Gravitation: Special Relativity - An Introduction to...
Transcript of Gravitation: Special Relativity - An Introduction to...
Gravitation: Special RelativityAn Introduction to General Relativity
Pablo Laguna
Center for Relativistic AstrophysicsSchool of Physics
Georgia Institute of Technology
Notes based on textbook: Spacetime and Geometry by S.M. CarrollSpring 2013
Pablo Laguna Gravitation: Special Relativity
Space and TimeSpace-time: is a 4-dim set, with elements labeled by 3-dim of space and one of time.
Event: An individual point in space-time.
Worldline: A path of events.
There is a difference between the paths that particles follow in Special Relativity (SR) and those in Newton’s theory.The absence of a preferred time-slicing foliation is fundamental in SR.
space at afixed time
t
x, y, z
Pablo Laguna Gravitation: Special Relativity
!
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!
y
x’
x
y
y’
x
x’
s
y’
!
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Geometrical facts about the spatial planes are independent of our choice of coordinates. E.g., the distance betweentwo points:
(∆s)2 = (∆x)2 + (∆y)2.
In a different Cartesian coordinate system x′ and y′ the distance is unaltered:
(∆s)2 = (∆x′)2 + (∆y′)2.
∆S is invariant under such changes of coordinates.
Pablo Laguna Gravitation: Special Relativity
Space-time interval
(∆s)2 = −c2(∆t)2 + (∆x)2 + (∆y)2 + (∆z)2.
Notice:
∆s can be positive, negative, or zero.
c is some fixed conversion factor between space and time.
c is the conversion factor that makes ∆s invariant.
The minus sign is necessary to preserve invariance.
In a more compact form, using the summation convention ( indices which appear both as superscripts andsubscripts are summed over), form.
(∆s)2 = ηµν∆xµ∆xν.
where ∆xµ = (∆t,∆x,∆y,∆z) and ηµν is a 4× 4 matrix called the metric:
ηµν =
−c2 0 0 0
0 1 0 00 0 1 00 0 0 1
.
Pablo Laguna Gravitation: Special Relativity
(∆s)2 = ηµν∆xµ∆xν
Light Cone:
Time-like separated if (∆s)2 < 0
Space-like separated if (∆s)2 > 0
Null or Light-like separated if(∆s)2 = 0
Proper Time τ : Measures the time elapsed as seen by anobserver moving on a straight path between events. That isc2(∆τ)2 = −(∆s)2
Notice: if ∆x i = 0 then c2(∆τ)2 = −ηµν (∆t)2, thus∆τ = ∆t .
Pablo Laguna Gravitation: Special Relativity
What kind of transformations leave ∆s invariant?
Translations:xµ → xµ′ = xµ + aµ ,
where aµ four fixed numbers.
“Rotations”:xµ′ = Λµ′
νxν,
orx′ = Λx .
(∆s)2 = (∆x)Tη(∆x) = (∆x′)T
η′ (∆x′)
= (∆x)TΛTη′ Λ(∆x) ,
and thereforeη = ΛT
η′ Λ ,
orηρσ = Λµ′
ρΛν′σηµ′ν′ .
Goal: To find Λµ′ν such that the components of ηµ′ν′ are the same as those of ηρσ .
Pablo Laguna Gravitation: Special Relativity
Lorentz Transformations: the matrices Λµ′ν that satisfy η = ΛTη′ Λ
Lorentz transformations categories:
Rotations:
Λµ′ν =
1 0 0 00 cos θ sin θ 00 − sin θ cos θ 00 0 0 1
.
Boosts or “rotations between space and time directions.”
Λµ′ν =
coshφ − sinhφ 0 0− sinhφ coshφ 0 0
0 0 1 00 0 0 1
.
The set of both translations and Lorentz transformations is a 10-parameter non-abelian group (4+3+3), called thePoincare group.
Pablo Laguna Gravitation: Special Relativity
For the boost transformation above,
t′ = t coshφ− x sinhφ
x′ = −t sinhφ + x coshφ .
Consider the point x′ = 0. Is is moving at
v =x
t=
sinhφ
coshφ= tanhφ .
