PHYS 366 Relativity
text: Mould: Basic Relativity

--Class 1--
read chapters 1&2
there is a 6 minute video connected to the below link; watch it!
http://www.bbc.com/earth/story/20170206-physics-suggests-that-the-future-is-already-set-in-stone

Next, "Do the math" on this video.
I believe we are to consider Andromeda and Earth to both be at rest in 
a particular frame of reference. (Fact Check: Andromeda is approaching
us at about 110 km/s; a "collision" is expected in about 4 billion years.
Additionally it is 2.5 million LY away, rather than the 2 million used in
the video.) Let's call this frame the "train".
Mike is moving away from Andromeda; Melissa is moving toward Andromeda.
That is from Mike's frame the train--with Andromeda as the engine and
Earth as the caboose--has passed by him. Whereas Melissa sees the train
approaching with Earth as engine and Andromeda as caboose.
Assuming a reasonable walking speed find the lack of synchronization
between Earth and Andromeda from both Mike's and Melissa's perspectives.
Is this lack approximately a day?
Show the math: from who's frame is the vote tomorrow?

--Class 2--
we will start chapter 3 next time
CM_trains.pdf

--Class 3--
chapter 3 Minkowski Diagrams
old exam 366t105.pdf:
#1---proceed by writing down the x,t equation for each photon.
Using the Lorentz Transformation results express x & t in terms of x' & t'.
Simplify the resulting equations to produce the result

#7--use minkowski.gamma=1.1.pdf for your Minkowski Diagram

--Class 4--
chapter 4: 4-vectors
old exam 366t105.pdf: 2 & 3
text problem: 4.3

--Class 5--
chapter 4: the grad is a vector, summation notation
we may start chapter 5, but we will skip to section 5.6
Remark: Mould endorses the concept of "relativisitic mass" = gamma * rest mass;
I do not. All "m" that I write are rest masses.

old exam 366t105.pdf: 4 & 6
text problem: 4.10

The "rapidity" \phi is related to velocity by:
tanh(\phi) = \beta
(or tanh^-1(\beta) = \phi...find that inverse hyperbolic tangent key on your calculator!)
Show that:
\gamma = cosh(\phi)
\beta \gamma = sinh(\phi)

Rapidity is analogous to a rotation angle (recall that a basic rotation matrix would
have cos(\theta) and sin(\theta) in it...now we've got cosh and sinh).
Velocity addition corresponds to staightforward addition of rapidity.

read: uniform_acceleration_05.pdf

--Class 6--
practical particle physics problems:
threshold energy: m1 + m2 -> m3
decay energy: m3->m1+m2+...
compton scattering: g + e -> g' + e'

start uniform acceleration

problems:
in pion decay: pi -> mu + neutrino (Particle Booklet: p.25, p.14) find the energy of the muon 
assuming the neutrino is massless
IF the neutrino mass was 1 eV (neutrinos have mass, but this is probably a huge over-estimate)
what would be the difference in muon energy? (Calculate the difference directly; its too small to calcualate by subtraction)
whait is the kinetic energy of the muon?
what is the velocity of the mu?
what would you calculated for the velocity if you assumed KE/c^2=.5 m beta^2
Note that in this 2-body decay every mu is produced with the same energy.

in muon decay: mu-> electron + neutrino + neutrino'
what is the maximum electron energy and velocity
what is the minimum electron energy and velocity
in the case of an electron produced at rest, what are the neutrino energies

a green photon (lambda=500nm) back-scatters off an electron, what is its final wavelength?

The delta(1232) (p.164) can be produced by shining light or pi+ on a (at rest) proton.
What light and pi energy is required?

--Class 7--
finish uniform acceleration
start chapter 6: EM

Given an infinite supply of anti-matter, the obvious way to visit other
stars is to accelerate at g for half the distance and then deaccelerate
at g for the remaining half of the distance.  With a constant proper 
acceleration of g, it takes only about a year to reach speeds approaching
c, so time dilation helps you to reach great distances. The process is clearly
symmetric, so (for example) the total proper time for the trip is just
twice the proper time to the turn-around point (which is itself half
the total distance). 
What is the total proper time to reach the following locations:
(LY=light year)
d=15.25 LY.....Gliese 876...a cool star known to have multiple planets
d=45.9 LY.......47 Ursae Majoris...a star similar to the Sun with multiple planets
d=25.6 kLY......black hole at the center of our Galaxy
d=2.5 MLY...."nearby" spiral galaxy: M31 in Andromeda

The most efficient imagined way of powering this spacecraft is direct
conversion of mass energy to light, and then using the light-exhaust to
accelerate the ship.  It isn't too difficult to show that if the final speed
is v, the mass ratio (M final/M initial) must be:
Sqrt[(c-v)/(c+v)]
(Think of it as a decay process: Minitial->Mfinal+light; the 4-vector of all the
emitted photons will still be null...prove this mass ratio result)
Of course, for the full trip this mass ratio applies twice: once to speed up
and then again to slow down.  Find the full-trip mass ratio to reach the 
above locations.

