Introduction to tensor calculus for general relativity

427 Pages·2000·3.174 MB·English

by  Bertschinger E.

Massachusetts Institute of Technology

Department of Physics

Physics (cid:0)(cid:1)(cid:2)(cid:3)(cid:4) Spring (cid:4)(cid:5)(cid:5)(cid:5)

Introduction to Tensor Calculus for

General Relativity
c (cid:4)(cid:5)(cid:5)(cid:5) Edmund Bertschinger(cid:1)
(cid:0)

(cid:0) Introduction

There arethree essentialideas underlyinggeneralrelativity(cid:6)GR(cid:7)(cid:1) The(cid:8)rst isthat space(cid:9)

time may be described as a curved(cid:10) four(cid:9)dimensional mathematical structure called a

pseudo(cid:9)Riemannian manifold(cid:1) In brief(cid:10) time and space together comprise a curved four(cid:9)

dimensional non(cid:9)Euclidean geometry(cid:1) Consequently(cid:10) the practitioner of GR must be

familiar with the fundamental geometrical properties of curved spacetime(cid:1) In particu(cid:9)

lar(cid:10) the laws of physics must be expressed in a form that is valid independently of any

coordinate system used to label points in spacetime(cid:1)

The second essential idea underlying GR is that at every spacetime point there exist

locallyinertialreferenceframes(cid:10)correspondingtolocally(cid:11)at coordinatescarriedbyfreely

falling observers(cid:10) in which the physics of GR is locally indistinguishable from that of

special relativity(cid:1) This is Einstein(cid:12)s famous strong equivalence principle and it makes

general relativity an extension of special relativity to a curved spacetime(cid:1) The third key

idea is that mass (cid:6)as well as mass and momentum (cid:11)ux(cid:7) curves spacetime in a manner

described by the tensor (cid:8)eld equations of Einstein(cid:1)

These three ideas are exempli(cid:8)edby contrasting GR with Newtonian gravity(cid:1) In the

Newtonian view(cid:10) gravity is a force accelerating particles through Euclidean space(cid:10) while

time is absolute(cid:1) From the viewpoint of GR as a theory of curved spacetime(cid:10) there is no

gravitational force(cid:1) Rather(cid:10) in the absence of electromagnetic and other forces(cid:10) particles

follow the straightest possible paths (cid:6)geodesics(cid:7) through a spacetime curved by mass(cid:1)

Freely falling particles de(cid:8)ne locally inertial reference frames(cid:1) Time and space are not

absolute but are combined into the four(cid:9)dimensional manifold called spacetime(cid:1)

In special relativity there exist global inertial frames(cid:1) This is no longer true in the
presence of gravity(cid:1) Ho w ever(cid:10) there are local inertial frames in GR(cid:10) such that within a

(cid:13)
suitably small spacetime volume around an event (cid:6)just how small is discussed e(cid:1)g(cid:1) in

MTW Chapter (cid:13)(cid:7)(cid:10) one may choose coordinates corresponding to a nearly(cid:9)(cid:11)at spacetime(cid:1)
Thus(cid:10) the local properties of special relativity c a r ry ov er to GR(cid:1) The mathematics of

vectorsand tensors appliesin GR muchas it does inSR(cid:10) with the restrictionthat vectors

and tensors are de(cid:8)ned independently at each spacetime event (cid:6)or within a su(cid:14)ciently

small neighborhood so that the spacetime is sensibly (cid:11)at(cid:7)(cid:1)

Working with GR(cid:10) particularly with the Einstein (cid:8)eld equations(cid:10) requires some un(cid:9)

derstanding of di(cid:15)erential geometry(cid:1) In these notes we will develop the essential math(cid:9)

ematics needed to describe physics in curved spacetime(cid:1) Many physicists receive their

introduction to this mathematics in the excellent book of Weinberg (cid:6)(cid:13)(cid:2)(cid:16)(cid:4)(cid:7)(cid:1) Weinberg

minimizes the geometrical content of the equations by representing tensors using com(cid:9)

ponent notation(cid:1) We believe that it is equally easy to work with a more geometrical

description(cid:10) with the additional bene(cid:8)t that geometrical notation makes it easier to dis(cid:9)

tinguish physical results that are true in any coordinate system (cid:6)e(cid:1)g(cid:1)(cid:10) those expressible

using vectors(cid:7) from those that are dependent on the coordinates(cid:1) Because the geometry

of spacetime is so intimately related to physics(cid:10) we believe that it is better to highlight

the geometryfrom the outset(cid:1) In fact(cid:10) using a geometricalapproach allows us to develop

the essential di(cid:15)erential geometry as an extension of vector calculus(cid:1) Our treatment

is closer to that Wald (cid:6)(cid:13)(cid:2)(cid:0)(cid:17)(cid:7) and closer still to Misner(cid:10) Thorne and Wheeler (cid:6)(cid:13)(cid:2)(cid:16)(cid:18)(cid:10)

MTW(cid:7)(cid:1) These books are rather advanced(cid:1) For the newcomer to general relativity we

warmly recommend Schutz (cid:6)(cid:13)(cid:2)(cid:0)(cid:19)(cid:7)(cid:1) Our notation and presentation is patterned largely

after Schutz(cid:1) It expands on MTW Chapters (cid:4)(cid:10) (cid:18)(cid:10) and (cid:0)(cid:1) The student wishing addi(cid:9)
tional practice problems in GR should consult Lightman et al(cid:0) (cid:6)(cid:13)(cid:2)(cid:16)(cid:19)(cid:7)(cid:1) A slightly more

advanced mathematical treatment is provided in the excellent notes of Carroll (cid:6)(cid:13)(cid:2)(cid:2)(cid:16)(cid:7)(cid:1)

These notes assume familiarity with special relativity(cid:1) We will adopt units in which
the speed of light c (cid:20) (cid:13) (cid:1) Greek indices (cid:6)(cid:0)(cid:10) (cid:1)(cid:10) etc(cid:1)(cid:10) which t a ke the range (cid:5)(cid:2) (cid:13)(cid:2) (cid:4)(cid:2) (cid:18) (cid:7)
f g
will be used to represent components of tensors(cid:1) The Einstein summation convention

is assumed(cid:21) repeated upper and lower indices are to be summed over their ranges(cid:10)
e(cid:1)g(cid:1)(cid:10) A(cid:0)B(cid:0) A(cid:0)B(cid:0) (cid:22) A(cid:1)B(cid:1) (cid:22) A(cid:2)B(cid:2) (cid:22) A(cid:3)B(cid:3)(cid:1) Four(cid:9)vectors will be represented with
(cid:1)
an arrow ov er the symbol(cid:10) e(cid:1)g(cid:1)(cid:10) A(cid:3)(cid:10) while one(cid:9)forms will be represented using a tilde(cid:10)
e(cid:1)g(cid:1)(cid:10) B(cid:23)(cid:1) Spacetime points will be denoted in boldface type(cid:24) e(cid:1)g(cid:1)(cid:10) x refers to a point
with coordinates x(cid:0) (cid:1) Our metric has signature (cid:22)(cid:4)(cid:24) the (cid:11)at spacetime Minkowski metric
components are (cid:4)(cid:0)(cid:1) (cid:20) diag(cid:6) (cid:13)(cid:2) (cid:22)(cid:13)(cid:2) (cid:22)(cid:13)(cid:2) (cid:22)(cid:13)(cid:7)(cid:1)
(cid:2)

