2. Geometry

2 roles of spacetime:

• Stage on which physical events play out.

• Cause of physical phenomenon, e.g., gravity or perhaps everything.

Main point of relativity theories:

Remove incorrect, simplifying, assumptions about the nature of spacetime.

Mathematics: Replace Euclidean space with

differentiable manifold (continuum)

+ affine connection (parallelism)

+ metric (length)

Bonus: Gravity emerges as natural consequence of curvature of spacetime.

2.0. The Special and General Theories of Relativity

2.1. Spacetime as a Differentiable Manifold

2.2. Tensors

2.3. Extra Geometrical Structures

2.4. What is the Structure of Our Spacetime?

2. Geometry

2.0. The Special and General Theories of Relativity

2.0.1. The Special Theory

2.0.2. The General Theory

2.0.1. The Special Theory

Newton's Laws (1686)

1. Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed on it.

2. The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

Eddington’s parody (1929) :

Every body tends to move in the track in which it actually does move, except insofar it is compelled by material impacts to follow some other track than that in which it would otherwise move.

→ Force on body & acceleration of observer seemingly indistinguishable.

Implicit assumption in Newton’s laws:

Forces arise only from interaction between bodies.

Inertial frame: Any frame in which Newton's 1st law holds.

Operational definition: Any frame in which every sufficiently isolated body is not accelerated.

Any 2 inertial frames must either be at rest or moving with uniform velocity with respect to each other.

Galilean Relativity

x x vt y y t t

Inertial frame S moving with velocity v along x-axis wrt inertial frame S.

t x x v , ,x vt y z

d

dt

xu d

dt

x

u dt

dt x v

u v , ,x y zu v u u

d

dt

u

a d

dt u v

d

dt

u a

z z

Failures of the Galilean Relativity

Vibration theory + Maxwell’s EM theory

→ Ether theory of light: speed of light = c wrt ether.

Galilean relativity → Change of speed of Earth relative to ether ~ orbital velocity.

Michelson & Morley experiment: No such change measured.

Related experiments:

• Kennedy-Thorndike experiment.

• Fizeau’s experiment.

• Transverse Doppler effect.

Special Relativity

Einstein's theory of special relativity:

1. The speed c of light in vaccum is the same in any inertial frame.

2. The laws of physics are the same in all inertial frames.

Application to Maxwell's equations led to the Lorentz transformation [Ex 2.1]:

x x vt

y y z z

ct ct x 2

2

1

1vc

2

1

1

v

c

whereupon: 2 22 2 2 2 2 2 2 2 2x y z c t x v t y z ct x

2 2 2x vt ct x x vt ct x y z

2 2 2 2x y z ct

2 2 21 1 1 1x ct x ct y z

2 2x ct x ct y z

Lorentz interpreted the Lorentz transformation as the effect of the ether upon the inter-molecular forces.

Einstein interpreted it as the embodiment of the fundamental structure of spacetime.Consider 2 events at ( x1 , t1 ) and ( x2 , t2 ) wrt S.

Galilean transformation: Δt = t2 t1 = t2 t1 = Δt

→ Simultaneity is preserved.

Distance is preserved.

~ Galilean spacetime

Lorentz transformation: c2 Δτ2 = c2 Δt2 Δx2 = c2 Δt 2 Δx 2 = c2 Δτ2

→ Only event (or proper time) intervals are preserved:

~ Minkowski spacetime

c is just the conversion factor between units of time & length.

Relativity is a theory about the structure of spacetime.

Principle of relativity (Einstein 1905):

The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.

Laws of physics must be covariant under transformations between inertial systems.

Newton’s theory of gravitation:

21 2 1

1 1 2 32

2 1

dm Gm m

dt

x x x

x x 11 2 2

Newton's gravitational constant

6.67 10

G

N m kg

is covariant under Galilean transformations,

but not so under Lorentz transformations.

The reverse is true for electromagnetic forces.

2.0.2. The General Theory

Constancy of c → special relativity.

Equality of inertial & gravitational masses → general relativity.

Experimental proof: Gravitational acceleration is the same for all bodies on earth:

Galileo, Newton, Eotvos ( accurate to 109 ), Dicke,…Analog: inertial (fictitious) forces such as Coriolis & centrifugal forces.

