Notes on Special and General Relativity Andrew Forrester

January 28, 2009

Contents 1 Questions and Ideas

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2 The Big Picture

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3 Notation 3.1 Ambiguous Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4

4 Theoretical Summary 4.1 Fundamental Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Developing the Special Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 6

5 Basic Quantities and Terms 5.1 Physical Quantities and Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Mathematical Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 6 9

6 Holors and Invariants

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7 Holor Calculus and Differential Geometry

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8 More Differential Geometry 8.1 Maps between Manifolds . . . . . . . 8.2 Stokes’ Theorem . . . . . . . . . . . 8.3 Parallel Propagation, Killing Vectors 8.4 Noncoordinate Bases . . . . . . . . . 8.5 Conformal Stuff... . . . . . . . . . . .

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17 17 17 17 17 17

9 Lagrangian Formalism

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10 Special Relativity

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11 Equations 18 11.1 Major Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 11.2 Minor Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 12 The Speed of Gravity, Etc.

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13 Other Relativity Theories

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14 Open Questions and Mysteries

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15 Applications

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16 Common Errors 20 16.1 Mistaking Similar Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 16.1.1 Applying this notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 16.2 Silly Mistake(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 17 Useful Programs

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1

Questions and Ideas

• (Carroll [1], p.39: Why does δ(∂µ Φi ) = ∂µ (δΦi )?) • (Can we say that our relativistic velocity always has the same “pseudo” magnitude?) • Prove that the Lorentz transformation makes the coordinates “scissor”. • Are there particular parametrizations that are necessary to describe paths of massless particles (which travel at the speed of light)? (affine parameters?)

Project Ideas • Connection − What is it? − How (many ways) is it defined? Why? − • Spacetime Inflation • Gyroscope precession

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The Big Picture

Subtopics The topic is most broadly divided into the following categories: Kinematics Statics Dynamics

Domains • Relativistic versus Non-relativistic • Relativity versus Quantum Mechanics • Newtonian Gravitation (Time-independent gravitational field, slow moving test particles) • Weak Linear Gravitation (Weak time-dependent gravitational field, no restriciton on test particle motion) • ...

2

3

Notation

Notation in These Notes p = p~

3-vector (usually representing a geometric object such as momentum) (I use the notation p~ in my handwriting and p in my electronic documents.) the 3-vector p in a particular coordinate system (three numbers that transform together as a spatial vector)

i pi = p1 , p2 , p3

pi

pi = {pi }

the ith component (or i-component) of the 3-vector p p2 = p · p = pi pi = (p1 )2 + (p2 )2 + (p3 )2

p pµ

4-vector (usually representing a geometric object such as 4-momentum or energy-momentum) the 4-vector p in a particular coordinate system (four numbers that transform together as a 4vector) the µ-component of the 4-vector p (one number) Saying “µth component” could be misleading, since the counting starts at zero. p2 = p · p = pµ pµ = −(p0 )2 + (p1 )2 + (p2 )2 + (p3 )2

pµ = {pµ }

bµ

4-tuple of 4-vectors (is this a 4-vector?) p = pµ eµ = p0 e0 + p1 e1 + p2 e2 + p3 e3

We use the “spacelike convention” of metric signature (−1, 1, 1, 1), meaning that the flat, Minkowski spacetime metric is η = diag(−1, 1, 1, 1), that is,   −1 0 0 0  0 1 0 0   ηµν =   0 0 1 0  0 0 0 1 µν and we therefore also use ∂2

1

= −∂0 2 + ∂1 2 + ∂2 2 + ∂3 2

= ∂µ ∂ µ

= − c12 ∂t 2 + ∂x 2 + ∂y 2 + ∂z 2

= −∂0 2 + ∇2 .

We let

µ0 1 and Km ≡ . 4πε0 4π ¯ to denote some form of contraction of the tensor h, for example, taking the “pseudo-trace” of hµν : We use h ¯ = hµµ = g µρ hρµ ; or the following contraction: h ¯ β γ  = hαβ γ α = gαδ hαβ γδ . We will thus put bars over the h ¯ and over the Einstein tensor symbol ¯ and Ricci scalar (R) Riemann tensor symbol (R) to denote the Ricci tensor (R) ¯ (G) to denote the Einstein scalar (G), showing that they are contractions. We will use GN for Newton’s gravitational constant (except when accompanied by a mass M ): Ke ≡

¯ = Gµµ 6= G 6= GN G GM ≡ GN M i

Also, we will use G and G to denote the gravito-electric field. With these notations, the only ambiguity is between the Einstein tensor G (when writing it without super- or subscripts) and the magnitude of the gravito-electric field |G| = G (which I think does not turn up in this document), so long as they are not multiplied by a mass M . Overline notation: to distinguish between tensorial and non-tensorial indices. . . for example,

Rτ µσν 1 Sometimes

prefer the

2

gτ ρ Rρµσν

Rτ µσν

=

Rτ µσν

= −Rτ µνσ

Rτ µσν

= −Rµτ σν

Rτ µσν + Rτ σνµ + Rτ νµσ

= Rσµτ ν = 0;

∂2

Rτ [µσν] = 0

is written as , in analogy to ∆, and sometimes it’s written as 2 , in analogy to ∇2 , usually in physics settings. I notation because it reveals its squared nature (while the box itself reveals its four-variable nature).