(cid:1) Vectors and one(cid:2)forms

The essential mathematics of general relativity is di(cid:15)erential geometry(cid:10) the branch of

mathematics dealing with smoothly curved surfaces (cid:6)di(cid:15)erentiable manifolds(cid:7)(cid:1) The

physicist does not need to master all of the subtleties of di(cid:15)erential geometry in order

(cid:4)
to use general relativity(cid:1) (cid:6)For those readers who want a deeper exposure to di(cid:15)erential

geometry(cid:10) see the introductory texts of Lovelock and Rund (cid:13)(cid:2)(cid:16)(cid:19)(cid:10) Bishop and Goldberg

(cid:13)(cid:2)(cid:0)(cid:5)(cid:10) or Schutz (cid:13)(cid:2)(cid:0)(cid:5)(cid:1)(cid:7) It is su(cid:14)cient to develop the needed di(cid:15)erential geometry as a

straightforward extension of linear algebra and vector calculus(cid:1) However(cid:10) it is important

to keep in mind the geometrical interpretation of physical quantities(cid:1) For this reason(cid:10)

we will not shy from using abstract concepts like points(cid:10) curves and vectors(cid:10) and we will
distinguish between a vector A(cid:3) and its components A(cid:0) (cid:1) U nl i ke some other authors (cid:6)e(cid:1)g(cid:1)(cid:10)

Weinberg (cid:13)(cid:2)(cid:16)(cid:4)(cid:7)(cid:10) we will introduce geometrical objects in a coordinate(cid:9)free manner(cid:10) only

later introducing coordinates for the purpose of simplifying calculations(cid:1) This approach

requires that we distinguish vectors from the related objects called one(cid:9)forms(cid:1) Once

the di(cid:15)erences and similarities between vectors(cid:10) one(cid:9)forms and tensors are clear(cid:10) we will

adopt a uni(cid:8)ed notation that makes computations easy(cid:1)

(cid:0)(cid:1)(cid:2) Vectors

We begin with vectors(cid:1) A vector is a quantity with a magnitude and a direction(cid:1) This

primitiveconcept(cid:10)familiarfromundergraduate physicsand mathematics(cid:10)appliesequally
in general relativity(cid:1) An example of a vector is d(cid:3)x(cid:10) the di(cid:15)erence vector between two

in(cid:8)nitesimally close points of spacetime(cid:1) Vectors form a linear algebra (cid:6)i(cid:1)e(cid:1)(cid:10) a vector
space(cid:7)(cid:1) If A(cid:3) is a vector and a is a real number (cid:6)scalar(cid:7) then aA(cid:3) is a vector with the
same direction (cid:6)or the opposite direction(cid:10) if a (cid:5) (cid:5)(cid:7) whose length is multiplied by a (cid:1) If
j j
A(cid:3) and B(cid:3) are vectors then so is A(cid:3) (cid:22) B(cid:3)(cid:1) These results are as valid for vectors in a curved

four(cid:9)dimensional spacetimeas they are for vectors in three(cid:9)dimensionalEuclidean space(cid:1)

Note that we have introduced vectors without mentioning coordinates or coordinate

transformations(cid:1) Scalars and vectors are invariant under coordinate transformations(cid:24)
vector components are not(cid:1) The whole point of writing the laws of physics (cid:6)e(cid:1)g(cid:1)(cid:10) F(cid:3) (cid:20) m(cid:3)a(cid:7)

using scalars and vectors is that these laws do not depend on the coordinate system

imposed by the physicist(cid:1)
We denote a spacetime point using a boldface symbo l (cid:21) x(cid:1) (cid:6)This notation is not meant
to imply coordinates(cid:1)(cid:7) Note that x refers to a point(cid:10) not a vector(cid:1) In a curved spacetime
the concept of a radius vector (cid:3)x pointing from some origin to each p oi nt x is not useful

because vectors de(cid:8)ned at two di(cid:15)erent points cannot be added straightforwardly as

they can in Euclidean space(cid:1) For example(cid:10) consider a sphere embedded in ordinary

three(cid:9)dimensional Euclidean space (cid:6)i(cid:1)e(cid:1)(cid:10) a two(cid:9)sphere(cid:7)(cid:1) A vector pointing east at one

point on the equator is seen to point radially outward at another point on the equator
�

whose longitude is greater by (cid:2)(cid:5) (cid:1) The radially outward direction is unde(cid:8)ned on the

sphere(cid:1)
Technically(cid:10) we are discussing tangent vectors that lie in the tangent space of the

manifold at each point(cid:1) For example(cid:10) a sphere may be embedded in a three(cid:9)dimensional

Euclideanspaceintowhichmaybeplacedaplanetangenttothesphereatapoint(cid:1) Atwo(cid:9)

(cid:18)
dimensional vector space exists at the point of tangency(cid:1) However(cid:10) such an embedding

is not required to de(cid:8)ne the tangent space of a manifold (cid:6)Wald (cid:13)(cid:2)(cid:0)(cid:17)(cid:7)(cid:1) As long as the

space is smooth (cid:6)as assumed in the formal de(cid:8)nition of a manifold(cid:7)(cid:10)the di(cid:15)erence vector
d(cid:3)x be tw een two in(cid:8)nitesimally close points may be de(cid:8)ned(cid:1) The set of all d(cid:3)x de(cid:8)nes
the tangent space at x(cid:1) By assigning a tangent ve ctor to every spacetime point(cid:10) we

can recover the usual concept of a vector (cid:8)eld(cid:1) However(cid:10) without additional preparation

one cannot compare vectors at di(cid:15)erent spacetime points(cid:10) because they lie in di(cid:15)erent

tangent spaces(cid:1) In later notes we introduce will parallel transport as a means of making
this comparison(cid:1) Until then(cid:10) we consider only tangent ve ctors at x(cid:1) To emphasize the
status of a tangent ve ctor(cid:10) we will occasionally use a subscript notation(cid:21) A(cid:3)X (cid:1)