Einstein’s (weak) principle of equivalence:

The effects of a gravitational field can be locally eliminated by using a free-falling frame of reference.

Strong principle of equivalence:

Gravity is the fictitious force resulted in using a Galilean or Minkowski spacetime to interpret motion in the spacetime curved by a massive body.

Event interval in a free-falling inertial frame, Cartesian coordinates:

2 2 22c c d t d x

Actual spacetime, general coordinates:

3

2

, 0

c g x dx dx

g is called the metric tensor. It contains the effects of gravitation.

For the Minkowski spacetime described by Cartesian coordinates,

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

g x

By means of coordinate transformation, any g can be made to equal to η at a single point in spacetime (locally flat).

2.1. Spacetime as a Differentiable Manifold

Approach to modelling spacetime:

Start with minimal assumptions ( n-D smooth continuum ~ differentiable manifold ).

Other properties such as distances & parallelism are added as needed.Comments:

Usually, one takes n 4, but some unified theory makes use of other values, e.g., n 10.

A set of points is a continuum if there are always other points between any two distinct points of the set.

Smoothness means that all points in the set can be described, at least locally, by some differentiable coordinate systems.

That spacetime be a smooth continuum is a prerequisite for formulating physical theories in terms of integral or differential equations.

At present, there is no experimental evidence that spacetime is granular.

The study of the properties of a continuum is called topology.

The fundamental object is the neighborhood of a point (open set).

Properties of interest are those invariant under continuous mappings (deformations).

Length is not defined at this level of geometry.

The connectivity of all regions is unchanged under deformation.

Any object that has a 1-1 onto correspondence with n is an n-D continuum.

2.1.1. Topology of the Real Line and d

= set of all (finite) real numbers.

can be represented by a straight line

( bijection between elements of set and points on the real line).

An open interval (a,b) is the set { x | a < x < b }.

is a continuum in the sense that

any point x (a,b) is contained in another open interval ( x ε, x +ε) (a,b).

Formally, topology deals with open sets.

An open set in is any union of open intervals.

2 = set of all (finite) ordered pair of real numbers [ 2-tuples ( x1 , x2 ) ].

2 can be represented by a plane.

An open set in 2 is any union of open rectangles/regions.

Extension to d is straightforward.

Continuous Functions

Consider the function f : X→ Y by x y = f(x), where X, Y are subsets of .

f is continuous on X if the inverse image f 1(S) is an open set of X for all open set S of Y.

Caution: f 1(S) is a single symbol & does not imply the existence of the inverse function f 1.

The empty set is open.

X & Y are open.

Definition holds for X m & Y n.

Distance on graph is meaningless if X & Y represent different quantities.

Topology

A system of subsets of a set X defines a topology on X if contains

a) and X.

b) the union of every one of its subsystems.

c) the intersection of every one of its finite subsystems.

The sets (subsystems) in are called open sets of the topological space , often abbreviated to X.

Let X be any non-empty set.

• The topology with open sets and X is called trivial.

• The topology with all subsets of X, and X included, is called discrete.

The topology on d using unions of open regions as open sets is called the usual / natural topology on d.

A topology imposes 2 kinds of structures on the space:

a) Local topology determines how open sets are fit together inside any globally small region.

Notions like continuity are defined here.

b) Global topology determines how the open sets cover the whole space.

For example, the plane 2, the sphere S2 and the torus T2 have the same local topology but different global ones.

The global topology of spacetime is not known.

Its local topology is usually taken to be that of 4.

2.1.2. Differentiable Spacetime Manifold

A set M of points is a (topological) manifold if it is locally like n .

Each point P has an open neighborhood U homeomorphic to some open set V in n .

a bi-continuous (map and inverse map both continuous) bijection (1-1 onto map)

:U V 1 , , nP P x P x P by

The n numbers x j (P) are called the coordinates of P and n is the dimension of M.

The pair ( U,φ) is called a chart, or a local coordinate system.

An atlas on M is a set { ( Uα ,φα ) }of charts so that the domains covers M,

i.e., every P is in some U .

For reasons of compatibility, an atlas of class requires the maps

1 : U U U U to be maps of class C k.

If k > 1, M is called a differentiable manifold.

is a map between open sets of n.1

It represents a coordinate transformation for points in the overlap region Uα Uβ of two coordinate systems given by φα& φβ.