Defining v = tanhφ and γ = 1/√
1− v2 one obtains the conventional expressions for Lorentz transformations:
t′ = γ(t − vx)
x′ = γ(x − vt)
Notice: We have set c = 1
Pablo Laguna Gravitation: Special Relativity
Inertial Frames
Inertial Frame: a frame in which Newton’s first law holds; that is,
d2x i
dt2= 0
Inertial Coordinates:
Build a rigid frame and label the grid points with coordinates (x, y, z); thus at each point
d2x i
dt2= 0
Place a clock at each grid point and synchronize them
Pablo Laguna Gravitation: Special Relativity
Clock Synchronization
Send light from your location to a neighboring point.
Reflect the light from the neighboring point back to your location.
Define the time at the neighboring point as
τ2 =τ3 + τ1
2
In this way, a single observer’s clock can be used to define temporalcoordinates everywhere.
Pablo Laguna Gravitation: Special Relativity
F1 and F2 are inertial frames
S are hypersurfaces of simultaneity
In Newtonian spacetimes, F1 and F2 are related byGalilean transformations
t′ = t
x′ = x − vt
y = y
z = z
In SR spacetimes, F1 and F2 are related by Lorentztransformations
t′ = γ(t − vx)
x′ = γ(x − vt)
y = y
z = z
Inertial frames can only differ by a rotation, translation and uniform motion
Pablo Laguna Gravitation: Special Relativity
O1 and O2 are at rest
O3 is in uniform motion
O4 is accelerating
O5 and O6 orbiting about P, which is at rest
O7 and O8 orbiting about P′, which is inuniform motion
O1 and O2 are in uniform motion
O3 is at rest
O4 is accelerating
O5 and O6 orbiting about P, which is inuniform motion
O7 and O8 orbiting about P′, which is at rest
Principle of Relativity
Identical experiments carried out in different inertial frames give identical results.
Pablo Laguna Gravitation: Special Relativity
Vectors
p
manifold
M
Tp
A vector is located at a given point in space-time
Tangent space Tp at p is the set of all possible vectorslocated at that point.
A vector space is a collection of vectors-objects such thatfor any two vectors V and W and real numbers a and b,
(a + b)(V + W ) = aV + bV + aW + bW .
Vector field is set of vectors with exactly one at each point inspacetime.
Tangent bundle T (M) is set of all the tangent spaces of amanifold M.
A basis is any set of vectors which both spans the vectorspace and is linearly independent .
Pablo Laguna Gravitation: Special Relativity
Consider at each Tp a basis e(µ) adapted to the coordinates xµ; that is, e(1) pointing along the x-axis, etc. Then,any abstract vector A can be written as
A = Aµe(µ) .
The coefficients Aµ are the components of the vector A.
The real vector is the abstract geometrical entity A, while the components Aµ are just the coefficients of thebasis vectors in some convenient basis.
The parentheses around the indices on the basis vectors e(µ) label collection of vectors, not components ofa single vector.
Pablo Laguna Gravitation: Special Relativity
Tangent to a vector curveConsider a parametrized curve or path xµ(λ). Its tangent vector V (λ) has components
Vµ =dxµ
dλ.
Thus V = Vµe(µ) . Under a Lorentz transformation
xµ → xµ′ = Λµ′νxν
thereforeVµ → Vµ′ = Λµ′
νVν.
However, the vector itself is invariant under Lorentz transformations. That is,
V = Vµe(µ) = Vν′ e(ν′) = Λν′µVµe(ν′) .
Therefore,
e(µ) = Λν′µe(ν′)
Let Λν′µ denote the inverse to Λν′
µ; that is, such that
Λν′µΛσ′
µ = δσ′ν′ , Λν′
µΛν′ρ = δ
µρ ,
where δµρ is the Kronecker delta. Thus,
e(ν′) = Λν′µ e(µ) .
Pablo Laguna Gravitation: Special Relativity
Dual Vectors or 1-forms
Co-tangent space T∗p : dual space to Tp .
Dual vector space: is the space of all linear maps from the original vector space to the real numbers. Thatis, if ω ∈ T∗p is a dual vector, then
ω(aV + bW ) = aω(V ) + bω(W ) ∈ R ,
where V , W are vectors and a, b are real numbers.
The dual vectors or one-forms yield also a vector space. That is if ω and η are dual vectors, we have
(a + b)(ω + η) = aω + bω + aη + bη .
Pablo Laguna Gravitation: Special Relativity
Dual Vectors or 1-forms
There exists a set of basis dual vectors θ(ν) by demanding
θ(ν)(e(µ)) = δ
νµ .
Thereforeω = ωµθ
(µ).