(cid:0)(cid:1)(cid:0) One(cid:3)forms and dual vector space

Nextweintroduceone(cid:9)forms(cid:1) Aone(cid:9)formisde(cid:8)nedas alinearscalarfunctionofavector(cid:1)
That is(cid:10) a one(cid:9)form takes a vector as input and outputs a scalar(cid:1) For the one(cid:9)form P(cid:23)(cid:10)
P(cid:23)(cid:6)V(cid:3)(cid:7) is also called the scalar product and may be denoted using angle brackets(cid:21)
P(cid:23)(cid:6)V(cid:3)(cid:7) (cid:20) P(cid:23)(cid:2)V(cid:3) (cid:6) (cid:6)(cid:13)(cid:7)
h i
The one(cid:9)form is a linear function(cid:10) meaning that for all scalars a and b and vectors V(cid:3) and
W(cid:3) (cid:10) the one(cid:9)form P(cid:23) satis(cid:8)es the following relations(cid:21)
P(cid:23)(cid:6)aV(cid:3) (cid:22) bW(cid:3) (cid:7) (cid:20) P(cid:23)(cid:2)aV(cid:3) (cid:22) bW(cid:3) (cid:20) a P(cid:23)(cid:2)V(cid:3) (cid:22) b P(cid:23)(cid:2)W(cid:3) (cid:20) aP(cid:23)(cid:6)V(cid:3)(cid:7)(cid:22) bP(cid:23)(cid:6)W (cid:3) (cid:7) (cid:6) (cid:6)(cid:4)(cid:7)
h i h i h i
Just as we m ay consider any function f(cid:6) (cid:7) as a mathematical entity independently of
any particular argument(cid:10) we m ay consider the one(cid:9)form P(cid:23) independently of any particular
vector V(cid:3)(cid:1) We m ay also associate a one(cid:9)form with each spacetime point(cid:10) resulting in a
one(cid:9)form (cid:8)eld P(cid:23) (cid:20) P(cid:23)X (cid:1) No w the distinction between a point a ve ctor is crucial(cid:21) P(cid:23)X is
a one(cid:9)form at point x while P(cid:23)(cid:6)V(cid:3)(cid:7) is a scalar(cid:10) de(cid:8)ned implicitly at point x(cid:1) The scalar
product notation with subscripts makes this more clear(cid:21) P(cid:23)X (cid:2)V(cid:3)X (cid:1)
h i
One(cid:9)forms obey their own linear algebra distinct fromthat of vectors(cid:1) Givenany two
scalars a and b and one(cid:9)forms P(cid:23) and Q(cid:23)(cid:10) we m ay de(cid:8)ne the one(cid:9)form aP(cid:23) (cid:22) bQ(cid:23) by
(cid:6)aP(cid:23) (cid:22) bQ(cid:23)(cid:7)(cid:6)V(cid:3)(cid:7) (cid:20) aP(cid:23) (cid:22) bQ(cid:23)(cid:2)V(cid:3) (cid:20) a P(cid:23)(cid:2)V(cid:3) (cid:22) b Q (cid:23)(cid:2)V(cid:3) (cid:20) aP(cid:23)(cid:6)V(cid:3)(cid:7)(cid:22) bQ(cid:23)(cid:6)V(cid:3)(cid:7) (cid:6) (cid:6)(cid:18)(cid:7)
h i h i h i

Comparing equations (cid:6)(cid:4)(cid:7) and (cid:6)(cid:18)(cid:7)(cid:10) we see that vectors and one(cid:9)forms are linear operators

on each other(cid:10) producing scalars(cid:1) It is often helpful to consider a vector as being a linear
scalar function of a one(cid:9)form(cid:1) Thus(cid:10) we ma y w ri t e P(cid:23)(cid:2)V(cid:3) (cid:20) P(cid:23)(cid:6)V(cid:3)(cid:7) (cid:20) V(cid:3)(cid:6)P(cid:23)(cid:7)(cid:1) The set of
h i
all one(cid:9)forms is a vector space distinct from(cid:10) but complementary to(cid:10) the linear vector
space of vectors(cid:1) The vector space of one(cid:9)forms is called the dual vector (cid:6)or cotangent(cid:7)

space to distinguish it from the linear space of vectors (cid:6)tangent space(cid:7)(cid:1)

(cid:17)
Although one(cid:9)forms may appear to be highly abstract(cid:10) the concept of dual vector

spaces is familiar to any student of quantum mechanics who has seen the Dirac bra(cid:9)ket

notation(cid:1) Recall that the fundamental object in quantum mechanics is the state vector(cid:10)
represented by a ke t (cid:7) in a linear vector space (cid:6)Hilbert space(cid:7)(cid:1) A distinct Hilbert
space is given by the sjet iof bra vectors (cid:8) (cid:1) Bra vectors and ket vectors are linear scalar
functions of each other(cid:1) The scalar prodhucjt (cid:8) (cid:7) maps a bra vector and a ket vector to a
h j i

scalar called a probability amplitude(cid:1) The distinction between bras and kets is necessary

because probability amplitudes are complex numbers(cid:1) As we will see(cid:10) the distinction

between vectors and one(cid:9)forms is necessary because spacetime is curved(cid:1)

(cid:3) Tensors
Having de(cid:8)ned vectors and one(cid:9)forms we can now de(cid:8)ne tensors(cid:1) A tensor of rank (cid:6)m(cid:2) n(cid:7)(cid:10)
also called a (cid:6)m(cid:2) n(cid:7) tensor(cid:10) is de(cid:8)ned to be a scalar function of m one(cid:9)forms and n vectors

that is linear in all of its arguments(cid:1) It follows at once that scalars are tensors of rank
(cid:6)(cid:5)(cid:2) (cid:5)(cid:7)(cid:10) vectors are tensors of rank (cid:6)(cid:13)(cid:2) (cid:5)(cid:7) and one(cid:9)forms are tensors of rank (cid:6)(cid:5)(cid:2) (cid:13)(cid:7)(cid:1) We
may denote a tensor of rank (cid:6)(cid:4)(cid:2) (cid:5)(cid:7) by T(cid:6)P(cid:23) (cid:2)Q(cid:23)(cid:7)(cid:24) one of rank (cid:6)(cid:4)(cid:2) (cid:13)(cid:7) by T(cid:6)P(cid:23) (cid:2) Q(cid:23)(cid:2)A(cid:3)(cid:7)(cid:10) etc(cid:1)
Our notation will not distinguish a (cid:6)(cid:4)(cid:2) (cid:5)(cid:7) tensor T from a (cid:6)(cid:4)(cid:2) (cid:13)(cid:7) tensor T(cid:10) although a
notational distinction could be made by placing m arrows and n tildes over the symbo l(cid:10)

