Special Relativity

Question 1

Inertial Reference Frame for Relativity

Definition

Answer 1

• 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)

Question 2

Line Element

Definition

Answer 2

-the infinitesimal separation of two points in spacetime is defined by the line element:
ds² = -c²dt² + dx² + dy² + dz²

Question 3

Minkowski Metric

Definition

Answer 3

ημν = diag{-1,1,1,1}

• a tensor
• note the sign/signature of the metric (-,+,+,+), opposite convention is used in some textbooks: (+,-,-,-)

Question 4

Indices Notation

Answer 4

• 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

Question 5

Line Element & Lightlike, Timelike and Spacelike Events

Answer 5

• lightlike events (photons) : ds² = 0

- timelike events (massive particles, vc) : ds²>0

Question 6

Proper Time

Answer 6

-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

Question 7

Einstein Summation Convention

Answer 7

-implied summation over 0,1,2,3 when two indices are the same

Question 8

Scalar

Definition

Answer 8

• 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)

Question 9

Vector

Definition

Answer 9

• 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)

Question 10

Vector Field

Definition

Answer 10

• has magnitude like a vector but ;in-built’ direction

- only one can be defined at any point

Question 11

Vector in 4D Space

Answer 11

-a 4-vector

Question 12

Define a Basis

Answer 12

{\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

Question 13

Basis Vectors in a Dot Product

Answer 13

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

Question 14

Dual Basis

Answer 14

there is nothing to stop us from writing:
X = Xα e^α
-where e is the dual basis

Question 15

Metric Tensor and Indices

Answer 15

Yα = ηαβ Y^β

-the metric tensor can be used to raise or lower an index

Question 16

Inverse Metric

Answer 16

ηαβ ηαγ = δ^γ_β
-where δ is a Kronecker delta
-in the special case where all indices are dummy:
ηαβ η^αβ = 4

Question 17

Dot Product and Spacelike, Timelike & Lightlike Vectors

Answer 17

• take X . X = |X|², an invariant scalar
• frame transformations require such behaviour
• spacelike: |X|² > 0
• timelike: |X|² < 0
• lightlike: |X| = 0

Question 18

Tensor

Definition

Answer 18

• the definition of a tensor is something that transforms like a tensor
• both vectors and scalars are special cases of tensors

Question 19

Tensor Rank

Answer 19

-a tensor is rank (m,n) if it has m upper indices (contravariant) and n lower indices (covariant)

Question 20

Tensor Quotient Theorem

Answer 20

-a rank (a,b) tensor multiplied by a rank (m,n) tensor results in a rank (m+a,n+b) tensor

Question 21

Symmetric and Antisymetric Tensors

Answer 21

-symmetric:
T^αβ = T^βα
-antisymmetric
T^αβ = - T^βα

Question 22

Expressing Tensors in Terms of Symmetric and Antisymmetric Components

Answer 22

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

Question 23

Frame Transformations

Invariance of ds²

Answer 23

-new frame: x^μ’, original frame: x^μ
-invariance:
ds² = ημν dx^μ dx^ν = η’αβ dx’^α dx’^β

Question 24

Frame Transformations

Special Relativity

Answer 24

-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

Question 25

Frame Transformations

Lorentz Transformations

Answer 25

-denote linear transformations by Λ^μ_ν:
Λ^μ_ν dx^ν = dx’^μ/dx^ν dx^ν
-these are the Lorentz transformations (LTs)
-only ‘proper’ LTs are allowed: detΛ=+1 and Λ^0_0≥1
-this gives constancy of magnitude of vectors etc. and ensures dot products behave correctly

Question 26

Proper Lorentz Transforms

Answer 26

• only two transformations satisfy the rules for a proper LT:
• -rotations e.g. by angle θ in the x-y plane
• -boosts e.g. to a frame with speed v in the x-direction

Question 27

β

Answer 27

β = v/c

-where v is the frame speed and c is the speed of light

Question 28

γ

Answer 28

γ = [1 - β²]^(-1/2)

Question 29

Lorentz Transform Inverse

Answer 29

-every LT must have an inverse
-if the LT is:
Λ^μ_ν = dx’^μ/dx^ν
-then the inverse is:
Λ_α^β = dx^α/dx’^β