Elements of Tp are also referred as contravariant vectors and elements of T∗p as covariant vectors.
The action of a one-form on a vector:
ω(V ) = ωµVνθ
(µ)(e(ν)) = ωµVνδµν = ωµVµ ∈ R .
Vectors are also linear maps on dual vectors, i.e.
V (ω) ≡ ω(V ) = ωµVµ.
Therefore, the dual space to the dual vector space is the original vector space itself.
Pablo Laguna Gravitation: Special Relativity
Transformation properties of dual vectors:
ωµ′ = Λµ′νων ,
and for basis dual vectors,
θ(ρ′) = Λρ′
σ θ(σ)
.
Simplest example of a dual vector is the gradient of a scalar function:
dφ =∂φ
∂xµθ
(µ).
with a transformation rule∂φ
∂xµ′=∂xµ
∂xµ′∂φ
∂xµ= Λµ′
µ ∂φ
∂xµ,
Shorthand notations for partial derivatives:
∂φ
∂xµ= ∂µφ = φ, µ .
Notice also
∂µφ∂xµ
∂λ=
dφ
dλ.
Pablo Laguna Gravitation: Special Relativity
Tensors
A tensor T of type (or rank) (k, l) is a multilinear map from a collection of dual vectors and vectors to R:
T : T∗p × · · · × T∗p × Tp × · · · × Tp → R
(k times) (l times)
“×” denotes the Cartesian product; e.g., Tp × Tp is the space of ordered pairs of vectors.
Multilinearity means that the tensor acts linearly in each of its arguments; e.g., for a tensor of type (1, 1),
T (aω + bη, cV + dW ) = acT (ω, V ) + adT (ω,W ) + bcT (η, V ) + bdT (η,W ) .
A scalar is a type (0, 0) tensor, a vector is a type (1, 0) tensor, and a dual vector is a type (0, 1) tensor.
The space of all tensors of a fixed type (k, l) forms a vector space; that is, they can be added together andmultiplied by real numbers.
Pablo Laguna Gravitation: Special Relativity
Tensor Product
The tensor product of a (k, l) tensor T with a (m, n) tensor S , denoted by T ⊗ S, is defined by
(T ⊗ S)(ω(1), . . . , ω
(k), . . . , ω
(k+m), V (1)
, . . . , V (l), . . . , V (l+n))
= T (ω(1), . . . , ω(k), V (1), . . . , V (l))S(ω(k+1), . . . , ω(k+m), V (l+1), . . . , V (l+n)) .
Steps:
act T on the appropriate set of dual vectors and vectors
act S on the remainder dual vectors and vectors
multiply the answers.
Notice, in general, T ⊗ S 6= S ⊗ T .
Examples
Given two dual-vectors U and V
(U ⊗ V )(e(µ), e(ν)) = U(e(µ))V (e(ν)) = Uαθ(α)(e(µ))Vβ θ
(β)(e(ν)) = UαδαµVβδ
βν = UµVν
Given a tensor T of rank (1,1) and a tensor S of rank (2,1), their tensor product is
(T ⊗ S)(θ(µ), e(ν), θ
(α), θ
(β), e(γ)) = Tµ
νSαβγ
Pablo Laguna Gravitation: Special Relativity
Tensor BasisRecall for a vector T
T (θ(µ)) = Tαe(α)(θ(µ)) = Tαδµα = Tµ
Also(e(α) ⊗ e(β))(θ(µ)
, θ(ν)) = e(α)(θ(µ))e(β)(θ(ν)) = δ
µαδ
νβ
We can then view (e(α) ⊗ e(β)) as the basis for rank (2,0) tensors. Thats is,
T = Tαβ (e(α) ⊗ e(β))
The components of T can be found from
T (θ(µ), θ
(ν))) = Tαβ (e(α) ⊗ e(β))(θ(µ), θ
(ν))) = Tαβδµαδ
νβ = Tµν
In general, the basis for the space of all (k, l) tensors is constructed by taking tensor products of basis vectors anddual vectors; namely
e(µ1) ⊗ · · · ⊗ e(µk ) ⊗ θ(ν1) ⊗ · · · ⊗ θ(νl )
.
An arbitrary (k, l) tensor is then written as
T = Tµ1···µk ν1···νl e(µ1) ⊗ · · · ⊗ e(µk ) ⊗ θ(ν1) ⊗ · · · ⊗ θ(νl )
.
and its components as
Tµ1···µk ν1···νl = T (θ(µ1), . . . , θ
(µk ), e(ν1), . . . , e(νl )) .
