Symmetry + Relativity Flashcards
To add: energy-momentum flow, stress tensor, spinors, wave-like equations
1
Q
What is an “inertial” reference frame?
A
One n which N1 is obeyed - a particle subjeted to no forces does not accelerate
2
Q
Postulate 1
A
‘Principle of Relativity’ The laws of physics are the same in all inertial frames of reference.
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.
4
Q
What is an ‘event’?
A
A point in spacetime
5
Q
What is the worldline?
A
The line of events which give the location of the patcile as a function of time
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
7
Q
an nxm matrix has how many rows?
A
n rows m columns
8
Q
What is the inverse of a 2x2 matrix (a b) (c d)
A
1/(ad-bc) *
(d -b)
(-c a)
9
Q
Lorentz factor gamma
A
1/(1-(v/c)^2)^1/2
10
Q
d/dv (gamma) =
A
gamma ^ 3 * v/c^2
11
Q
dt/dtau =
A
gamma
12
Q
d/dv (gamma*v) =
A
gamma^3
13
Q
Lorentz Transformations (all)
A
ct’ = gamma (ct - bx)
x’ = gamma (-bt + x)
y’ = y
z’ = z
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)
15
Q
Lorentz transformation matrix
A
g -gb 0 0
-gb g 0 0
0 0 1 0
0 0 0 1
16
Q
define rapidity p
A
tanh(p) = v/c = beta
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
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
19
Q
define 4-velocity what is its vector form
A
U=dX/dtau = (gc, gu)
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
21
Q
dg/dt =
A
g^3 * u.a/c^2
22
Q
what is the relation between 4-acceleration and 4-velocity
A
they are always orthogonal
23
Q
4-force F =
A
F = dP/dtau
= (g W/c, g f)
f is 3-force = dp/dt
W = dE/dt
24
Q
4-momentum P =
A
P = m0U
= g m0 dX/dt
= (g m0 c, g mo u)
= (E/c, p)
25
Wave vector K =
K = (w/c, k)
26
4-Force F=
F = dP/dtau =
(g\*W/c, g\*f)
f = 3-force = dp/dt
27
what is the scalar product of 2 4-vectors
a Lorentz-invariant quantity
28
for a pure force W =
dE/dt = f.u
29
f[para]’ =
f[perp]’ =
(f[pa] - (v/c^2)\*dE/dt)/(1-u.v/c^2)
f[pe]/g(1-u.v/c^2)
30
phase velocity vp =
vp = w/k
31
dro/dtau =
a0/c ao = proper acceleration
32
[1] [2]-\>v\_21 [3]-\>v\_32 find the rapidity of [3] in the frame of [1] (v\_31)
p\_31 = p\_21 + p+32
33
To derive the Dopplar Effect
K.U
34
to derive the headlight effect or aberration
K = (LAMBDA)^-1 K\_0
35
What is a pure force?
U.F=0
dm0/dtau = 0
dE/dt = f.u
36
particle moves with velocity u in S S' is moving with velocity v wrt S what is gamma\_u' = gamma for particle in S'
gamma\_u gamma\_v (1-u.v/c^2)
37
how does the pressure change from frame to frame
it stays the same
38
for a pure force dg/dt =
f.v/mc^2
39
an instantaneous rest frame is
an inertial frame
40
constant proper acceleration leads to
hyperbolic motion
41
equation of a hyperbola
(x-b)^2 - (ct)^2 = (c^2 / a\_0)^2
42
for constant proper acceleration tau =
c\*ro/a\_0 ro = rapidity
43
A.A =
a\_0 ^2
44
zero component lemma
if one component of a 4-vector is zero in all reference frames, then the whole 4-vector is zero
45
what do the “Zero component lemma” and 4-momentum conservation imply
4-momentum conservation energy conservation
46
P.P =
-m^2 c^2
47
energy momentum invariant
E^2 - (pc)^2 = m^2 c^4
48
velocity in terms of momentum and energy
v = pc^2 / E
49
P = P\_1 + P\_2 how to eliminate P\_2 ?