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or to distinguish between the “full object” tensor and the “parts” or merates

3.1

Ambiguous Notation p

p2

This character is ambiguous because it could be the magnitude of p or it could be the “pseudomagnitude” of p. ? √ p = p·p ? √ p = p·p Both of these uses seem to be common. If either of these notations is used, it should be clearly and loudly defined. If p is used, then one may wonder whether this is the 2-component of pµ (or pi ) or the square of p. ?

p2 = p · e2 ?

p2 = p · p ?

p2 = p · p If p is used, then a preferrable notation for the square of p would be (p)2 to distinguish it from the 2-component of p (or pi ). Perhaps I should use the MTW notation for tensors: bold sans serif. µ µ  x0 ct  x1   x     =  x2  =  y  z x3   ct  x    OR x =  y  z 

xµ

  x0  x1     x=  x2  =  x3 

OR

  x0  x1    2 = x x3

 0 µ  µ ct  XXX x  Xx   x X1 X µ      X=  x =  2  x  XX XyX X  X 3 z x   0 x ct  x1 x   OR x=  x2 y  z x3

 ct   x   =   y  z 



(With this notation, beware of the ambiguity between x or x the four-vector and x the component of the three-vector ~x. In handwriting, you can’t distinguish between x and x, unless you use x...)   0   x ct  x1   x     x=  x2  =  y  z x3 Notation from Theory of Holors:   x 0   x 1 µ {x } =  x2    x 3

= ct =x =y =z

if µ = 0 if µ = 1 if µ = 2 if µ = 3

Possible improvement:   x0    x 1 xµ =  x2    x 3

= ct =x =y =z

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if µ = 0 if µ = 1 if µ = 2 if µ = 3

Abbreviations in These Notes • GR = General Relativity • SR = Special Relativity • EM = electromagnetic or electromagnetism

4

Theoretical Summary

4.1

Fundamental Principles

• General Equivalence? • Principle of General Covariance “There are to be no preferred coordinates, but also, if we have two different spacetimes, representing two physically distinct gravitational fields, then there is to be no naturally preferred pointwise identification between the two – so we cannot say which particular spacetime point of one is to be regarded as the same point as some particular spacetime point of the other! This philosophical issue will concern us later, regarding how Einstein’s theory relates to the principles of quantum mechanics. It is for this reason, most particularly, that the tensor formalism is central to Einstein’s theory.” – Penrose [4] pg 459 • Postulate of Equivalence (from Goldstein): for Galilean transformations requires that physical laws must be phrased in an identical manner for all uniformly moving systems, i.e., be covariant when subjected to a Galilean transfomation. Measurements made entirely within a given system must be incapable of distinguishing that system from all others moving uniformly with respect to it. • Equivalence Principles (from Wikipedia) − Weak Equivalence (universality of free fall): The trajectory of a falling test body depends only on its initial position and velocity, and is independent of its composition. or All bodies at the same spacetime point in a given gravitational field will undergo the same acceleration. (Are these the same?) The principle does not apply to large bodies, which might experience tidal forces, or heavy bodies, whose presence will substantially change the gravitational field around them. This form of the equivalence principle is closest to Einstein’s original statement: in fact, his statements imply this one. − Einstien Equivalence: The result of a local non-gravitational experiment in an inertial frame of reference is independent of the velocity or location in the universe of the experiment. This is a kind of Copernican extension of Einstein’s original formulation, which requires that suitable frames of reference all over the universe behave identically. It is an extension of the postulates of special relativity in that it requires that dimensionless physical values such as the fine-structure constant and electron-to-proton mass ratio be constant. Many physicists believe that any Lorentz invariant theory that satisfies the weak equivalence principle also satisfies the Einstein equivalence principle. − Strong Equivalence: The results of any local experiment, gravitational or not, in an inertial frame of reference are independent of where and when in the universe it is conducted. This is the only form of the equivalence principle that applies to self-gravitating objects (such as stars), which have substantial internal gravitational interactions. It requires that the gravitational constant be the same everywhere in the universe and is incompatible with a fifth force. It is much more restrictive than the Einstein equivalence principle. General relativity is the only known theory of gravity compatible with this form of the equivalence principle. − “we [. . . ] assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.” (Einstein 1907) − . . . the inertial mass in Newton’s second law, F = ma, mysteriously equals the gravitational mass in Newton’s

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law of universal gravitation. Under the equivalence principle, this mystery is solved because gravity is an acceleration from inertial motion caused by the mechanical resistance of the Earth’s surface. These considerations suggest the following corollary to the equivalence principle, which Einstein formulated precisely in 1911: “Whenever an observer detects the local presence of a force that acts on all objects in direct proportion to the inertial mass of each object, that observer is in an accelerated frame of reference.” − Einstein also referred to two reference frames, K and K 0 . K is a uniform gravitational field, whereas K 0 has no gravitational field but is uniformly accelerated such that objects in the two frames experience identical forces: “We arrive at a very satisfactory interpretation of this law of experience, if we assume that the systems K and K 0 are physically exactly equivalent, that is, if we assume that we may just as well regard the system K as being in a space free from gravitational fields, if we then regard K as uniformly accelerated. This assumption of exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system of reference, just as the usual theory of relativity forbids us to talk of the absolute velocity of a system; and it makes the equal falling of all bodies in a gravitational field seem a matter of course.” (Einstein 1911) − This observation was the start of a process that culminated in general relativity. Einstein suggested that it should be elevated to the status of a general principle when constructing his theory of relativity: “As long as we restrict ourselves to purely mechanical processes in the realm where Newton’s mechanics holds sway, we are certain of the equivalence of the systems K and K 0 . But this view of ours will not have any deeper significance unless the systems K and K 0 are equivalent with respect to all physical processes, that is, unless the laws of nature with respect to K are in entire agreement with those with respect to K 0 . By assuming this to be so, we arrive at a principle which, if it is really true, has great heuristic importance. For by theoretical consideration of processes which take place relatively to a system of reference with uniform acceleration, we obtain information as to the career of processes in a homogeneous gravitational field.” (Einstein 1911) • “Covariance” of Physical Laws • Geodesics / Spacetime manifold curvature • Principle of Maximal Aging (or Stationary/Critical Proper Time) for geodesics