Hints: A good starting point is to express g in units of LY/year^2 (of course
c=1 LY/year; check the result should be about 1).  
I hope its clear that one wants to find \tau such that:
x(\tau/2)-x(0)=d/2

textbook: 4.9 (FYI: I work these in Mathemtica)

--Class 8--
start chapter 6: EM

homework:
textbook: 3.25
invariants_Fuv.pdf: #2 & #3

The plan is to essentially do #1 in class
Strongly suggest you work these in Mathematica on a Linux box as there LorentzBoost.m
should be available

--Class 9--
more chapter 6: EM

textbook: 6.8 There is potential confusion in going from current, i,  in an
infinitesimal wire to J and charge per length, lambda, to rho.  The upshot is you
can just pretend the 4-vector is J=(i,I c lambda) [note the units work].  In greater detail:
in making the actual 4-vector you would use: J=i/A, rho=lambda/A,
but since we're considering boosts in the direction of i, A is invariant,
and so it would cancel in the transformation of J'->J.

textbook: 6.10 (I'm not sure what exactly part c is asking for, but note that in S'
there is no force --- magnetic fields are required for forces on currents.  Calculate
the force in S -- where there are E & rho and B & J and so you might think there would
be a force and show instead the resulting force is zero)

In frame S there is a uniform electric field vector E=(0,a,0) and a uniform
magnetic field vector B=(0,0,5a/3c).  Find a frame S' in which the electric field
is zero.  Find B' in the S' frame.  Check the values of the invariants: 
E.B and E^2-c^2B^2
are the results the same in S and S'?

stress.pdf                                                                                                                                        

--Class 10--
classfvac.pdf read--provides a nice summary of special relativity using the real (non-complex)
metric.  Note it also uses a more usual ordering where the zeroth (rather than fourth)
component is used for time

start chapter 7: differential geometry

Consider the usual spherical coordinates:
(x,y,z)=(r \sin\theta \cos\phi, r \sin\theta \sin\phi, r \cos\theta )

find the superscript basis vectors: e^r, e^\theta, e^\phi
find the subscripted basis vectors: e_r, e_\theta, e_\phi
calculate the metric tensor gij

--Class 11--
mid-term exam in one week...help?

More on the usual spherical coordinates:
Using the definition of Christoffel symbols: \partial_i e_j =\Gamma^k_ij e_k
find the 6 distinct Christoffel symbols:
\Gamma^1_22, \Gamma^1_33, \Gamma^2_12, \Gamma^2_33, \Gamma^3_13, \Gamma^3_23
where 1=r, 2=\theta, 3=\phi

Consider the distance: ds^2=(da^2 + sin(a)^2 db^2)
with coordinates (a,b).  Find the metric tensor g_ij.  Find the Christoffel
symbols from the metric tensor.
Answer check: \Gamma^2_12 = cot(a)
Write down the geodesic diff equation and use Mathematica to NDSolve it; use ParametricPlot
to display an example that does not look like a straight line

The Mathematica code handout details the 2D parabolic coordinate case.
In particular near the top right find the Christoffel symbols.
Using those results show how equ and eqv (equations for a geodesic)
come about.  Select some initial conditions and NDSolve.
ParametricPlot the results on the (u,v) plane; convert to
show the result is a straight line in (x,y)

--Class 12--
we skip several chapters and begin chapter 11
(Chapter 11 will not be on the mid-term exam)

ch7HW.pdf #6
note "flat" means gij= diagonal(1,1,1,-1)

--Class 13--
mid-term exam covers thru chapter 7
help Sunday 2pm
we start chapter 11: curvature tensor
(skipping chapters 8-10)

--class 14--
mid-term exam

--class 15--
no homework
review thru curvature, ricci tensors
finished chapter 11; starting 12 thru 13.2
my plan is to spend more time than the book does on the details of schwarzschild solution

--class 16--
calculating R for schwarzschild solution

In Class 11 you found the Christoffel symbols for the 2d metric tensor:
ds^2=(da^2 + sin(a)^2 db^2)

The non-zero Christoffels are:
Gamma^1_22= {1/22}= -(Cos[a] Sin[a])
Gamma^2_12= {2/12}={2/21}= Cot[a]

Find the Riemann Curvature Tensor (RDabc), Ricci Tensor (Rab), and the curvature scaler R
Do note that most of the 2^4 elements of RDabc are zero by symmetry; in 2d there is
just one independent curvature (Gauss)


--class 17--
motion in schwarzschild solution

The Kerr solution is for a rotating black hole; following the Mathematica procedure
used for Schwarzschild (schwarzschildR.m), show that it solves the GR equations for free space.
A start: entering the metric (textbook equation 13.6):

d[r_]=r^2 + a^2 -2 M r
rho2[r_,t_]=r^2+a^2 Cos[t]^2
gij=DiagonalMatrix[{rho2[r,t]/d[r], rho2[r,t], Sin[t]^2(r^2+a^2+ 2 M r a^2/rho2[r,t] Sin[t]^2),-(1-2 M r/rho2[r,t])}]
gij[[3,4]]= -2 M r a/rho2[r,t] Sin[t]^2
gij[[4,3]]=gij[[3,4]]

In words describe why the (d phi d tau) term in the textbook has a 4, whereas this gij has a 2

--class 18--
motion in schwarzschild solution
read chapters 12 & 13: my take on dynamics (section 12.8) has focused on numerical solutions

following the Mathematica procedure used for Schwarzschild orbits (schwarzschildO.m), 
find an "interesting" orbit, print out the trajectory and parameters, and (words) describe
why you found it "interesting"

Using the Christoffel symbols for the Kerr solution that you found as part of Class 17,
show that theta=Pi/2 (constant) is a solution of the theta geodesic equation.
Note: theta=Pi/2 solutions are a special case (rather than a general case as in Schwarzschild);
inclined orbits are not planer.

--class 19--
motion in schwarzschild 
classic tests of GR
read interstellar.pdf

following the Mathematica procedure kerrO.m, 
find an "interesting" orbit, print out the trajectory and parameters, and (words) describe
why you found it "interesting"

--class 20--
motion in kerr solution

Consider the closest-in stable circular orbit of an extreme {a-> .999999, M->1} Kerr hole.
Find the time dilation factor (dt/d tau)

--class 21--
motion in kerr solution

Help: study day 2pm