or by appropriate use of dummy indices (cid:6)Wald (cid:13)(cid:2)(cid:0)(cid:17)(cid:7)(cid:1)
The scalar product is a tensor of rank (cid:6)(cid:13)(cid:2) (cid:13)(cid:7)(cid:10) which we will denote I and call the

identity tensor(cid:21)
I(cid:6)P(cid:23) (cid:2)V(cid:3) (cid:7) P (cid:23) (cid:2)V(cid:3) (cid:20) P(cid:23) (cid:6)V(cid:3) (cid:7) (cid:20) V(cid:3) (cid:6)P(cid:23)(cid:7) (cid:6) (cid:6)(cid:17)(cid:7)
(cid:1) h i
We call I the identity because(cid:10) when applied to a (cid:8)xed one(cid:9)form P(cid:23) and any vector V(cid:3) (cid:10) it
returns P(cid:23)(cid:6)V(cid:3)(cid:7)(cid:1) Although the identity t e ns o r wa s de(cid:8)ned as a mapping from a one(cid:9)form

and a vector to a scalar(cid:10) we see that it may equally be interpreted as a mapping from a
one(cid:9)form to the same one(cid:9)form(cid:21) I(cid:6)P(cid:23) (cid:2) (cid:7) (cid:20) P(cid:23)(cid:10) where the dot indicates that an argument
(cid:6)a vector(cid:7) is needed to give a scalar(cid:1) A(cid:3) similar argument shows that I may be considered
the identity operator on the space of vectors V(cid:3) (cid:21) I(cid:6) (cid:2)V(cid:3) (cid:7) (cid:20) V(cid:3)(cid:1)
(cid:3)
A tensor of rank (cid:6)m(cid:2) n(cid:7) is linear in all its arguments(cid:1) For example(cid:10) for (cid:6)m (cid:20) (cid:4) (cid:2)n (cid:20) (cid:5)(cid:7)

we have a straightforward extension of equation (cid:6)(cid:4)(cid:7)(cid:21)
T(cid:6)aP(cid:23) (cid:22) bQ (cid:23)(cid:2) c R(cid:23) (cid:22) dS(cid:23)(cid:7) (cid:20) ac T(cid:6)P(cid:23) (cid:2)R(cid:23)(cid:7)(cid:22) ad T(cid:6)P(cid:23) (cid:2)S(cid:23)(cid:7)(cid:22) bc T(cid:6)Q(cid:23)(cid:2)R(cid:23)(cid:7)(cid:22) bd T(cid:6) q(cid:23)(cid:2)S(cid:23)(cid:7) (cid:6) (cid:6)(cid:19)(cid:7)

Tensors of a given rank form a linear algebra(cid:10) meaning that a linear combination of
tensors of rank (cid:6)m(cid:2) n(cid:7) is also a tensor of rank (cid:6)m(cid:2) n(cid:7)(cid:10) de(cid:8)ned by straightforward extension

of equation (cid:6)(cid:18)(cid:7)(cid:1) Two tensors (cid:6)of the same rank(cid:7) are equal if and only if they return the

samescalarwhen appliedto allpossible input vectorsand one(cid:9)forms(cid:1) Tensors of di(cid:15)erent

rank cannot be added or compared(cid:10) so it is important to keep track of the rank of each

(cid:19)
tensor(cid:1) Just as in the case of scalars(cid:10) vectors and one(cid:9)forms(cid:10) tensor (cid:8)elds TX are de(cid:8)ned

by associating a tensor with each spacetime point(cid:1)

There are three ways to change the rank of a tensor(cid:1) The (cid:8)rst(cid:10) called the tensor (cid:6)or
outer(cid:7) product(cid:10) combines two tensors of ranks (cid:6)m(cid:1)(cid:2)n (cid:1)(cid:7) and (cid:6) m(cid:2)(cid:2)n (cid:2)(cid:7) to form a tensor
of rank (cid:6)m(cid:1) (cid:22) m(cid:2)(cid:2)n(cid:1) (cid:22) n(cid:2)(cid:7) by simply combining the argument lists of the two tensors

and thereby expanding the dimensionality of the tensor space(cid:1) For example(cid:10) the tensor
product of two ve ctors A(cid:3) and B(cid:3) gives a rank (cid:6)(cid:4)(cid:2) (cid:5)(cid:7) tensor
T (cid:20) A(cid:3) B(cid:3) (cid:2) T(cid:6)P(cid:23) (cid:2)Q(cid:23)(cid:7) A(cid:3)(cid:6)P(cid:23)(cid:7) B(cid:3)(cid:6)Q(cid:23)(cid:7) (cid:6) (cid:6)(cid:3)(cid:7)
(cid:4) (cid:1)

We use the symbol to denote the tensor product(cid:24) later we will drop this symbol for
(cid:4)
notational conveniencewhen it is clear fromthe contextthat a tensor product is implied(cid:1)
Note that the tensor product is non(cid:9)commutative(cid:21) A(cid:3) B(cid:3) (cid:20) B(cid:3) A(cid:3) (cid:6)unless B(cid:3) (cid:20) cA(cid:3) for
(cid:4) (cid:5) (cid:4)
some scalar c(cid:7) because A(cid:3)(cid:6)P(cid:23)(cid:7) B(cid:3)(cid:6)Q(cid:23)(cid:7) (cid:20) A(cid:3)(cid:6)Q(cid:23)(cid:7) B(cid:3)(cid:6)P(cid:23)(cid:7) for all P(cid:23) and Q(cid:23)(cid:1) We use the symbo l
(cid:5) (cid:4)
to denote the tensor product of any tw o tensors(cid:10) e(cid:1)g(cid:1)(cid:10) P(cid:23) T (cid:20) P(cid:23) A(cid:3) B(cid:3) is a tensor
of rank (cid:6)(cid:4)(cid:2) (cid:13)(cid:7)(cid:1) The second way t o ch ange the rank of a te(cid:4)nsor is by(cid:4) c on (cid:4)t raction(cid:10) which
reduces the rank of a (cid:6)m(cid:2) n(cid:7) tensor to (cid:6)m (cid:13)(cid:2)n (cid:13)(cid:7)(cid:1) The third way is the gradient(cid:1) We
(cid:2) (cid:2)

will discuss contraction and gradients later(cid:1)