4.2

Developing the Special Theory of Relativity

Improve/correct this: (using Goldstein, 7-1) • Newton’s Laws (and various formulations of mechanics), which are Galilean-invariant, work • Maxwell’s Equations, which are Lorentz-invariant, work There are many derivations of the Lorentz transformation. . . list (and learn) some of them. • ... • Speed of light • ... • It is postulated that the known physical laws can be improved by generalizing them somehow into Lorentzinvariant forms • This postulate is verified by experiment

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Basic Quantities and Terms

5.1

Physical Quantities and Terms

• Frames − Inertial Frame − Lorentz Frame − Mathematical Frame (basis of vectors or forms? coordinate frame? local trivialization?) − Proper Frame (Rest frame, Self Frame) − “geodesic frame”, where ∂µ gνρ = 0, so {ρµν } = 0 but ∂σ {ρµν } = 6 0

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• Spacetime • Minkowski Space • Spacetime Position and Coordinates - a.k.a. Spacetime Event or simply Event • Proper Time - Self time dt 1 ≡γ =p dτ 1 − β2 β = v/c is called (the “relativistic beta”)? and γ = γβ is called the Lorentz factor. • Spacetime Diagram − Worldline − Light Cone Past - the lower half-cone Future - the upper half-cone Elsewhere - the outside of the light cone • Spacetime Interval - ... also Proper Interval Flat spacetime, straight line path: (∆s)2 = −(c∆t)2 + (∆x)2 + (∆y)2 + (∆z)2 = ηµν (∆x)µ (∆x)ν (∆τ )2 = −(∆s)2 = −ηµν (∆x)µ (∆x)ν Infinitesimal Spacetime Interval - ... or Line Element ds2 = ηµν dxµ dxν Metric: g = gµν dxµ dxν

or

g = gµν (dxµ ⊗ dxν )

Rp ηµν dxµ dxν is supposed to mean, we parametrize the path Any spacetime, path: (since it’s unclear what µ using the dimensionless parameter λ, x (λ), and write the following...) Z q µ dxν ∆s = ηµν dx dλ dλ dλ Z q µ dxν ∆τ = −ηµν dx dλ dλ dλ • Separation - (in flat spacetime only?) Property

Separation Type

Light Cone Position

(∆s)2 > 0 (∆s)2 = 0 (∆s)2 < 0

Space-like Light-like or Null Time-like

“Outside” the light cone On the light cone “Inside” the light cone

• Dust • Perfect Fluid - In cosmology, the matter filling hte universe is typically modelled as a perfect fluid. T µν = (ρ + p)U µ U ν + pg µν where ρ is the energy density, p is the pressure, and U µ is the four-velocity of the fluid.

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• Action - S, Lagrangian L, Lagrangian (Density) L Z S = d4 x L(Φi , ∂µ Φi ) Z L=

d3 x L(Φi , ∂µ Φi ) Z

S=

dt L

• Energy − “Inertial” energy: E 2 = p2 c2 + m2 c4 ; E = γmc2 − Slow-moving limit: rest energy Er = mc2 ; kinetic energy Ek = 21 mv 2 • Time-Position 4-Vector (Position Four-Vector, Events) Four-Position xµ ≡ (ct, x)µ • Speed-Velocity 4-Vector (Velocity Four-Vector) Four-Velocity dxµ = γ(c, v)µ = (γc, u)µ uµ ≡ dτ dx dx v≡ u≡ = γv dt dτ where the path of the particle (or particle(s)/object) through some coordinate system is parametrized by the particle’s proper time τ : xµ = xµ (τ ) • Energy-Momentum 4-Vector Four-Momentum pµ =

For massive particles:

dxµ = (E/c, p) dλ dx p≡ dλ

dxµ dxµ =m = muµ = γ(mc, pN )µ = (γmc, p)µ dλ dτ dx dx λ = τ /m p=m = γpN pN = mv = m dτ dt pµ =

- What are the limitations on λ for massless particles? • Number-Flux 4-Vector Four-Number-Flux-Density N µ = nuµ • Charge-Current 4-Vector Four-Current Jµ = ρuµ where ρ is the proper charge density (charge density in the rest frame of the (local?) charge distribution) • Work/Power-Force 4-Vector Four-Force fµ = m

dpµ d2 xµ = dτ 2 dτ

- Lorentz force f µ = quλ Fλ µ

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• Frequency-Wave 4-Vector (Frequency-Wave-Vector Four-Vector, Wave Four-Vector) Four-Wave-Vector Relativistic Doppler shift • Stress Tensor - Energy-Momentum-Stress Tensor Spin tensor Warning: In solid state physics and fluid mechanics, the stress tensor is defined to be the spatial components of the stress-energy tensor in the comoving frame of reference. In other words, the stress energy tensor in engineering differs from the stress energy tensor here by a momentum convective term. • EM Four-Potential Aµ =



Φ x y y ,A ,A ,A c

µ

• EM Field-Strength Tensor Field-Strength Tensor (and Auxiliary) 0 B B F αβ = B @

0 Ex Ey Ez

−Ex 0 Bz −By

−Ey −Bz 0 Bx

−Ez By −Bx 0

1αβ C C C A

0 ≡ (E, B)αβ

E i = ∂ i A0 − ∂ 0 Ai

B B Gαβ = B @

0 Dx Dy Dz

−Dx 0 Hz −Hy

−Dy −Hz 0 Hx

−Dz Hy −Hx 0

1αβ C C C A

≡ (D, H)αβ

B i = −˜ εijk (∂ i Aj − ∂ k Aj )

