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3rd Year Oxford Physics > Symmetry + Relativity > Flashcards

Symmetry + Relativity Flashcards

To add: energy-momentum flow, stress tensor, spinors, wave-like equations

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1
Q

What is an “inertial” reference frame?

A

One n which N1 is obeyed - a particle subjeted to no forces does not accelerate

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2
Q

Postulate 1

A

‘Principle of Relativity’ The laws of physics are the same in all inertial frames of reference.

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3
Q

Postulate 2

A

The Constancy of Speed of Light in Vacuum The speed of light in vacuum has the same value c in all inertial frames of reference.

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4
Q

What is an ‘event’?

A

A point in spacetime

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5
Q

What is the worldline?

A

The line of events which give the location of the patcile as a function of time

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6
Q

What is the spacetime interval? What is a timelike interval (explain in words too)?

A

s^2 = -c^2(t2-t1)^2 + (x2-x1)^2 + …

s^2 negative

a particle can travel from one event to another

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7
Q

an nxm matrix has how many rows?

A

n rows m columns

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8
Q

What is the inverse of a 2x2 matrix (a b) (c d)

A

1/(ad-bc) *

(d -b)

(-c a)

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9
Q

Lorentz factor gamma

A

1/(1-(v/c)^2)^1/2

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10
Q

d/dv (gamma) =

A

gamma ^ 3 * v/c^2

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11
Q

dt/dtau =

A

gamma

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12
Q

d/dv (gamma*v) =

A

gamma^3

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13
Q

Lorentz Transformations (all)

A

ct’ = gamma (ct - bx)

x’ = gamma (-bt + x)

y’ = y

z’ = z

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14
Q

velocity addition (all) also do vector version

A

ux` = (ux - v)/(1-ux*v/c^2)

uy` = uy/gamma(1-ux*v/c^2)

u[parallel]’ = (u[pa] - v)/(1-u.v/c^2)

u[perp]` = u[pe]/gamma(1-u.v/c^2)

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15
Q

Lorentz transformation matrix

A

g -gb 0 0

-gb g 0 0

0 0 1 0

0 0 0 1

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16
Q

define rapidity p

A

tanh(p) = v/c = beta

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17
Q

cosh(p)

sinh(p)

exp(p)

lorentz matrix in p

A

g

bg

(1+b/1-b)^1/2

cosh(p) -sinh(p) 0 0

-sinh(p) cosh(p) 0 0

0 0 1 0

0 0 0 1

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18
Q

mathematically, what is a lorentz transform. How does this effect 4-vectors

A

a rotation in spacetime the length of a 4-vector is conserved between frames

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19
Q

define 4-velocity what is its vector form

A

U=dX/dtau = (gc, gu)

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20
Q

4-acceleration?

A

A = dU/dtau

= g dU/dt

= g (dg/dt * c, dg/dt * u +ga)

= g^2 (u.a/c * g^2 , u.a/c^2 * g^2 * u + a)

a is three acceleration

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21
Q

dg/dt =

A

g^3 * u.a/c^2

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22
Q

what is the relation between 4-acceleration and 4-velocity

A

they are always orthogonal

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23
Q

4-force F =

A

F = dP/dtau

= (g W/c, g f)

f is 3-force = dp/dt

W = dE/dt

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24
Q

4-momentum P =

A

P = m0U

= g m0 dX/dt

= (g m0 c, g mo u)

= (E/c, p)

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25
Q

Wave vector K =

A

K = (w/c, k)

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26
Q

4-Force F=

A

F = dP/dtau =

(g*W/c, g*f)

f = 3-force = dp/dt

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27
Q

what is the scalar product of 2 4-vectors

A

a Lorentz-invariant quantity

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28
Q

for a pure force W =

A

dE/dt = f.u

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29
Q

f[para]’ =

f[perp]’ =

A

(f[pa] - (v/c^2)*dE/dt)/(1-u.v/c^2)

f[pe]/g(1-u.v/c^2)

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30
Q

phase velocity vp =

A

vp = w/k

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31
Q

dro/dtau =

A

a0/c ao = proper acceleration

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32
Q

[1] [2]->v_21 [3]->v_32 find the rapidity of [3] in the frame of [1] (v_31)

A

p_31 = p_21 + p+32

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33
Q

To derive the Dopplar Effect

A

K.U

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34
Q

to derive the headlight effect or aberration

A

K = (LAMBDA)^-1 K_0

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35
Q

What is a pure force?