(cid:4)(cid:1)(cid:2) Metric tensor

The scalar product (cid:6)eq(cid:1) (cid:13)(cid:7) requires a vector and a one(cid:9)form(cid:1) Is it possible to obtain a

scalar from two vectors or two one(cid:9)forms(cid:25) From the de(cid:8)nition of tensors(cid:10) the answer is
clearly yes(cid:1) Any tensor of rank (cid:6)(cid:5)(cid:2) (cid:4)(cid:7) will give a scalar from two ve ctors and any tensor
of rank (cid:6)(cid:4)(cid:2) (cid:5)(cid:7) combines two one(cid:9)forms to give a scalar(cid:1) However(cid:10) there is a particular
(cid:6)(cid:5)(cid:2) (cid:4)(cid:7) tensor (cid:8)eld gX called the metric tensor and a related (cid:6)(cid:4)(cid:2) (cid:5)(cid:7) tensor (cid:8)eld g;X (cid:1) called

the inverse metric tensor for which special distinction is reserved(cid:1) The metric tensor is
a symmetric bilinear scalar function of two ve ctors(cid:1) That is(cid:10) given vectors V(cid:3) and W(cid:3) (cid:10) g

returns a scalar(cid:10) called the dot product(cid:21)
g(cid:6)V(cid:3)(cid:2) W(cid:3) (cid:7) (cid:20) V(cid:3) W(cid:3) (cid:20) W(cid:3) V(cid:3) (cid:20) g(cid:6)W(cid:3) (cid:2) V(cid:3) (cid:7) (cid:6) (cid:6)(cid:16)(cid:7)
(cid:3) (cid:3)
Similarly(cid:10) g; (cid:1) returns a scalar from two one(cid:9)forms P(cid:23) and Q(cid:23)(cid:10) w h ich we also call the dot

product(cid:21)
g; (cid:1)(cid:6)P(cid:23) (cid:2)Q(cid:23)(cid:7) (cid:20) P(cid:23) Q(cid:23) (cid:20) P(cid:23) Q(cid:23) (cid:20) g ;(cid:1)(cid:6)P(cid:23) (cid:2)Q(cid:23)(cid:7) (cid:6) (cid:6)(cid:0)(cid:7)
(cid:3) (cid:3)
Although a dot is used in both cases(cid:10) it should be clear from the context whether g or g ;(cid:1)

is implied(cid:1) We reservethe dot product notation for the metricand inversemetrictensors

just as we reserve the angle brackets scalar product notation for the identity tensor (cid:6)eq(cid:1)
(cid:17)(cid:7)(cid:1) Later (cid:6)in eq(cid:1) (cid:17)(cid:13)(cid:7) we will see what distinguishes g from other (cid:6)(cid:5)(cid:2) (cid:4)(cid:7) tensors and g ;(cid:1)
from other symmetric (cid:6)(cid:4)(cid:2) (cid:5)(cid:7) tensors(cid:1)

(cid:3)
One of the most important properties of the metric is that it allows us to convert
vectors to one(cid:9)forms(cid:1) If we forget to include W(cid:3) in equation (cid:6)(cid:16)(cid:7) we get a quantity(cid:10) denoted
V(cid:23)(cid:10) that behaves like a one(cid:9)form(cid:21)
V(cid:23)(cid:6) (cid:7) g(cid:6)V(cid:3)(cid:2) (cid:7) (cid:20) g(cid:6) (cid:2)V(cid:3)(cid:7) (cid:2) (cid:6)(cid:2)(cid:7)
(cid:3) (cid:1) (cid:3) (cid:3)

where we have inserted a dot to remind ourselves that a vector must be inserted to give

a scalar(cid:1) (cid:6)Recall that a one(cid:9)form is a scalar function of a vector(cid:26)(cid:7) We use the same letter
to indicate the relation of V(cid:3) and V(cid:23)(cid:1)
Thus(cid:10) the metric g is a mapping from the space of vectors to the space of one(cid:9)forms(cid:21)
g (cid:21) V(cid:3) V(cid:23)(cid:1) By de(cid:8)nition(cid:10) the inverse metric g ;(cid:1) is the inverse mapping(cid:21) g; (cid:1) (cid:21) V(cid:23) V(cid:3)(cid:1)
(cid:6)The i(cid:6)nverse always exists for nonsingular spacetimes(cid:1)(cid:7) Thus(cid:10) if V(cid:23) is de(cid:8)ned for a(cid:6)ny V(cid:3)

by equation (cid:6)(cid:2)(cid:7)(cid:10) the inverse metric tensor is de(cid:8)ned by
V(cid:3)(cid:6) (cid:7) g ;(cid:1)(cid:6)V(cid:23)(cid:2) (cid:7) (cid:20) g; (cid:1)(cid:6) (cid:2)V(cid:23)(cid:7) (cid:6) (cid:6)(cid:13)(cid:5)(cid:7)
(cid:3) (cid:1) (cid:3) (cid:3)

Equations (cid:6)(cid:17)(cid:7) and (cid:6)(cid:16)(cid:7)(cid:27)(cid:6)(cid:13)(cid:5)(cid:7) give us several equivalentways to obtain scalars from vectors
V(cid:3) and W(cid:3) and their associated one(cid:9)forms V(cid:23) and W(cid:23) (cid:21)
V(cid:23) (cid:2)W(cid:3) (cid:20) W(cid:23) (cid:2)V(cid:3) (cid:20) V(cid:3) W (cid:3) (cid:20) V(cid:23) W (cid:23) (cid:20) I(cid:6)V(cid:23) (cid:2)W(cid:3) (cid:7) (cid:20) I(cid:6)W(cid:23) (cid:2)V(cid:3)(cid:7) (cid:20) g(cid:6)V(cid:3)(cid:2) W(cid:3) (cid:7) (cid:20) g ;(cid:1)(cid:6)V(cid:23) (cid:2)W(cid:23) (cid:7) (cid:6) (cid:6)(cid:13)(cid:13)(cid:7)
h i h i (cid:3) (cid:3)