(check signs). . . with two covariant indices 0 B B Fαβ = gαγ F γδ gδβ = B @

0 −Ex −Ey −Ez

Ex 0 Bz −By

Ey −Bz 0 Bx

Ez By −Bx 0

1 C C C A

≡ (E, B)αβ αβ

. . . and polarization-magnetization tensor (P, −M) Dual Field-Strength Tensor 0 B B Feαβ = 12 αβγδ Fγδ = B @

5.2

0 Bx By Bz

−Bx 0 −Ez Ey

−By Ez 0 −Ex

−Bz −Ey Ex 0

1αβ C C C A

Mathematical Terms

• Vector Space - (“Vector” here does not mean a physical vector, it means a mathematical vector. We may be talking about a four-vector space or a tensor space, etc.) • Dimension • Inner Product - (pseudo-inner-product) Norm, Metric (“distance” function), Inverse Metric • Poincar´ e Transformation - ... Lorentz Transformation − Translation − Rotation − Boosting - Boost Parameter or Rapidity ζ: β = tanh ζ, γv = cosh ζ, γv β = sinh ζ − Proper/Improper - Proper Lorentz transformations are connected continuously to the identity transformation and so have det Λ = 1. Improper Lorentz transformations may have det Λ = 1 or det Λ = −1 (a fact related to the indefiniteness of the metric). Both Λ = η (space inversion) and Λ = −I (space-time inversion) are improper. If Λ is proper, −Λ is improper.

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− Orthogonal/Non-orthogonal − Orthochronous/Non-orthochronous Group Structure: Lorentz Group (Homogeneous Lorentz Group), Poincar´e Group (Inhomogeneous Lorentz Group) x0 = Λx 0

0

xµ = Λµ 0



µ

xµ

0

dxµ = Λµ µ dxµ   cosh ζv 0 0  − sinh ζv 0 0  = 0 1 0   0 0 1

− sinh ζv γv −γv βx  −γv βx cosh ζv γ v Λ=  0 0 0 0 0 0  0  x0 = γv (x0 − βx x1 ) = x0 cosh ζv − x1 sinh ζv    0  0 x1 = γv (x1 − βx x0 ) = x1 cosh ζv − x0 sinh ζv xµ : 0 x2 = x2    x30 = x3

 0 0 0 0   1 0  0 1

(See Lorentz Transformation paper and/or Jackson) • Tangent Space • Tangent Bundle The tangent space of a point consists of the possible directional derivatives at that point, and has the same dimension n as the manifold does. The collection of all tangent spaces can in turn be made into a manifold, the tangent bundle, whose dimension is 2n. • Cotangent Bundle Elements of the cotangent space can be thought of as infinitesimal displacements: if f is a differentiable function we can define at each point p a cotangent vector df p which sends a tangent vector Xp to the derivative of f associated with Xp . However, not every covector field can be expressed this way. • Bundles − Tangent, Cotangent − Tensor Bundle - The tensor bundle is the direct sum of all tensor products of the tangent bundle and the cotangent bundle. Each element of the bundle is a tensor field, which can act as a multilinear operator on vector fields, or on other tensor fields. The tensor bundle cannot be a differentiable manifold, since it is infinite dimensional. It is however an algebra over the ring of scalar functions. − Fiber Bundle − Jet Bundle A connection is a tensor on the jet bundle. • Field • Dual Space • One-Forms - Dual Vector (Dual Tensor, e.g. dual field-strength tensor) “Four-Position” (Four-Position with index lowered) xµ Partial Derivatives (Four-Gradiant) ∂µ Four-Potential ((Four)-Vector Potential) Aµ • Partial Derivative - Four-Derivative? Not a tensor (tensorial transformation) in general, right? (We’ll have to create the connection to make it tensorial.) ∂µ ≡

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∂ ∂ xµ

 ∂µ = h∂0 , ∂1 , ∂2 , ∂3 iµ =

1 ∂ ∂ ∂ ∂ , , , c ∂t ∂x ∂y ∂z

 µ

  E D 1 ∂ ∂ ∂ ∂ 0 1 2 3 = − ∂t , ∂x , ∂y , ∂z ∂µ = ∂ , ∂ , ∂ , ∂ c µ µ “Notice that, unlike the partial derivative, it makes sense to raise an index on the covariant derivative, due to metric compatibility.” g µλ ∇λ = ∇µ but g µλ ∂λ 6= ∂ µ ? (What about η µλ ∂λ = ∂ µ ?) • Gradient - Four-Gradient (Derivative on a scalar field that yields a “vector” field...) ∂i f = h∂1 f, ∂2 f, ∂3 f i i = (∇f )i

(∇i f ) ?

∂µ f = h∂0 f, ∂1 f, ∂2 f, ∂3 f ii • Divergence - Four-Divergence (Derivative on a “vector” field that yields a scalar field...) ∇ · x = ∂i xi = ∂1 x1 + ∂2 x2 + ∂3 x3 ∂ · x = ∂µ xµ = ∂0 x0 + ∂1 x1 + ∂2 x2 + ∂3 x3 • Curl - (Four-Curl?) (Goldstein [3] p.582: “Clearly, Fµν is a sort of four-dimensional curl of the vector Aµ ”: Fµν = ∂µ Aν − ∂ν Aµ ) • d’Alembertian - (Four-Laplacian or as Jackson says (pg 555) “invariant four-dimensional Laplacian”; And what about a “vector” four-Laplacian?) 2 2 ≡ ∂ 2 = ∂ · ∂ = η µν ∂µ ∂ν = −c2 ∂t 2 + ∇2