‘Isolate and Square’
P\_2 = P - P\_1
P\_2 . P\_2 = (P - P\_1).(P - P\_1)
= -(m\_2 c)^2
50
what is an elastic collision in special relativity
one in which the rest masses do not change
51
for a photon relate energy and momentum
E = pc
52
what is []'phi what does this imply?
a lorentz-invariant scalar field
[]'phi = /\ [] phi
hence []phi is a 4-vector
53
[] =
-1/c \* d/dt
d/dx
d/dy
d/dz
all partial
54
4-Divergence [].F =
[]^T g F = 1/c dF0/dt + nabla(f)
55
D'Alembertian []^2 =
[].[] = []^T g []
= -1/c^2 d^2/dt^2 +nabla^2
nabla^2 is the Laplacian
56
wave equation in terms of a scalar field
[]^2 phi = 0
57
[].X =
4
58
[].(K.X) = K is a constant 4-vector
K
59
p + p --\> p + p + pion
p hits stationary p
how to find threshold energy of incident p for pion creation
compare LABF before and ZMF after
equate energy invariants threshold energy
=\> particles are stationary in ZMF
60
electromagnetic force f =
q ( E + v ^ B )
61
transformation of electromagnetic field (parallel and perp)
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
electric field in a capacitor
charge per unit area / e0
63
J 4-vector
(pc, j) p is charge density j is current density
64
maxwell 1
div E = charge density/e0
65
maxwell 2
div B = 0
66
maxwell 3
curl E = -dB/dt
67
maxwell 4
curl B = mu0 j + mu0 e0 dE/dt
j is current density vector
68
B in terms of vector potential
B = curl A
69
E in terms of vector and scalar potential
E = - grad phi - dA/dt (partial)
70
quote the gauge transformations
A -\> A + grad chi
phi -\> phi - dchi/dt (partial)
71
grad(1/r)
-1/r^2 r[hat]
72
4-vector potential A and its gauge transformation
A = (phi/c , A)
A -\> A + []chi
73
maxwell equations in 4-vector potential
[]^2 A = -1/(e0 c^2) J
with [].A=0
74
B in terms of E
B = v ^ E / c^2
75
Liénard-Wiechert Potential of an arbitrarily moving charged particle
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
poisson equation
nabla^2 phi = -p/eo
p is charge density
77
for a 4-vector A A' =
A' = /\ A
78
M1 integral form
integral (E.dS) = Qenc / e0
integral over a gaussian surface = E \* surface area
79
M3 integral form
int over closed curve (E.dl)
= -d/dt surface int (B.dS) open surface integral
80
M4 integral form
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
T' =
T is a tensor
T' is that tensor in a new frame
T' = /\ T /\^T
82
what is a 0th rank tensor
a scalar invariant
83
what is a 1st rank tensor
a 4-vector
84
what is a 2nd rank tensor
something that transforms like T' = /\ T /\^T
85
what is ab^T
eg:
(1, 2, 3)(10; 20; 30) =
an outer product = a 2nd rank tensor
=
(10, 20, 30;
20, 40, 60;
30, 60, 90)
86
T . B =
T is a tensor
B is a 4-vector
= T g B
g is the metric
87
index notation summation convention
(Tensor).(4-vector)
A^ab X\_b =
SUM\_(b=0 -\>3) [A^ab X\_b]
88
define the metric tensor
A^T g B = A'^T g' B'
=\> g' = (/\^-1)^T g /\^-1
89
how does a contravariant 4-vector transform
(thing)' = /\ (thing)
the kind of 4-vector we are used to
90
how does a covariant 4-vector transform
give an example
(thing)' = (/\^-1)^T (thing)
gX
91
index lowering for 4-vectors
F\_a =
hence A.B =
F\_a = g\_ab F^b
A.B = A^T g B
= A^a g\_ab B^b
= A^a B\_b
92
index raising for 4-vectors F^a =
F^a = g^ab F\_b
where g^ab = (g\_ab)^-1
93
# define g^ab
what does this imply
g^ab g\_ab = delta^a \_b
delta is the kronecker delta
=\> g^ab is the inverse of g\_ab
94
partial
delta\_a = []\_a
delta^a = []^a
which is the normal []
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
rules for tensor sum
same number of indices
same valence - number up and down
96
rules for tensor outer product
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
rules for tensor contraction
A^a B\_b
F^ab
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
in matrix notation what are:
A^ab B\_b
A^ba B\_b
A.B
B\_b A^ba = B.A
remember the dot implies the metric is present
99
what is T^a\_a
to start, take T^ab and post-multiply by the metric
-a0b0 + a1b1 + a2b2 + a3b3 + a4b4
100
[] in index notation
delta^a
101
product rule
partial delta^a (U^b V^c) =
(delta^a U^b) V^c + U^b (delta^a V^c)
102
4-curl, what is the outcome?