(cid:4)(cid:1)(cid:0) Basis vectors and one(cid:3)forms

Itispossibletoformulatethemathematicsofgeneralrelativityentirelyusingtheabstract

formalismofvectors(cid:10)formsandtensors(cid:1) However(cid:10)whilethegeometrical(cid:6)coordinate(cid:9)free(cid:7)

interpretationof quantitiesshould always be keptinmind(cid:10)theabstract approach often is

not the most practical way to perform calculations(cid:1) To simplifycalculations it is helpful

to introduce a set of linearly independent basis vector and one(cid:9)form (cid:8)elds spanning

our vector and dual vector spaces(cid:1) In the same way(cid:10) practical calculations in quantum

mechanics often start by expanding the ket vector in a set of basis kets(cid:10) e(cid:1)g(cid:1)(cid:10) energy

eigenstates(cid:1) By de(cid:8)nition(cid:10) the dimensionality of spacetime (cid:6)four(cid:7) equals the number of

linearly independent basis vectors and one(cid:9)forms(cid:1)
We denote our set of basis vector (cid:8)elds by (cid:3)e(cid:0) X (cid:10) where (cid:0) labels the basis vector
(cid:6)e(cid:1)g(cid:1)(cid:10) (cid:0) (cid:20) (cid:5) (cid:2)(cid:13)(cid:2)(cid:4)(cid:2)(cid:18)(cid:7) and x labels the spacetime pofint(cid:1) Agny four linearly independent basis

vectors at each spacetime point will work(cid:24) we do not not impose orthonormality or any

other conditions in general(cid:10) nor have we implied any relation to coordinates (cid:6)although
later we will(cid:7)(cid:1) Given a basis(cid:10) we ma y expand any ve ctor (cid:8)eld A(cid:3) as a linear combination

of basis vectors(cid:21)
A(cid:3)X (cid:20) A(cid:0)X (cid:3)e(cid:0) X (cid:20) A(cid:0)X (cid:3)e(cid:0) X (cid:22) A(cid:1)X (cid:3)e(cid:1) X (cid:22) A(cid:2)X (cid:3)e(cid:2) X (cid:22) A(cid:3)X (cid:3)e(cid:3) X (cid:6) (cid:6)(cid:13)(cid:4)(cid:7)

(cid:16)
Note our placement of subscripts and superscripts(cid:10) chosen for consistency with the Ein(cid:9)

stein summation convention(cid:10) which requires pairing one subscript with one superscript(cid:1)
The coe(cid:14)cients A(cid:0) are called the components of the vector (cid:6)often(cid:10) the contravariant
components(cid:7)(cid:1) Note well that the coe(cid:14)cients A(cid:0) depend on the basis vectors but A(cid:3) does

not(cid:26)

Similarly(cid:10) we may choose a basis of one(cid:9)form (cid:8)elds in which to expand one(cid:9)forms
like A(cid:23)X(cid:1) Although any set of four linearly independent one(cid:9)forms will su(cid:14)ce for each
spacetime point(cid:10) we prefer to choose a special one(cid:9)form basis called the dual basis and
(cid:0)
denoted e(cid:23)X (cid:1) Note that the placement of subscripts and superscripts is signi(cid:8)cant(cid:24)
f g
we never use a subscript to label a basis one(cid:9)form while we never use a superscript
to label a basis vector(cid:1) Therefore(cid:10) e(cid:23)(cid:0) is not related to (cid:3)e(cid:0) through the metric (cid:6)eq(cid:1) (cid:2)(cid:7)(cid:21)
e(cid:23)(cid:0)(cid:6) (cid:7) (cid:20) g(cid:6)(cid:3)e(cid:0)(cid:2) (cid:7)(cid:1) Rather(cid:10) the dual basis one(cid:9)forms are de(cid:8)ned by imposing the following
(cid:3) (cid:5) (cid:3)
(cid:13)(cid:3) requirements at each spacetime point(cid:21)
e(cid:23)(cid:0)X (cid:2) (cid:3)e(cid:1) X (cid:20) (cid:9)(cid:0)(cid:1) (cid:2) (cid:6)(cid:13)(cid:18)(cid:7)
h i
where (cid:9)(cid:0)(cid:1) is the Kronecker delta(cid:10) (cid:9)(cid:0)(cid:1) (cid:20) (cid:13) if (cid:0) (cid:20) (cid:1) and (cid:9)(cid:0)(cid:1) (cid:20) (cid:5) otherwise(cid:10) with the

same values for each spacetime point(cid:1) (cid:6)We must always distinguish subscripts from

superscripts(cid:24) the Kronecker delta always has one of each(cid:1)(cid:7) Equation (cid:6)(cid:13)(cid:18)(cid:7) is a system of
four linear equations at each spacetime point for each of the four quantities e(cid:23)(cid:0) and it

has a unique solution(cid:1) (cid:6)The reader may show that any nontrivial transformation of the
dual basis one(cid:9)forms will violate eq(cid:1) (cid:13)(cid:18)(cid:1)(cid:7) Now we may expand any one(cid:9)form (cid:8)eld P(cid:23)X in

the basis of one(cid:9)forms(cid:21)
(cid:0)
P(cid:23)X (cid:20) P(cid:0) X e(cid:23) X (cid:6) (cid:6)(cid:13)(cid:17)(cid:7)
The component P(cid:0) of the one(cid:9)form P(cid:23) is often called the covariant component to distin(cid:9)
guish it from the contravariant component P (cid:0) of the vector P(cid:3) (cid:1) In fact(cid:10) because we h av e

consistently treated vectors and one(cid:9)forms as distinct(cid:10) we should not think of these as

being distinct (cid:28)components(cid:28) of the same entity at all(cid:1)

There is a simpleway to get the components of vectors and one(cid:9)forms(cid:10) using the fact

that vectors are scalar functions of one(cid:9)forms and vice versa(cid:1) One simply evaluates the

vector using the appropriate basis one(cid:9)form(cid:21)
A(cid:3)(cid:6) e(cid:23)(cid:0)(cid:7) (cid:20) e(cid:23)(cid:0)(cid:2)A(cid:3) (cid:20) e(cid:23)(cid:0)(cid:2)A(cid:1) (cid:3)e(cid:1) (cid:20) e(cid:23)(cid:0)(cid:2) (cid:3)e(cid:1) A(cid:1) (cid:20) (cid:9)(cid:0)(cid:1) A(cid:1) (cid:20) A(cid:0) (cid:2) (cid:6)(cid:13)(cid:19)(cid:7)
h i h i h i

and conversely for a one(cid:9)form(cid:21)
(cid:1) (cid:1) (cid:1)
P(cid:23)(cid:6)(cid:3)e(cid:0)(cid:7) (cid:20) P(cid:23) (cid:2) (cid:3)e(cid:0) (cid:20) P(cid:1) e(cid:23) (cid:2) (cid:3)e(cid:0) (cid:20) e(cid:23) (cid:2) (cid:3)e(cid:0) P(cid:1) (cid:20) (cid:9) (cid:0)P(cid:1) (cid:20) P(cid:0) (cid:6) (cid:6)(cid:13)(cid:3)(cid:7)
h i h i h i
We h av e suppressed the spacetime point x for clarity(cid:10) but it is always implied(cid:1)