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Holors and Invariants • Holors, scalars, 4-vectors, Tensors Kinds of “vectors” or “invariant vectors” • Relativistic Invariant - Scalars, Tensors Speed of Light Spacetime Interval Plane-Wave Phase (pseudo-speed): square root of u2 = uµ uµ = −γ 2 c2 + v2 = (−c2 + v 2 )/(1 − v 2 /c2 ) = −c2 (Laue’s scalar? T µµ ) • Four-Scalar (No, scalars don’t have four components) Four-Tensor? (versus Cartesian tensor, spherical tensor) • Covariance Various meanings: invariance of the form of an equation or law; (co)varying in tandem with another varying quantity in such a way as to keep a third quantity (that depends on the other two) constant; • Vectors (shall we call any object that transforms as the space-time coordinate does under Lorentz transformations a 4-vector? and shall we call any object that transforms as the space-time coordinate does under general coordinate transformations a “general vector”?) − mathematical vector (an element of a vector space) − vector under some transformation group − vector under some physical transformation group (transforms the same as some “physical” object such as spacetime coordinates. . . or velocity?) physical or geometric vector 2 See

the Notation section (footnotes) for comments on the Box notation that appear here.

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∗ spatial vector (rotations) pseudo vector, axial vector (parity, i.e. spatial inversion) ∗ four-vector (Lorentz transformations) ∗ general vector (spacetime diffeomorphisms: general coordinate transformations) • Four-Vector Contravariant Vector Covariant Vector Does this image help explain the meaning of contravariant and covariant?

• Tensors Tensor Space, Tensor Product, Tensor Field Tensor products: p ⊗ q(, ) = h, pi h, qi (“invariant spatial tensor” versus tensor made from a tensor product versus holor) Kronecker Tensor δνµ γ - δ µν , δ αβ δη ? ij ijk - δab ≡ δai δbj , δabc ≡ δai δbj δck , and so forth. Minkowski Metric ηµν Metric (Tensor) gµν : g(x, v) = gµν (x) dxµ (x, v) dxν (x, v) Electromagnetic Field Strenth Tensor F µν Stress Tensor (Stress-Energy-Momentum Tensor) T µν (energy density, pressure, stress, strain?) Projection (Tensor) Operator P σ ν = δνσ + c12 uσ uν Einstein Tensor Gµν Levi-Civita Tensor ε ε = εµ1 ···µn dxµ1 ⊗ · · · ⊗ dxµn 1 εµ ···µ dxµ1 ∧ · · · ∧ dxµn = n! 1 n q 1 gµν ε˜µ1 ···µn dxµ1 ∧ · · · ∧ dxµn = n! q gµν dx0 ∧ · · · ∧ dxn−1 = q gµν dn x ≡ So dn x ≡ dx0 ∧ · · · ∧ dxn−1 is a tensor density field (of weight 1?) Levi-Civita Symbol ε˜ (a tensor density field of weight 1) ε˜ijk ε˜µνρσ

: ε˜123 = 1, : ε˜0123 = 1,

ε˜ even permutation = 1, ε˜ even permutation = 1,

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ε˜ odd permutation = −1 ε˜ odd permutation = −1

ε˜ijk

= ε˜ija δak jk jk = ε˜iab (δab − δba ) ijk ijk ijk ijk ijk ijk = ε˜abc (δabc − δacb + δcab − δcba + δbca − δbac )

ε˜µνρσ

= ε˜µνρα δασ ρσ ρσ = ε˜µναβ (δαβ − δβα ) νρσ νρσ νρσ νρσ νρσ νρσ = ε˜µαβγ (δαβγ − δαγβ + δγαβ − δγβα + δβγα − δβαγ ) µνρσ µνρσ µνρσ µνρσ µνρσ µνρσ = ε˜αβγδ (δαβγδ − δαβδγ + δαδβγ − δαδγβ + δαγδβ − δαγβδ µνρσ µνρσ µνρσ µνρσ µνρσ µνρσ + δγαβδ − δγαδβ + δγδαβ − δγδβα + δγβδα − δγβαδ µνρσ µνρσ µνρσ µνρσ µνρσ µνρσ + δβγαδ − δβγδα + δβδγα − δβδαγ + δβαδγ − δβαγδ µνρσ µνρσ µνρσ µνρσ µνρσ µνρσ + δδβαγ − δδβγα + δδγβα − δδγαβ + δδαγβ − δδαβγ )

Might this notation ever be useful?: ε˜µνρσ Tνρσ = ε˜µνρσ (Tνρσ − Tνσρ + Tσνρ − Tσρν + Tρσν − Tρνσ ) ε˜µνρσ Tνρσ = ε˜µνρσ (Tνρσ − Tνσρ + Tσνρ − Tσρν + Tρσν − Tρνσ ) AνAρAσ ) AνAρAσ − δβαγ AνAρAσ + δβγα AνAρAσ − δγβα AνAρAσ + δγαβ AνAρAσ − δαγβ ε˜µAνAρAσ = ε˜µαβγ (δαβγ νρσ νρσ νρσ νρσ νρσ νρσ H H H H H νρσ H H H H H H H = ε˜µαβγ (δH ε˜µH αβγ − δαγβ + δγαβ − δγβα + δβγα − δβαγ )

Alternators, or Generalized Kronecker Deltas (from Holors [5] book) δi δi k ij l δkl = j δk δlj δi δi δi l n m ijk j δlmn = δlj δm δnj k k k δl δm δn r1 1 δs1 δsr21 · · · δsrm r2 r2 r2 · · · δ δ δ s1 sm s2 ···rm δsr11sr22···s = . .. .. .. m . . . . . rm m δs1 δsr2m · · · δsrm