U = Uα, V = Uβ

f = φα , g = φβ

2.1.3. Summary and Examples

Polar coordinates : ( x1, x2 ) = ( r , φ ). 0 r 4, 0 φ 2 π.

Map in chart must be betweeb open sets → boundary lines ( φ=0, φ=2π) omitted.

→ M can’t be covered by one chart.

2.2. Tensors

Planar Euclidean geometry is acceptable for street maps but not continental maps.

Classical mechanics treats Earth as an inertial frame when corrected for its orbital & rotation motions.

General relativity says that whether a frame is inertial depends on the gravitational fields present.

Hence, Newton’s phrasing of his laws wrt inertial systems is not valid.

Physical laws must be independent of frames (coordinate systems).

This is achieved by writing equations of motion as tensor equations.

There’re 2 ways to proceed

1. Tensor analysis: Tensor equations are written in component form & explicitly covariant under all coordinate transformations.

2. Differential geometry: Tensors are treated as a whole with no reference to any coordinate system nor components.

Note: not all physical objects are tensors, e.g., spinors.

Tensors are classified by their ranks ( ab ) , which indicate how they behave

under coordinate transformations.A tensor field assigns a tensor to every point of the manifold.

Rank ( 00 ) tensors are called scalars.

A scalar field is a real-valued function

f : M → with P f(P) or x μ f ( x μ)

such that, under a coordinate transformation x μ→ x μ, we have

f ( x μ ) = f ( x μ) = f ( P ) = f ( P )

Examples:

In Newtonian physics, electric potentials & fluid density are scalar fields.

In relativistic physics, true scalar fields are usually obtained by contraction / inner products.

Rank ( 10 ) tensors are called (contravariant) vectors.

Geometrically, it represents the tangent to a curve passing through the point.

A curve is the map C : → M by λ xμ (λ),

where λ is the parameter of the curve.

In differential geometry, the tangent vector V to C is defined as

d d x

d d x

V

where d xV

d

are the components of the vector wrt the basis x

V

Transformation to coordinate system { x μ } is accomplished by the chain rule:'

'

d d x

d d x

V'

'

d x x

d x x

→'

' d xV

d

'd x x

d x

'xV

x

which is the definition of a contravariant vector in tensor analysis.

'V

Contravariant Vectors / Vectors

Technicalities:

• The set of all contravariant vectors at a point PM form a linear vector space called the tangent space TP(M) at P.

• Parametrization of a curve (path) is not unique.

• For computational reasons, different parametrizations of the same geometric path are treated as different curves. This means d / dλ and d / dμ represents different vectors even when λ & μ parametrize the same geometric path.

• Two parametrizations λ & μ of the same path are affinely related if μ = a λ + b, where a, b are constants,

• The transformation matrix Λ satisfies the relation

''

' '

x

x

x

x

'

'x

x

''

Rank ( 01 ) tensors are called covariant vectors, covectors, or 1-forms.

Geometrically, it represents a set of hyperplanes parallel to the contours of a function over the manifold.

Algebraically, it is a linear function that maps vectors into numbers.

Thus, all the covectors at a point P form a vector space, called the cotangent space TP *(M), that is dual to the tangent space TP (M) there.

The coordinate basis for 1-forms is { d xμ } so that

ω = ωμ d xμ ωTP *(M)

Let V TP (M), then

ω(V) = V(ω) = ωμV μ

is a scalar called the contraction of V & ω.

The prototype of 1-form field is the gradient of a function f over M.f

d f d xx

ω d x

Covariant Vectors / 1-forms

Coordinate transformations of 1-forms are given by

''

xd x

x

ω → ' ''

x

x

so that ω(V) is indeed a scalar.

Thus, ' '' 'V

V ω V ' '

' 'V '

' V

If ω = d f and V = d / dλ , then

dd f

d

ω Vf d x

d xx d x

f d x

x d

d f

d

where, by definition, d xx

Coordinate free definitions:

A vector is a linear function that maps a 1-form to a number.

A 1-form is a linear function that maps a vector to a number.

Coordinate free definitions:

A rank (ab) tensor is a multi-linear function that maps a 1-forms and b vectors to a

number.

