Inertial Reference Frame for Relativity
Definition
- spacetime separations are invariant (gives constant speed of light)
- clocks can be synchronised (allows observers to compare time in different reference frames)
- (loccally) Euclidian spatial geometry (true everywhere for SR but not GR)
- unaccelerated (ensures physical laws are the same between frames)
Line Element
Definition
-the infinitesimal separation of two points in spacetime is defined by the line element:
ds² = -c²dt² + dx² + dy² + dz²
Minkowski Metric
Definition
ημν = diag{-1,1,1,1}
- a tensor
- note the sign/signature of the metric (-,+,+,+), opposite convention is used in some textbooks: (+,-,-,-)
Indices Notation
- for this course, greek indices take the value of all possible coordinates (time and space)
- Roman indices, e.g. i, take values relating only to the spatial coordinates
Line Element & Lightlike, Timelike and Spacelike Events
- lightlike events (photons) : ds² = 0
- timelike events (massive particles, vc) : ds²>0
Proper Time
-massive particles and timelike events allow definition of proper time:
-c²dτ² = -c²dt² + dx² + dy² + dz² = ds²
c²dτ² = -ds²
-this is time as measured by an observer, it can differ from coordinate time
-it is the clock time if you have a clock
-where τ is an observable and t, x, y, z are coordinates
Einstein Summation Convention
-implied summation over 0,1,2,3 when two indices are the same
Scalar
Definition
- a scalar in relativity is anything which has a value but no direction
- invariant under transformation
- not all ‘real’ scalars are relativistic scalars (e.g. temperature isn’t)
Vector
Definition
- a geometrical object independent of any coordinate system
- can be defined as the tangent to the manifold
- there are an infinite number of tangents to that point (same magnitude but different direction)
Vector Field
Definition
- has magnitude like a vector but ;in-built’ direction
- only one can be defined at any point
Vector in 4D Space
-a 4-vector
Define a Basis
{\bf X} = X^α eα
- this is NOT a scalar (inner, dot) product, it is a vector sum
- as shorthand the basis can be omitted and this is just written as X^α, which is a contravariant vector
Basis Vectors in a Dot Product
X . Y = X^α Y^β eα . eβ
-the real dot product is between the basis vectors
-in special relativity, these basis vectors are orthonormal:
eα . eβ = ηαβ
=>
X . Y = ηαβ X^α Y^β
-the dot product must result in a scalar
Dual Basis
there is nothing to stop us from writing:
X = Xα e^α
-where e is the dual basis
Metric Tensor and Indices
Yα = ηαβ Y^β
-the metric tensor can be used to raise or lower an index
Inverse Metric
ηαβ ηαγ = δ^γ_β
-where δ is a Kronecker delta
-in the special case where all indices are dummy:
ηαβ η^αβ = 4
Dot Product and Spacelike, Timelike & Lightlike Vectors
- take X . X = |X|², an invariant scalar
- frame transformations require such behaviour
- spacelike: |X|² > 0
- timelike: |X|² < 0
- lightlike: |X| = 0
Tensor
Definition
- the definition of a tensor is something that transforms like a tensor
- both vectors and scalars are special cases of tensors
Tensor Rank
-a tensor is rank (m,n) if it has m upper indices (contravariant) and n lower indices (covariant)
Tensor Quotient Theorem
-a rank (a,b) tensor multiplied by a rank (m,n) tensor results in a rank (m+a,n+b) tensor
Symmetric and Antisymetric Tensors
-symmetric:
T^αβ = T^βα
-antisymmetric
T^αβ = - T^βα
Expressing Tensors in Terms of Symmetric and Antisymmetric Components
M^αβ = 1/2 [ M^αβ + M^βα ] + 1/2 [ M^αβ - M^βα ]
= M^(αβ) + M^[αβ]
-where () brackets indicate symmetry and [] brackets antisymmetry
Frame Transformations
Invariance of ds²
-new frame: x^μ’, original frame: x^μ
-invariance:
ds² = ημν dx^μ dx^ν = η’αβ dx’^α dx’^β
Frame Transformations
Special Relativity
-for special relativity η’=η
-differentiate with respect to x^σ
=>
∂²x’^α / ∂x^μ ∂x^σ = 0
-special relativity requires a transform that is linear in the coordinates