Signature: 3D Lorentz signature (+, +, −), Majorana (−, +, +) Space with Minkowski-Lorentz signature: (ds)2 = (dx)2 + (dy)2 − (dz)2 Space with Minkowski-Majorana signature: (ds)2 = −(dx)2 − (dy)2 + (dz)2 Landau-Lifshitz convention (terminology from D’Hoker) West coast convention (Bjorken-Drell) (terminology from D’Hoker) Bjorken metric: diag(1,-1,-1,-1) (terminology from Rajpoot) Pauli metric: diag(-1,1,1,1) (terminology from Rajpoot) • Index Gymnastics Summation Conventions Dummy Indices X

ε˜ijk ε˜ilm = δjl δkm − δjm δkl

jk jk ε˜ijk ε˜ilm = δlm − δml

i

X

ε˜ijk ε˜ijl = 2δkl

ij

13

ε˜ijk ε˜ijl = 2δlk

Full Object versus Elements: something like this... T αβγ δ Sαβ δκ ≡

X

T αβγ δ Sαβ δκ = Rγκ

αβδ

T αβγ δ Sαβ δκ ≡

X

T αβγ δ Sαβ δκ = Bγκ (α, δ)

β

Contraction Raising and Lowering Indices Symmetrization T(µ1 µ2 ···µn )ρ σ =

1 (Tµ1 µ2 ···µn ρ σ + sum over permutations of indices µ1 · · · µn ) n! 1 (Tµνρσ + Tµρνσ + Tρµνσ + Tρνµσ + Tνρµσ + Tνµρσ ) 6

T(µνρ)σ = Antisymmetrization T[µ1 µ2 ···µn ]ρ σ =

1 (Tµ1 µ2 ···µn ρ σ + alternating sum over permutations of indices µ1 · · · µn ) n!

T[µνρ]σ =

1 (Tµνρσ − Tµρνσ + Tρµνσ − Tρνµσ + Tνρµσ − Tνµρσ ) 6

Decomposition into symmetric and antisymmetric parts Trace Partial Derivatives Projection (Tensor) Operator P σ ν = δνσ + c12 uσ uν

7

Holor Calculus and Differential Geometry • Covariant Derivative - transforms like a tensor, etc... ∇T The covariant derivative is defined to have the following properties: 1. Linearity: ∇(aT + bS) = a∇T + b∇S 2. Leibniz (product) rule: ∇(T ⊗ S) = (∇T ) ⊗ S + T ⊗ (∇S) 3. Commutes with contractions: ∇µ (T λλρ ) = (∇T )µλλρ 4. Reduces to the partial derivative on scalars: ∇µ φ = ∂µ φ

(∇µ )α β

=

∂µ δβα + Γα µβ

∇µ V ν

=

∂µ V ν + Γνµλ V λ

or

(∇µ V )ν

∇µ ω ν

=

∂µ ων − Γλµν ωλ

or

(∇µ ω)ν

∇σ T µ1 µ2 ···µk ν1 ν2 ···νl

=

∂σ T µ1 µ2 ···µk ν1 ν2 ···νl

 ∂µ δλν + Γνµλ V λ   = ∂µ δνλ − Γλµν ωλ =



+ Γµσλ1 T λµ2 ···µk ν1 ν2 ···νl + Γµσλ2 T µ1 λ···µk ν1 ν2 ···νl + · · · − Γλσν1 T µ1 µ2 ···µk λν2 ···νl − Γλσν1 T µ1 µ2 ···µk ν1 λ···νl − · · ·

14

• Connections A connection is a tensor on the jet bundle. (Wikipedia: Differentiable Manifold) Christoffel Connection (Christoffel Symbol(s)) Spin Connections (Cartan Structure Equations) (Gauge Connections?) (structure group, a Lie group; gauge transformations, gauge theories) − Kinds ∗ Christoffel symbols of the first kind ∗ Christoffel symbols of the second kind − Torsion-free: Γλµν = Γλ(µν) − Metric-compatible: ∇σ gµν = 0 Nice properties of a metric-compatible connection: ∗ ∇λ µνρσ = 0 ∗ ∇λ g µν = 0 ∗ gµλ ∇σ V λ = ∇σ (gµλ V λ ) = ∇σ Vµ (Does “the metric is compatible with the connection” make sense?) Connection induced by a metric gµν (Christoffel Levi-Civita Connection): Γλµν = 21 g λρ (∂µ gνρ + ∂ν gρµ − ∂ρ gµν ) Torsion tensor: T λ µν = Γλµν − Γλνµ = 2Γλ[µν] • Geodesic Equation D dλ



dxµ dλ

σ ρ d2 xµ µ dx dx + Γ =0 ρσ dλ2 dλ dλ

 =0

⇔

For time-like particle paths: pµ ∇µ pµ = 0 dpµ + Γµρσ pρ pσ = 0 dλ • Covariant Directional Derivative

D dxµ = ∇µ dλ dλ  dxµ    α dxµ  d α dxµ α dxµ α α α α α ∂ D = (∇ ) = ∂ δ + Γ = δ + Γ = δ + Γ µ µ µ β β µβ µβ dλ dλ dλ dλ ∂ x β dλ β dλ µβ β

• Parallel Transport - Geodesic Equation of Parallel Transport 

D dλ T

µ1 µ2 ···µk ν1 ν2 ···νl

=

D µ1 µ2 ···µk T ν1 ν2 ···νl = 0 dλ

D µ dxµ d µ V = ∇µ V µ = V + Γµσρ dλ dλ dλ
D µ dxν (λ) V = ∇ν V µ x(λ) = 0 dλ dλ For a spacetime geodesic, the parallel transport of a vector must be valid when the parameter is the proper time λ = τ , thus any valid parametrization must be by a parameter that is related to the proper time by an affine transformation. Any valid parameter is therefore called an affine parameter. Other parametrizations will yield a sort of “non-inertial frame” equation (my terminology) with a fictitious force f (α) d2 xµ dxρ dxσ dxµ + Γµρσ = f (α) 2 dα dα dα dα