Tensors

Definition from tensor analysis:

A rank (ab) tensor is an object that transforms like

1 1 1 1

1 1 1 1

' ' ' '

' ' ' 'a a b a

b a b bT T

Note: A tensor equation is automatically covariant under all coordinate transformations.

The contraction of 1 pair of indices reduces a (ab) tensor to a (a1

b1) tensor.

T S E.g., is a (2

1) tensor.

2.3. Extra Geometrical Structures

Two useful geometric structures:

1. Affine connection to define parallelism.

2. Metric to define length and angle.

An affine connection that is derived from a given metric is called a metric connection.

Examples:

Spacetime is described by a metric connection.

Gauge theories employ only an affine connection.

2.3.1. The Affine Connection

Four geometrical tools provided by an affine connection:

1. Parallelism.

2. Curvature.

3. Covariant derivative.

4. Geodesic.

Applications:

(a) In general relativity, Newton’s 1st law is replaced by the statement:

A test particle follows a geodesic unless acted on by a non-gravitational force.

A geodesic is a curve the tangents of which are all parallel to each other.

(b) Except for the scalar field, the partial derivatives of a tensor field do not form a tensor, e.g.,

' '' 'V V

' '' 'V V

The covariant derivative

connection termsT T

is designed to transform like a tensor.

(c) Derivatives of vector V along a curve C(λ).

dV d x V

d d x

0

limV Q V P

Vμ(Q) TQ(M) and Vμ(P) TP(M) are vectors in different vector spaces.

Therefore, their difference, and hence dVμ/ dλ, are not vectors.

Parallel Transport

Covariant derivative of vector V along curve C(λ):

0

limV Q V P QDV

d

Vμ(P→Q) is the parallel transport of Vμ(P) to Q.

d xV P Q V P P V P

d

Parallel transport depends on the (finite) route travelled if the manifold is curved.

Sphere embedded in Euclidean 3-space.

For infinitesimal displacements, Vμ(P→Q) is route independent because

d xd x

d

Given an affine connection Γ,

Sphere embedded in Euclidean 3-space.

Covariant Derivative

SettingDV d x

d dV

0

limV Q V PDV d x

P V Pd d

V d xV

x d

we have V VV

By definition, σ Vμ is a ( 11 ) tensor:

''

''V V

This means Γ must transform as : '' '

' '' ' ' '

' ' '' ' ' ' ' '

'

' ' 0

( Proof is tedious but straightforward.) Hint:

Assuming obeys the usual Leibniz’s rule

a b a b a b

it is straightforward to deduce from V V

that

In general, T T T T

More concise notations:

;T T

,T T

2.3.2. Geodesics

Geodesic: generalization of Euclidean straight line to curved spaces.

→ All of its tangents are parallel, e.g.,

1P Q Q

d x d xf

d d

2

2

P P P P P P

d x d x d x d x d x d xP f

d d d d d d

2

21

P P

d x d xf

d d

2

2

d x d x d x d xf

d d d d

Geodesic

equation

Changing parameter to μ gives

d d d

d d d

22 2 2

2 2 2

d d d d d

d d d d d

The geodesic equation becomes

2 22 2

2 2

d d x d d x d d x d x d d xf

d d d d d d d d d

22 2

2 2

d x d x d x d d d d xf

d d d d d d d

d x

gd

μ is an affine parameter if2

20

d x d x d x

d d d

If μ is an affine parameter, so is λ = a μ + b.

2.3.3. The Riemann Curvature Tensor

Parallel Transport Over Different Paths:Consider 2 points P and Q with coordinates xμ and xμ +δxμ, respectively. Without loss of generality, we can assume δxμ to be nonzero only for μ= a or b,i.e., δxμ = δμa δxa + δμb δxb .

Let R and S be two "mid-points" with coordinates xμ + δμa δxa and xμ + δμb δxb , respectively.

aaV P R V P P V P x

baV P R Q V P R R V P R x

a a ba b aV P P V P x R V P P V P x x

Similarly,

bbV P S V P P V P x

b b ab a bV P S Q V P P V P x S V P P V P x x

Using ,a

b b b aR P P x

,b

a a a bS P P x

we have

V P S Q V P R Q

,

,

a ba b a b

b a b a

x x P V P P P V P

P V P P P V P

, ,a b

b a a b a b b ax x V

a babx x R V

where

, ,ab b a a b a b b aR

is the Riemann tensor.