[]^A =
[](A^T) - ( [] (A^T) )^T = delta^a A^b - delta^b A^a
an antisymmetric tensor
103
a combination of 2 lorentz boosts leads to...
a lorentz boost AND a rotation
the rotation comes from the geometry of spacetime
104
an object moving in a circle anticlockwise will precess in what direction
clockwise
105
Faraday Tensor
(0, Ex/c, Ey/c, Ez/c)
(-Ex/c, 0, Bz, -By)
(-Ey/c, -Bz, 0, Bx)
(-Ez/c, By, -Bx, 0)
106
the faraday tensor is antisymmetric, what does this imply
delta\_a delta\_b F^ab = 0
=\> delta\_a J^a = 0
=\> charge conservation
107
1/2 F^ab F\_ab =
B^2 - E^2 /c == D (invariant)
108
4 angular momentum
L = XP^T - PX^T
L^ab = X^a P^b - X^b P^a
109
the centroid x\_c =
SUM (x\_i E\_i)/E\_tot
110
dual tensor F(tilde)\_ab
with example F(tilde)\_12
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
g^ab g\_bc =
kronecker delta^a\_c = identity
112
levi civita tensor definition
E\_abcd =
+1 for 0, 1, 2, 3 and even permutations
-1 for odd
0 if two the same
113
E\_01cd=
0 0 0 0
0 0 0 0
0 0 0 1
0 0 -1 0
114
shortcut for dual of an antisymmetric tensor F
F =
0 ax ay az
- ax 0 bz -by
- ay -bz 0 bx
- az by -bx 0
a -\> -b b -\> a
115
delta\_a x^b =
1 if a=b
0 if a/=b
so = delta^a\_b
116
delta\_a x\_b =
1 if a=b but -1 for a=0
0 if a/=b so = metric = g\_ab
117
delta\_a (k^a x\_a) =
for constant 4 vector k
k^a delta\_a (x\_a) =
k^a g\_ab =
k\_a
118
delta^a sin(k^a x\_a) =
[delta^a(k^a x\_a)] cos(k^a x\_a) =
k^a cos(k^a x\_a)
119
nabla^ 1/r
laplacian (1/r) = -4pi diracdelta^(3) (r)
shorthand for d(x)d(y)d(z) - there are 3 deltas
120
div( _a_ exp[i(_k_._r_-wt)] ) =
i _k_._a_ exp[i(_k_._r_-wt)]
121
curl( _a_ exp[i(_k_._r_-wt)] ) =
i _k_^_a_ exp[i(_k_._r_-wt)]
122
relate
k, c, w
w = kc
123
|E0| =
c |B0|
124
the Lorenz gauge
[].A = 0
125
in electrostatics ...
there are fixed charges and no currents
126
General form of a retarded spherical wave.
How is this different to an advanced wave?
What do these look like
f = g(t - r/c)/r
- -\> +
advanced wave collapses in to centre, retarded is the opposite
127
What do the Liènard-Wiechert potentials imply?
A depends on U and R but not acceleration.
128
radiated power:
energy flux
given by the poynting vector
N = e0c2 E^B
129
Larmor's formula
PL = 2/3 \* q2/4πe0 \* a2/c3
130
Pauli-Lubaski 4-vector
131
indices "see-saw" trick
A^a\_b B^b
= A^ab B\_b
132
what are polar an axial vectors?
polar change sign under parity transformation. These are normal vectors like x, p. Axial vectors (like L, B) do not change sign
133
(polar vector) ^ (polar vector) =
axial vector
134
Lagrangian
what is it a function of
L = T - V
L( {qi}, {dqi/dt}, t)
135
Euler Lagrange Equations
d/dt (partial dL/dqi[dot] ) = partial dL/dqi
136
canonical momentum p(tilde) =
partial dL/dqi[dot]
137
generalised force
partial dL/dqi
138
Hamiltonian
H( q, p~, t) = SUM( p~i qi(dot) ) - Lagrangian
139
the action
how does this relate to euler lagrange eqns?
path integral (Lagrangian dt)
the paths of min/max action satisfy euler lagrange equations
140
Hamilton's canonical equations
dq/dt = dH/dp
dp/dt = - dH/dq
for all qi
and p is pi~
141
lagrangian of a freely moving particle
-mc2/gamma
142
particle hits stationary particle, they stick together. How to find mass of combined partcle and its velocity?
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