15

Keeping an affine parametrization but adding non-gravitational forces yields a sort of “general Newton’s second law” (my terminology): ρ σ d2 xµ q µ dxν µ dx dx + Γ = F ρσ dλ2 dλ dλ m ν dλ More ideas: − Exponential map − Geodesisically incomplete manifolds • Riemann Tensor - Curvature Tensor Rρσµν = ∂µ Γρνσ − ∂ν Γρµσ + Γρµλ Γλνσ − Γρνλ Γλµσ Rρµσν = ∂σ Γρνµ − ∂ν Γρσµ + Γρσλ Γλνµ − Γρνλ Γλσµ Number of independent components in n dimensions: 1 2 2 n (n − 1) 12 In 4-D, it has 20 independent components, “precisely the 20 degrees of freedom in the second derivatives of the metric that we could not set to zero by a clever choice of coordinates when we first discussed the locally inertial coordinates in Chapter 2.” Rρσµν = −Rρσνµ Rτ µσν

= gτ ρ Rρµσν

Rτ µσν

= −Rτ µνσ

Rτ µσν

= −Rµτ σν

Rτ µσν

= Rσµτ ν

Rτ µσν + Rτ σνµ + Rτ νµσ

=

0;

Rτ [µσν] = 0

• Ricci Tensor Rµν = Rλ µλν Rµν = Rνµ • Ricci Scalar - Curvature Scalar R = Rµ µ = g µν Rνµ • Weyl Tensor - Conformal Tensor - invariant under conformal transformations...basically the Riemann tensor with all of its contractions removed. In n dimensions (where n ≥ 3), it is   2 2 Cρσµν = Rρσµν − gρ[µ Rν]σ − gσ[µ Rν]ρ + gρ[µ gν]σ R (n − 2) (n − 1)(n − 2) “This messy formula is designed so that all possible contractions of Cρσµν vanish, wihle it retains the symmetries of the Riemann tensor:” Cρσµν

= C[ρσ][µν]

Cρσµν

= Cµνρσ

Cρ[σµν]

=

0

• Einstein Tensor Gµν = Rµν − 21 Rgµν Gµν = Gνµ due to the symmetry of the Ricci tensor and the metric “In 4D, the Einstein tensor can be thought of as a trace-reversed version of the Ricci tensor.” (Carroll pg 131) Einstein scalar?: G = Gµ µ

16

• Bianchi Identity ∇[µ Rρσ]µν = 0 For a general connection there would be additional terms involving the torsion tensor. It implies (after contracting twice on (3.139)) ∇µ Rρµ = 12 ∇ρ R (3.150) The twice-contracted Bianchi identity is equivalent to ∇µ Gµν = 0 From http://www.mathpages.com/home/kmath528/kmath528.htm . . . “The field equations of general relativity provide a good illustration of how non-linear laws imply constraints (e.g., the Bianchi identities) on the allowable initial conditions, so we are not free to specify a system at an arbitrary point in the phase space. This type of constraint applies to the Lorentz-Dirac equation as well, since it too is non-linear.”

8

More Differential Geometry

8.1

Maps between Manifolds

8.2

Stokes’ Theorem

8.3

Parallel Propagation, Killing Vectors

8.4

Noncoordinate Bases

8.5

Conformal Stuff...

9

Lagrangian Formalism Z

L(Φi , ∇µ Φi ) dn x

S=

where dn x and L are densities (tensor densities?) and their product is a tensor p L = − |g| Lb where Lb is a tensor (scalar). Euler-Lagrange equations: ∂ Lb − ∇µ ∂Φ

∂ Lb ∂ (∇µ Φ)

! =0

Hilbert-Einstein Action SH = S=

Z p

− |g| R dn x

1 SH + SM 16πGN

where SM is the action for matter. A definition of the stress tensor: Tµν ≡ −2 p

δSM − |g| δg µν

17

1

Palatini Formalism Z S[g, Γ] =

10

d4 x

p − |g| g µν Rµν (Γ)

Special Relativity

Principles • Equivalence • Velocity of Light

Other Things Flat Spacetime: g µν (xσ ) = η µν for all four-positions xσ Length Contraction: L(v) = γv L0 Time Dialation: (∆t)(v) = (∆t)0 /γv ? Z τ2 dτ p = γ(τ ) dτ ∆t = 1 − β 2 (τ ) τ1 τ1 Z

τ2

Relativistic Doppler Shift: Some equation here transverse relativistic doppler shift Relativistic addition of velocity u0 =

dx0 cγv (dx1 − βx dx0 ) = 0 dt γv (dx0 − βx dx1 )

Newton’s Law F = γm(13 + γ 2 vvT )a where 13 is the 3×3 identity matrix and vT is the velocity row-vector ( http://math.ucr.edu/home/baez/physics/ Relativity/SR/mass.html ) Anything else in particular? (From Jackson, say?)

11 11.1

Equations Major Equations

• Conservation of Energy and Momentum ∂µ T µν = 0ν • Maxwell’s Equations ∂ν F µν = J µ εµνρσ ∂µ Fνρ = 0σ

or

∂ν Feµν = 0µ

or something to that effect. Or, in other notation, the second equation is ∂[µ Fνλ] = 0(µ, ν, λ)

18

1 (∂µ Fνλ − ∂µ Fλν + ∂λ Fµν − ∂λ Fνµ + ∂ν Fλµ − ∂ν Fµλ ) 6 1 = (∂µ Fνλ + ∂µ Fνλ + ∂λ Fµν + ∂λ Fµν + ∂ν Fλµ + ∂ν Fλµ ) 6 1 = (∂µ Fνλ + ∂λ Fµν + ∂ν Fλµ ) 3 = 0(µ, ν, λ)

∂[µ Fνλ] =

Four-Potential Fµν = ∂µ Aν − ∂ν Aµ Gauge Transformation Aµ0

Aµ → Aµ + ∂µ λ(x)
= Aµ + ∂µ λ(x) δµµ0 ???