Commutator of the Covariant Derivatives

Consider the commutator ,

Its effect on a vector is a rank 3 tensors , V V V

Since τVμ is a (11) tensor, we have

V V V V

V V V V V

,V V V V V V

Taking μ ↔ τ gives

V V V V

,V V V V V V

Hence,

, ,, V V V V V V V

, , V V

R V V

, V R V V

Properties of the Riemann Tensor

As a rank 4 tensor, Rμνσ τ has 44 = 256 components in a 4-D space.

However, owing to various symmetries, e.g.,

it can shown that the number of independent components is only 80. If is a metric connection, additional symmetries further reduces this number to 20.

R R , , since

In most applications in physics, e.g., general relativity, one needs only deal with contractions of Rμ

νστ , e.g., the rank 2 Ricci tensor defined by

R R , ,

or the Ricci scalar defined by R R

Although is not a tensor, Γμστ Γν

τσ is a tensor called the torsion tensor.

The torsion tensor of the spacetime continuum is often set to zero.

However, there is not yet any means to test this assumption experimentally.

2.3.4. The Metric

The distance ds between two points with coordinates xμ and xμ+d xμ is given by

2ds g x dx dx

where g is the metric tensor (field).

Since the antisymmetric part of g cannot contribute to ds, g is taken to be symmetric.

Finite distance between two points P and Q is given in terms of the arc length of some specific path joining them:Q

PQ

P

d ss d

d

Q

P

d x d xg x d

d d

For the Euclidean 3-space, we have gμν = δμν or

1 0 0

0 1 0

0 0 1

g

so that 2 2 22 1 2 3ds dx dx dx

The scalar / inner product of vectors U and V is defined as g U V U V

The length of a vector is defined as g V V V V V

The angle θ between vectors U and V is defined as cos U V U V

Note: the length & angle may be imaginary if g is not positive definite.

By identifying the inner product with the contraction between vector & 1-form,

we see that g relates vectors & 1-forms.U g U

g

where g g

In general, g can be used to lower or raise the indices of any tensor.

In a metric space, vectors & 1-forms are just different versions of the same thing.

To see which version is more fundamental, one must take out the metric.

2.3.5. The Metric Connection

Affine connection (parallelism) and metric (lengths and angles) must be compatible.

E.g., if two vectors are parallel transported together along a curve, the angle between them should remain unchanged.

An affine connection that is compatible with a metric is called a metric connection.

Consider the parallel transport of 2 vectors V and W at point P = xμ(λ) along curve C(λ) to point Q = xμ(λ+δλ ) .

Define field V(x) s.t. V(Q) = V(P → Q) Q on C ; and analogously a field W(x).

The derivative of V along C is thereforeDU

d

VV

0lim

Q P Q

V V0

where U is the tangent to C. Similarly, 0D

d

W

Compatibility condition for a metric connection can be stated as the conservation of the scalar product VW = gμνVμWν under parallel transport, i.e.,

0D

U g V Wd

V W

Leibnitz rule : g V W g V W g V W g V W

→ 0U g V W

Since this must hold for arbitrary V and W on arbitrary curve or U, we must have

0g or , 0g g g

i.e., g is covariantly constant.

, 0g g g

, 0g g g

, , ,g g g g g g

By interchanging indices, we have

If the connection is also symmetric in the lower indices, we have

, , ,g g g g g g

, , , 2g g g g

, , ,

1

2g g g g

Christoffel symbol

A metric connection expresses the parallelism implied by the metric.

A manifold can have other affine connections and their associated parallelism.

It may even possess several different metrics.

→ there can be several different kinds of "distance" and meanings of "parallel".

Fortunately, our space-time continuum seems to have only one metric.

A measure of the curvature of the manifold is given by the Ricci curvature scalar

R g R R ~ radius of curvature

2.4. What is the Structure of Our Spacetime?

1. Galilean Spacetime:Space is the 3-D Euclidean space S. Time is an 1-D metric space T.Spacetime is the direct product space called a fibre bundle.

2. Minkowski Spacetime:

Spacetime is a 4-D metric flat space with signature -2.

3. Curved Spacetime:

Spacetime is a 4-D space with a metric tensor that depends on gravity.