A

U.F=0

dm0/dtau = 0

dE/dt = f.u

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36
Q

particle moves with velocity u in S S’ is moving with velocity v wrt S what is gamma_u’ = gamma for particle in S’

A

gamma_u gamma_v (1-u.v/c^2)

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37
Q

how does the pressure change from frame to frame

A

it stays the same

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38
Q

for a pure force dg/dt =

A

f.v/mc^2

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39
Q

an instantaneous rest frame is

A

an inertial frame

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40
Q

constant proper acceleration leads to

A

hyperbolic motion

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41
Q

equation of a hyperbola

A

(x-b)^2 - (ct)^2 = (c^2 / a_0)^2

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42
Q

for constant proper acceleration tau =

A

c*ro/a_0 ro = rapidity

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43
Q

A.A =

A

a_0 ^2

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44
Q

zero component lemma

A

if one component of a 4-vector is zero in all reference frames, then the whole 4-vector is zero

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45
Q

what do the “Zero component lemma” and 4-momentum conservation imply

A

4-momentum conservation energy conservation

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46
Q

P.P =

A

-m^2 c^2

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47
Q

energy momentum invariant

A

E^2 - (pc)^2 = m^2 c^4

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48
Q

velocity in terms of momentum and energy

A

v = pc^2 / E

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49
Q

P = P_1 + P_2 how to eliminate P_2 ?

A

‘Isolate and Square’

P_2 = P - P_1

P_2 . P_2 = (P - P_1).(P - P_1)

= -(m_2 c)^2

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50
Q

what is an elastic collision in special relativity

A

one in which the rest masses do not change

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51
Q

for a photon relate energy and momentum

A

E = pc

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52
Q

what is []’phi what does this imply?

A

a lorentz-invariant scalar field

[]’phi = /\ [] phi

hence []phi is a 4-vector

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53
Q

[] =

A

-1/c * d/dt

d/dx

d/dy

d/dz

all partial

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54
Q

4-Divergence [].F =

A

[]^T g F = 1/c dF0/dt + nabla(f)

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55
Q

D’Alembertian []^2 =

A

[].[] = []^T g []

= -1/c^2 d^2/dt^2 +nabla^2

nabla^2 is the Laplacian

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56
Q

wave equation in terms of a scalar field

A

[]^2 phi = 0

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57
Q

[].X =

A

4

58
Q

[].(K.X) = K is a constant 4-vector

A

K

59
Q

p + p –> p + p + pion

p hits stationary p

how to find threshold energy of incident p for pion creation

A

compare LABF before and ZMF after

equate energy invariants threshold energy

=> particles are stationary in ZMF

60
Q

electromagnetic force f =

A

q ( E + v ^ B )

61
Q

transformation of electromagnetic field (parallel and perp)

A

E[pa]’ = E[pa]

E[pe]’ = ga (E[pe] + v^B)

B[pa]’ = B[pa]

B[pe]’ = ga (B[pe] -v^E/c^2)

62
Q

electric field in a capacitor

A

charge per unit area / e0

63
Q

J 4-vector

A

(pc, j) p is charge density j is current density

64
Q

maxwell 1

A

div E = charge density/e0

65
Q

maxwell 2

A

div B = 0

66
Q

maxwell 3

A

curl E = -dB/dt

67
Q

maxwell 4

A

curl B = mu0 j + mu0 e0 dE/dt

j is current density vector

68
Q

B in terms of vector potential

A

B = curl A

69
Q

E in terms of vector and scalar potential

A

E = - grad phi - dA/dt (partial)

70
Q

quote the gauge transformations

A

A -> A + grad chi

phi -> phi - dchi/dt (partial)

71
Q

grad(1/r)

A

-1/r^2 r[hat]

72
Q

4-vector potential A and its gauge transformation

A

A = (phi/c , A)

A -> A + []chi

73
Q

maxwell equations in 4-vector potential

A

[]^2 A = -1/(e0 c^2) J

with [].A=0

74
Q

B in terms of E

A

B = v ^ E / c^2

75
Q

Liénard-Wiechert Potential of an arbitrarily moving charged particle

A

A = q/(4 pi e0) * (U/c) / (-R.U)