Pablo Laguna Gravitation: Special Relativity
More Tensor Properties
A (k, l) tensor has k upper indices and l lower indices; that is, Tµ1···µk ν1···νl
The order of the indices is important; that is, Tµνρσν 6= Tµρν
σν
Lorentz transformations:
Tµ′1···µ′kν′1···ν
′l
= Λµ′1 µ1 · · · Λ
µ′k µk Λν′1ν1 · · · Λν′l
νl Tµ1···µk ν1···νl
Tensor Contractions
Tµν : Vν → Uµ = Tµ
νVν
Tµρσ : Sσ
ρν → Uµν = Tµρ
σSσρν
Pablo Laguna Gravitation: Special Relativity
More Tensor Properties
Inner or dot product: Given the metric η,
η(V ,W ) = ηµνVµWν = V · W
If η(V ,W ) = 0, the vectors are orthogonal.
Since η(V ,W ) = V · W is a scalar, it is left invariant under Lorentz transformations.
norm of a vector is given by V · V .
if ηµνVµVν is
< 0 , Vµ is timelike= 0 , Vµ is lightlike or null> 0 , Vµ is spacelike .
Pablo Laguna Gravitation: Special Relativity
Important Tensors
Kronecker delta δµν = diag(1, 1, 1, 1), of type (1, 1),
Inverse metric ηµν , a type (2, 0) such that ηµνηνρ = ηρνηνµ = δµρ
Levi-Civita tensor, a (0, 4) tensor:
εµνρσ =
+1 if µνρσ is an even permutation of 0123−1 if µνρσ is an odd permutation of 01230 otherwise .
Electromagnetic field strength tensor, a (0, 2) tensor:
Fµν =
0 −E1 −E2 −E3
E1 0 B3 −B2E2 −B3 0 B1E3 B2 −B1 0
= −Fνµ
Pablo Laguna Gravitation: Special Relativity
Manipulating Tensors
Sµρσ = Tµνρ
σν
Tαβµδ = η
µγTαβγδ
Tµβγδ = ηµαTαβ
γδ
Tµνρσ = ηµαηνβη
ργησδTαβ
γδ
Vµ = ηµνVν
ωµ = η
µνων
Pablo Laguna Gravitation: Special Relativity
Maxwell’s Equations
∇× B− ∂t E = 4πJ
∇ · E = 4πρ
∇× E + ∂t B = 0
∇ · B = 0
in component notation
εijk∂j Bk − ∂0E i = 4πJ i
∂i Ei = 4πJ0
εijk∂j Ek + ∂0Bi = 0
∂i Bi = 0 .
recall
F0i = E i
F ij = εijk Bk
then, the first two Maxwell eqs. read
∂j Fij − ∂0F0i = 4πJ i
∂i F0i = 4πJ0
.
or equivalently
∂µFνµ = 4πJν
similarly the last two Maxwell eqs. can be rewritten as
∂[µFνλ] = 0
Pablo Laguna Gravitation: Special Relativity
Energy and Momentum
Four-velocity
Uµ =dxµ
dτ.
Since dτ2 = −ηµνdxµdxν , the four-velocity is time-like, i.e., ηµνUµUν = −1
In the rest frame of a particle, its four-velocity has components Uµ = (1, 0, 0, 0).
Four-momentum: pµ = mUµ with m the rest-mass of the particle.
Energy of the particle is given by E = po = m. Recall E = m c2.
Notice that
pµ = mUµ = mdxµ
dτ= m
dt
dτ
dxµ
dt= mγVµ
where γ = dt/dτ and Vµ = (1, dx i/dt) = (1,~v)
Notice also, from
−1 = ηµνUµUν = γ2ηµνVµVν = γ
2(−1 + v2)
thus γ = 1/√
1− v2 the Lorentz boost factor.
Pablo Laguna Gravitation: Special Relativity
Newton’s 2nd Lay and Special Relativity
Newton’s Second Law:
~f = m~a =d~p
dt
in SR
fµ = md2
dτ2xµ(τ) =
d
dτpµ(τ)
Lorentz Law:~f = q(~E +~v × ~B)
in SRfµ = qUλFλ
µ.
Pablo Laguna Gravitation: Special Relativity