(cid:0)
(cid:4)(cid:1)(cid:4) Tensor algebra

We can use the same ideas to expand tensors as products of components and basis
tensors(cid:1) First we note that a basis for a tensor of rank (cid:6)m(cid:2)n(cid:7) i s pr ov ided by the tensor
product of m vectors and n one(cid:9)forms(cid:1) For example(cid:10) a (cid:6)(cid:5)(cid:2)(cid:4)(cid:7) tensor like the metric tensor
can be decomposed into basis tensors e(cid:23)(cid:0) e(cid:23)(cid:1)(cid:1) T he components of a tensor of rank (cid:6)m(cid:2)n(cid:7)(cid:10)
(cid:4)
labeled with m superscripts and n subscripts(cid:10) are obtained by ev a luating the tensor using
m basis one(cid:9)forms and n basis vectors(cid:1) For example(cid:10) the components of the (cid:6)(cid:5)(cid:2)(cid:4)(cid:7) metric
tensor(cid:10) the (cid:6)(cid:4)(cid:2)(cid:5)(cid:7) inverse metric tensor and the (cid:6)(cid:13)(cid:2)(cid:13)(cid:7) identity tensor are
g(cid:0)(cid:1) g(cid:6)(cid:3)e(cid:0)(cid:2) (cid:3)e(cid:1) (cid:7) (cid:20) (cid:3)e(cid:0) (cid:3)e(cid:1) (cid:2) g (cid:0)(cid:1) g; (cid:1)(cid:6) e(cid:23)(cid:0)(cid:2)e(cid:23)(cid:1)(cid:7) (cid:20) e(cid:23)(cid:0) e(cid:23)(cid:1) (cid:2) (cid:9)(cid:0)(cid:1) (cid:20) I(cid:6) e(cid:23)(cid:0)(cid:2) (cid:3)e(cid:1) (cid:7) (cid:20) e(cid:23)(cid:0) (cid:2) (cid:3)e(cid:1) (cid:6) (cid:6)(cid:13)(cid:16)(cid:7)
(cid:1) (cid:3) (cid:1) (cid:3) h i

(cid:6)The last equation follows from eqs(cid:1) (cid:17) and (cid:13)(cid:18)(cid:1)(cid:7) The tensors are given by summing over

the tensor product of basis vectors and one(cid:9)forms(cid:21)
g (cid:20) g(cid:0)(cid:1) e(cid:23)(cid:0) e(cid:23)(cid:1) (cid:2) g; (cid:1) (cid:20) g (cid:0)(cid:1) (cid:3)e(cid:0) (cid:3)e(cid:1) (cid:2) I (cid:20) (cid:9)(cid:0)(cid:1) (cid:3)e(cid:0) e(cid:23)(cid:1) (cid:6) (cid:6)(cid:13)(cid:0)(cid:7)
(cid:4) (cid:4) (cid:4)

The reader should check that equation (cid:6)(cid:13)(cid:0)(cid:7) follows from equations (cid:6)(cid:13)(cid:16)(cid:7) and the duality

condition equation (cid:6)(cid:13)(cid:18)(cid:7)(cid:1)

Basis vectors and one(cid:9)forms allow us to represent any tensor equations using com(cid:9)

ponents(cid:1) For example(cid:10) the dot product between two vectors or two one(cid:9)forms and the

scalar product between a one(cid:9)form and a vector may be written using components as
A(cid:3) B(cid:3) (cid:20) g(cid:0)(cid:1) A(cid:0)A(cid:1) (cid:2) P(cid:23)(cid:2)A(cid:3) (cid:20) P(cid:0) A(cid:0) (cid:2) P(cid:23) Q(cid:23) (cid:20) g(cid:0)(cid:1) P(cid:0) P(cid:1) (cid:6) (cid:6)(cid:13)(cid:2)(cid:7)
(cid:3) h i (cid:3)

The reader should prove these important results(cid:1)

If two tensors of the same rank are equal in one basis(cid:10) i(cid:1)e(cid:1)(cid:10) if all of their components

are equal(cid:10) then they are equal in any basis(cid:1) While this mathematical result is obvious

from the basis(cid:9)free meaning of a tensor(cid:10) it will have important physical implications in

GR arising from the Equivalence Principle(cid:1)

As we discussed above(cid:10) the metric and inverse metric tensors allow us to transform
vectors into one(cid:9)forms and vice versa(cid:1) If we e va luate the components of V(cid:3) and the
one(cid:9)form V(cid:23) de(cid:8)ned by equations (cid:6)(cid:2)(cid:7) and (cid:6)(cid:13)(cid:5)(cid:7)(cid:10) we ge t
V(cid:0) (cid:20) g(cid:6)(cid:3)e(cid:0)(cid:2)V(cid:3)(cid:7) (cid:20) g(cid:0)(cid:1) V(cid:1) (cid:2) V(cid:0) (cid:20) g ;(cid:1)(cid:6) e(cid:23)(cid:0)(cid:2)V(cid:23)(cid:7) (cid:20) g (cid:0)(cid:1) V(cid:1) (cid:6) (cid:6)(cid:4)(cid:5)(cid:7)
Because these two equations must hold for any ve ctor V(cid:3)(cid:10) we conclude that the matrix
de(cid:8)ned by g(cid:0)(cid:1) is the inverse of the matrix de(cid:8)ned by g(cid:0)(cid:1) (cid:21)
(cid:0)(cid:2) (cid:0)
g g(cid:2)(cid:1) (cid:20) (cid:9) (cid:1) (cid:6) (cid:6)(cid:4)(cid:13)(cid:7)

We also see that the metricand its inverse are used to lower and raise indices on compo(cid:9)
nents(cid:1) Thus(cid:10) given two ve ctors V(cid:3) and W(cid:3) (cid:10) we m ay e va luate the dot product any of four

equivalent ways (cid:6)cf(cid:1) eq(cid:1) (cid:13)(cid:13)(cid:7)(cid:21)
V(cid:3) W(cid:3) (cid:20) g(cid:0)(cid:1) V(cid:0)W(cid:1) (cid:20) V(cid:0) W(cid:0) (cid:20) V(cid:0)W(cid:0) (cid:20) g(cid:0)(cid:1) V(cid:0) W(cid:1) (cid:6) (cid:6)(cid:4)(cid:4)(cid:7)
(cid:3)