• Einstein’s Equation 8πG 1 Rµν − Rgµν = 4 Tµν 2 c
Rµν = 8πG Tµν − 12 T gµν In free space (no matter) Rµν = 0 With cosmological constant 1 Rµν − Rgµν + (factor?)Λgµν = 8πGTµν 2 (Add a const to a Lagrangian and usually that doesn’t matter, but in gravity it does since even “constants” interact with gravity/matter. Explain that.) The cosmological constant is important but tiny. • Geodesic Equation D dλ



dxµ dλ

ρ σ d2 xµ µ dx dx + Γ =0 ρσ dλ2 dλ dλ

 =0

⇔

For time-like particle paths: pµ ∇µ pµ = 0 dpµ + Γµρσ pρ pσ = 0 dλ • Euler-Lagrange Equations for a field theory in flat spacetime δS ∂L = − ∂µ i δΦ ∂ Φi

∂L ∂ (∂µ Φi )

! =0

• Klein-Gordon Equation ( − m2 )φ = 0

11.2

Minor Equations µν Tdust = pµ N ν = mnuµ uν = ρuµ uν µν µ ν µν Tperfect fluid = (ρ + p)u u + pη h i µν µλ νσ µν 1 λσ Tscalar = η η ∂ φ ∂ φ − η η ∂ φ ∂ φ + V (φ) λ σ λ σ field theory 2

1 µν TEM = F µλ F ν λ − η µν F λσ Fλσ 4

19

12

The Speed of Gravity, Etc.

13

Other Relativity Theories

• Weyl’s theory • Einstein-Cartan theory • Supergravity • String theory – 26D • Active Research − String theory – 10D − Loop gravity

14

Open Questions and Mysteries

15

Applications

• GPS − GR − SR − Sagnac effect

16

Common Errors

16.1

Mistaking Similar Quantities

There are several quantities of the same type that must be correctly distinguished: • Velocities − Newtonian velocity: v − Relativistic velocity: u = γv − 4-velocity: uµ = γ(c, v)µ = (γc, u)µ • Velocity magnitudes (speeds) − Newtonian speed: v = kvk − Relativistic speed: u = kuk − “Relativistic pseudo-speed”:

√

u2 =

√

uµ uµ =

√

−c2 = ic or =

√

c2 = c (depending on signature)

∗ Note that we will use the notation u2 ≡ uµ uµ and u2 = u · u 2 (u2 6= u · u, even though u · u = kuk and kuk = u) ∗ In quantum field theory (and perhaps in these notes somewhere) u may also refer to the 4-velocity in expressions such as. . . • Momenta − Generalized, canonical, or conjugate momentum − Mechanical momentum: pm = mv (also, “Newtonian momentum” and “kinetic momentum”) − Relativistic momentum: p = γpm − 4-momentum: pµ = (E/c, p)µ For massive particles pµ = (E/c, p)µ = γ(mc, pm )µ = (γmc, p)µ • Momenta magnitudes

20

− magnitude of generalized momentum − magnitude of mewtonian momentum: pm = mv − magnitude of relativistic momentum: p = γpm  √  p p √ − “pseudo-magnitude” of 4-momentum: p2 = pµ pµ = −E 2 /c2 + p2 = −m2 c2 ∗ Note that we will use the notation p2 ≡ pµ pµ and p2 = p · p 2 (p2 6= p · p, even though p · p = kpk and kpk = p) ∗ In quantum field theory (and perhaps in these notes somewhere) p may also refer to the 4-momentum in expressions such as p · x = ηµν pµ xν = −p0 x0 + p · x. • Mass − Invariant mass: m − Relativistic mass: γm − Relation to point-particle energies (E = total energy; P = potential energy; K = kinetic energy): P = mc2 + Premaining γmc2 = mc2 + K E = P + K = Premaining + γmc2 16.1.1

Applying this notation

• Relativistic Energy − Given that p2 is the square of the 4-momentum and p is the relativistic momentum and given that E = γmc2 E 2 6= p2 c2 + m2 c4 E 2 = p2 c2 + m2 c4 E 2 = (p2 − p2 )c2

We also have

p2 = −m2 c2

and

16.2

Silly Mistake(s)

• Confusing the forms of the velocity addition formula and gamma: − (putting a plus/minus sign in the addition formula when you mean minus/plus:) − This one is hard to catch once it’s been written (with an equals sign): γv 6= p

1 1 + v 2 /c2

I made the error of equating the two when trying to prove. . .

17

Useful Programs

Programs (that use second party computer algebra systems) • GRTensorM, Cartan (both use Mathematica) • GRTensor II (Maple) I haven’t used them (yet), though.

21

References [1] Sean M. Carroll: Spacetime and Geometry: An Introduction to General Relativity, Addison-Wesley (2004) [2] Charles W. Misner, Kip S. Thorne, John Archibald Wheeler: Gravitation, W. H. Freeman and Company (1973) [3] Herbert Goldstien: Classical Mechanics, Second Edition, Addison-Wesley (1980) [4] Roger Penrose: The Road to Reality: A Complete Guide to the Laws of the Universe, Alfred A. Knopf, a division of Random House (2004) [5] Parry Moon, Domina E. Spencer: Theory of Holors: A generalization of tensors, Cambridge University Press (1986)

22