A is 4 vector potential

U is 4 velocity of particle

R is 4 displacement from particle to point of interest (where we want to know the potential)

76
Q

poisson equation

A

nabla^2 phi = -p/eo

p is charge density

77
Q

for a 4-vector A A’ =

A

A’ = /\ A

78
Q

M1 integral form

A

integral (E.dS) = Qenc / e0

integral over a gaussian surface = E * surface area

79
Q

M3 integral form

A

int over closed curve (E.dl)

= -d/dt surface int (B.dS) open surface integral

80
Q

M4 integral form

A

int over closed curve (B.dl) =

mu0 surface int (J.dS) + mu0 e0 d/dt surface int (E.dS)

both are open surface integrals

81
Q

T’ =

T is a tensor

T’ is that tensor in a new frame

A

T’ = /\ T /\^T

82
Q

what is a 0th rank tensor

A

a scalar invariant

83
Q

what is a 1st rank tensor

A

a 4-vector

84
Q

what is a 2nd rank tensor

A

something that transforms like T’ = /\ T /\^T

85
Q

what is ab^T

eg:

(1, 2, 3)(10; 20; 30) =

A

an outer product = a 2nd rank tensor

=

(10, 20, 30;

20, 40, 60;

30, 60, 90)

86
Q

T . B =

T is a tensor

B is a 4-vector

A

= T g B

g is the metric

87
Q

index notation summation convention

(Tensor).(4-vector)

A

A^ab X_b =

SUM_(b=0 ->3) [A^ab X_b]

88
Q

define the metric tensor

A

A^T g B = A’^T g’ B’

=> g’ = (/\^-1)^T g /\^-1

89
Q

how does a contravariant 4-vector transform

A

(thing)’ = /\ (thing)

the kind of 4-vector we are used to

90
Q

how does a covariant 4-vector transform

give an example

A

(thing)’ = (/\^-1)^T (thing)

gX

91
Q

index lowering for 4-vectors

F_a =

hence A.B =

A

F_a = g_ab F^b

A.B = A^T g B

= A^a g_ab B^b

= A^a B_b

92
Q

index raising for 4-vectors F^a =

A

F^a = g^ab F_b

where g^ab = (g_ab)^-1

93
Q

define g^ab

what does this imply

A

g^ab g_ab = delta^a _b

delta is the kronecker delta

=> g^ab is the inverse of g_ab

94
Q

partial

delta_a = []_a

delta^a = []^a

which is the normal []

A

d/(dx^a) = (1/c d/dt, d/dx, d/dy, d/dz)

delta^a = g^ab delta_b

= same as above but first term is negative

^a is the normal []

95
Q

rules for tensor sum

A

same number of indices

same valence - number up and down

96
Q

rules for tensor outer product

A

for 4-vectors, examples:

A^a B^b = T^ab

A^a B_b = T^a_b

A^a_b B^c_de = C^ac_bde

97
Q

rules for tensor contraction

A^a B_b

F^ab

A

there must be one up and one down

examples

A^a B_b -> A^a B_a

F^ab -> g_ac F^cb = F_a^b -> F_a^a = a scalar

98
Q

in matrix notation what are:

A^ab B_b

A^ba B_b

A

A.B

B_b A^ba = B.A

remember the dot implies the metric is present

99
Q

what is T^a_a

A

to start, take T^ab and post-multiply by the metric

-a0b0 + a1b1 + a2b2 + a3b3 + a4b4

100
Q

[] in index notation

A

delta^a

101
Q

product rule

partial delta^a (U^b V^c) =

A

(delta^a U^b) V^c + U^b (delta^a V^c)

102
Q

4-curl, what is the outcome?

[]^A =

A

- ( [] (A^T) )^T = delta^a A^b - delta^b A^a

an antisymmetric tensor

103
Q

a combination of 2 lorentz boosts leads to…

A

a lorentz boost AND a rotation

the rotation comes from the geometry of spacetime

104
Q

an object moving in a circle anticlockwise will precess in what direction

A

clockwise

105
Q

Faraday Tensor

A

(0, Ex/c, Ey/c, Ez/c)

(-Ex/c, 0, Bz, -By)

(-Ey/c, -Bz, 0, Bx)

(-Ez/c, By, -Bx, 0)

106
Q

the faraday tensor is antisymmetric, what does this imply

A

delta_a delta_b F^ab = 0

=> delta_a J^a = 0

=> charge conservation

107
Q

1/2 F^ab F_ab =

A

B^2 - E^2 /c == D (invariant)