(cid:2)
In fact(cid:10) the metric and its inverse may be used to transform tensors of rank (cid:6)m(cid:2) n(cid:7)
into tensors of any rank (cid:6) m (cid:22) k (cid:2)n k(cid:7) where k (cid:20) m(cid:2) m (cid:22) (cid:13) (cid:2)(cid:6)(cid:6)(cid:6)(cid:2)n (cid:1) Consider(cid:10) for
example(cid:10) a (cid:6)(cid:13)(cid:2) (cid:4)(cid:7) tensor T with com(cid:2)ponents (cid:2) (cid:2)
(cid:0) (cid:0)
T (cid:1)(cid:3) T(cid:6) e(cid:23) (cid:2) (cid:3)e(cid:1) (cid:2) (cid:3)e(cid:3)(cid:7) (cid:6) (cid:6)(cid:4)(cid:18)(cid:7)
(cid:1)
If we fail to plug in the one(cid:9)form e(cid:23)(cid:0)(cid:10) the result is the vector T (cid:2)(cid:1) (cid:3)(cid:3) e(cid:2)(cid:1) (cid:6)A one(cid:9)form must be
inserted to return the quantity T (cid:2)(cid:1) (cid:3) (cid:1)(cid:7) This vector may then be inserted into the metric
tensor to give the components of a (cid:6)(cid:5)(cid:2) (cid:18)(cid:7) tensor(cid:21)
T(cid:0)(cid:1)(cid:3) g(cid:6)(cid:3)e(cid:0)(cid:2)T (cid:2)(cid:1) (cid:3) (cid:3)e(cid:2)(cid:7) (cid:20) g(cid:0)(cid:2) T (cid:2)(cid:1) (cid:3) (cid:6) (cid:6)(cid:4)(cid:17)(cid:7)
(cid:1)

Wecouldnowusetheinversemetrictoraisethethirdindex(cid:10)say(cid:10)givingusthecomponent
of a (cid:6)(cid:13)(cid:2) (cid:4)(cid:7) tensor distinct from equation (cid:6)(cid:4)(cid:18)(cid:7)(cid:21)
T(cid:0)(cid:1) (cid:3) g; (cid:1)(cid:6) e(cid:23)(cid:3)(cid:2)T (cid:0)(cid:1)(cid:2) e(cid:23)(cid:2)(cid:7) (cid:20) g (cid:3)(cid:2) T(cid:0)(cid:1)(cid:2) (cid:20) g(cid:3)(cid:2)g(cid:0)(cid:4) T (cid:4)(cid:1) (cid:2) (cid:6) (cid:6)(cid:4)(cid:19)(cid:7)
(cid:1)
In fact(cid:10) there are (cid:4)m(cid:4)n di(cid:15)erent tensor spaces with ranks summing to m (cid:22) n(cid:1) The metric

or inverse metric tensor allow all of these tensors to be transformed into each other(cid:1)

Returningtoequation(cid:6)(cid:4)(cid:4)(cid:7)(cid:10) weseewhywemustdistinguishvectors(cid:6)withcomponents
V (cid:0) (cid:7) from one(cid:9)forms (cid:6)with components V(cid:0) (cid:7)(cid:1) The scalar product of two ve ctors requires

the metric tensor while that of two one(cid:9)forms requires the inverse metric tensor(cid:1) In
general(cid:10) g(cid:0)(cid:1) (cid:20) g(cid:0)(cid:1) (cid:1) The only case in which the distinction is unnecessary is in (cid:11)at
(cid:5)
(cid:6)Lorentz(cid:7) spacetime with orthonormal Cartesian basis vectors(cid:10) in which cas e g(cid:0)(cid:1) (cid:20) (cid:4)(cid:0)(cid:1)
is everywhere the diagonal matrix with entries (cid:6) (cid:13)(cid:2) (cid:22)(cid:13)(cid:2) (cid:22)(cid:13)(cid:2) (cid:22)(cid:13)(cid:7)(cid:1) However(cid:10) gravity curves
(cid:2)
spacetime(cid:1) (cid:6)Besides(cid:10) we may wish to use curvilinear coordinates even in (cid:11)at spacetime(cid:1)(cid:7)
As a result(cid:10) it is impossible to de(cid:8)ne a coordinate system for which g(cid:0)(cid:1) (cid:20) g(cid:0)(cid:1) everywhere(cid:1)

We must therefore distinguish vectors from one(cid:9)forms and we must be careful about the

placement of subscripts and superscripts on components(cid:1)

At this stage it is useful to introduce a classi(cid:8)cation of vectors and one(cid:9)forms drawn
from special relativity with its Minkowski metric (cid:4)(cid:0)(cid:1) (cid:1) Recall that a vector A(cid:3) (cid:20) A(cid:0)(cid:3)e(cid:0)
is called spacelike(cid:10) timelike o r nu ll according to whether A(cid:3) A(cid:3) (cid:20) (cid:4)(cid:0)(cid:1) A(cid:0)A(cid:1) is positive(cid:10)
(cid:3)
negative or zero(cid:10) respectively(cid:1) In a Euclidean space(cid:10) with positive de(cid:8)nite metric(cid:10) A(cid:3) A(cid:3)
(cid:3)
is never negative(cid:1) However(cid:10) in the Lorentzian spacetime geometry of special relativity(cid:10)
time enters the metric with opposite sign so that it is possible to have A(cid:3) A(cid:3) (cid:5) (cid:5)(cid:1) In
particular(cid:10) the four(cid:9)velocity u(cid:0) (cid:20) dx(cid:0)(cid:10)d(cid:11) of a massive particle (cid:6)where d(cid:11) is p(cid:3)roper time(cid:7)

is a timelike vector(cid:1) This is seen most simply by performing a Lorentz boost to the
rest frame of the particle in which c a s e ut (cid:20) (cid:13)(cid:10) ux (cid:20) uy (cid:20) uz (cid:20) (cid:5) and (cid:4)(cid:0)(cid:1) u(cid:0)u(cid:1) (cid:20) (cid:13)(cid:1)
Now(cid:10) (cid:4)(cid:0)(cid:1) u(cid:0)u(cid:1) is a Lorentz scalar so that u(cid:3) (cid:3)u (cid:20) (cid:13) i n an y Lorentz frame(cid:1) Often thi(cid:2)s is
written p(cid:3) p(cid:3) (cid:20) m(cid:2) where p(cid:0) (cid:20) mu(cid:0) is th(cid:3)e four(cid:2)(cid:9)momentum for a particle of mass m(cid:1)
(cid:3) (cid:2)
For a massless particle(cid:6)e(cid:1)g(cid:1)(cid:10) a photon(cid:7) the proper timevanishes but the four(cid:9)momentum

(cid:13)(cid:5)