4-Force
In special relativity, a particle moves through 4-dimensional spacetime along a curve (its world line) which we shall denote, in frame-independent notation, by ~x(τ ). Here τ is time as measured by an ideal clock that the particle carries (the particle’s proper time), and ~x is the location of the particle in spacetime when its clock reads τ (or, equivalently, the vector from the arbitrary origin to that location). The particle typically will experience an acceleration as it moves—e.g., an acceleration produced by an external electromagnetic field. This raises the question of how the acceler- ation affects the ticking rate of the particle’s clock. We define the accelerated clock to be ideal if its ticking rate is totally unaffected by its acceleration, i.e., if it ticks at the same rate as a freely moving (inertial) ideal clock that is momentarily at rest with respect to it. The builders of inertial guidance systems for airplanes and missiles always try to make their clocks as acceleration-independent, i.e., as ideal, as possible. We shall refer to the inertial frame in which a particle is momentarily at rest as its momentarily comoving inertial frame or momentary rest frame. Now, the particle’s clock (which measures τ ) is ideal and so are the inertial frame’s clocks (which measure coordinate time t). Therefore, a tiny interval ∆τ of the particle’s proper time is equal to the lapse of coordinate time in the particle’s momentary rest frame, ∆τ = ∆t. Moreover, since the two events ~x(τ ) and ~x(τ + ∆τ ) on the clock’s world line occur at the same spatial location in its momentary rest frame, ∆xi^ = 0 (where i = 1, 2 , 3), the invariant interval between those events is (∆s)^2 = −(∆t)^2 +
i,j ∆x
i∆xj (^) δij = −(∆t) (^2) = −(∆τ ) (^2). This shows that the
particle’s proper time τ is equal to the square root of the invariant interval, τ =
−s^2 , along its world line.
τ =
x y
t
u →
u →
Fig. 1.7: Spacetime diagram showing the world line ~x(τ ) and 4-velocity ~u of an accelerated particle. Note that the 4-velocity is tangent to the world line.
Figure 1.7 shows the world line of the accelerated particle in a spacetime diagram where the axes are coordinates of an arbitrary Lorentz frame. This diagram is intended to emphasize
the world line as a frame-independent, geometric object. Also shown in the figure is the particle’s 4-velocity ~u, which (by analogy with the velocity in 3-space) is the time derivative of its position:
~u ≡ d~x/dτ. (1.15)
This derivative is defined by the usual limiting process
d~x dτ
≡ lim ∆τ → 0
~x(τ + ∆τ ) − ~x(τ ) ∆τ
The squared length of the particle’s 4-velocity is easily seen to be −1:
~u^2 ≡ g(~u, ~u) =
d~x dτ
d~x dτ
d~x · d~x (dτ )^2
The last equality follows from the fact that d~x · d~x is the squared length of d~x which equals the invariant interval (∆s)^2 along it, and (dτ )^2 is minus that invariant interval. The particle’s 4-momentum is the product of its 4-velocity and rest mass
p ~ ≡ m~u = md~x/dτ ≡ d~x/dζ. (1.18)
Here the parameter ζ is a renormalized version of proper time,
ζ ≡ τ /m. (1.19)
This ζ, and any other renormalized version of proper time with position-independent renor- malization factor, are called affine parameters for the particle’s world line. Expression (1.18), together with the unit length of the 4-velocity ~u^2 = −1, implies that the squared length of the 4-momentum is ~p 2 = −m^2. (1.20) In quantum theory a particle is described by a relativistic wave function which, in the geometric optics limit (Chapter 6), has a wave vector ~k that is related to the classical particle’s 4-momentum by ~k = ~p/ℏ. (1.21)
The above formalism is valid only for particles with nonzero rest mass, m 6 = 0. The corresponding formalism for a particle with zero rest mass (e.g. a photon or a graviton^7 ) can be obtained from the above by taking the limit as m → 0 and dτ → 0 with the quotient dζ = dτ /m held finite. More specifically, the 4-momentum of a zero-rest-mass particle is well defined (and participates in the conservation law to be discussed below), and it is expressible in terms of the particle’s affine parameter ζ by Eq. (1.18)
~p =
d~x dζ
(^7) We do not know for sure that photons and gravitons are massless, but the laws of physics as currently
undertood require them to be massless and there are tight experimental limits on their rest masses.