108
Q

4 angular momentum

A

L = XP^T - PX^T

L^ab = X^a P^b - X^b P^a

109
Q

the centroid x_c =

A

SUM (x_i E_i)/E_tot

110
Q

dual tensor F(tilde)_ab

with example F(tilde)_12

A

1/2 E_abcd F^cd

E is levi-civita tensor

= 1/2(E_1230 F^30 + E_1203 F^03)

= 1/2( -F^30 + F^03 )

111
Q

g^ab g_bc =

A

kronecker delta^a_c = identity

112
Q

levi civita tensor definition

A

E_abcd =

+1 for 0, 1, 2, 3 and even permutations

-1 for odd

0 if two the same

113
Q

E_01cd=

A

0 0 0 0

0 0 0 0

0 0 0 1

0 0 -1 0

114
Q

shortcut for dual of an antisymmetric tensor F

A

F =

0 ax ay az

  • ax 0 bz -by
  • ay -bz 0 bx
  • az by -bx 0

a -> -b b -> a

115
Q

delta_a x^b =

A

1 if a=b

0 if a/=b

so = delta^a_b

116
Q

delta_a x_b =

A

1 if a=b but -1 for a=0

0 if a/=b so = metric = g_ab

117
Q

delta_a (k^a x_a) =

for constant 4 vector k

A

k^a delta_a (x_a) =

k^a g_ab =

k_a

118
Q

delta^a sin(k^a x_a) =

A

[delta^a(k^a x_a)] cos(k^a x_a) =

k^a cos(k^a x_a)

119
Q

nabla^ 1/r

A

laplacian (1/r) = -4pi diracdelta^(3) (r)

shorthand for d(x)d(y)d(z) - there are 3 deltas

120
Q

div( a exp[i(k.r-wt)] ) =

A

i k.a exp[i(k.r-wt)]

121
Q

curl( a exp[i(k.r-wt)] ) =

A

i k^a exp[i(k.r-wt)]

122
Q

relate

k, c, w

A

w = kc

123
Q

|E0| =

A

c |B0|

124
Q

the Lorenz gauge

A

[].A = 0

125
Q

in electrostatics …

A

there are fixed charges and no currents

126
Q

General form of a retarded spherical wave.

How is this different to an advanced wave?

What do these look like

A

f = g(t - r/c)/r

  • -> +

advanced wave collapses in to centre, retarded is the opposite

127
Q

What do the Liènard-Wiechert potentials imply?

A

A depends on U and R but not acceleration.

128
Q

radiated power:

energy flux

A

given by the poynting vector

N = e0c2 E^B

129
Q

Larmor’s formula

A

PL = 2/3 * q2/4πe0 * a2/c3

130
Q

Pauli-Lubaski 4-vector

A
131
Q

indices “see-saw” trick

A

A^a_b B^b

= A^ab B_b

132
Q

what are polar an axial vectors?

A

polar change sign under parity transformation. These are normal vectors like x, p. Axial vectors (like L, B) do not change sign

133
Q

(polar vector) ^ (polar vector) =

A

axial vector

134
Q

Lagrangian

what is it a function of

A

L = T - V

L( {qi}, {dqi/dt}, t)

135
Q

Euler Lagrange Equations

A

d/dt (partial dL/dqi[dot] ) = partial dL/dqi

136
Q

canonical momentum p(tilde) =

A

partial dL/dqi[dot]

137
Q

generalised force

A

partial dL/dqi

138
Q

Hamiltonian

A

H( q, p~, t) = SUM( p~i qi(dot) ) - Lagrangian

139
Q

the action

how does this relate to euler lagrange eqns?

A

path integral (Lagrangian dt)

the paths of min/max action satisfy euler lagrange equations

140
Q

Hamilton’s canonical equations

A

dq/dt = dH/dp

dp/dt = - dH/dq

for all qi

and p is pi~

141
Q

lagrangian of a freely moving particle

A

-mc2/gamma

142
Q

particle hits stationary particle, they stick together. How to find mass of combined partcle and its velocity?

A

compare lab initial and ZMF after

look for E2-p2=m2

find combined mass M

energy conservation from lab initial to lab final

E = gamma M

find gamma

3rd Year Oxford Physics (4 decks)

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