We are asked to show that L(SO(4)) = L(SU(2)) (+) L(SU(2))
where L is the Lie algebra and (+) is the direct product.
We are given the hint to consider the antisymmetric 4 by 4 matrices where each row and column has a single 1/2 or -1/2 in it.
By doing this we generate four matrices...
We have a suggested Lagrangian
epsilon(abcd) F^(ab) F^(cd)
and are asked to comment if this is a sensible EM Lagrangian. The only thing i can think of is that its still gauge invariant in the normal way but otherwise im stumped. would appreciate any suggestions. thanks
I'm studying a QFT course, and we've been asked to consider why classical physicists found it useful to introduce electric and magnetic fields, but not fields for electrons or other particles. I'm completely stumped, and would appreciate any hints. thanks
Assuming the Lorentz force law and also that in the rest frame of the particle the 3 acceleration is zero, we need to explain why the following equations hold:
E.v = 0 and E + v.B = 0
where v is the velocity.
I think this is because g(A,A) = -a squared is invariant. Therefore if a=0, I...
We have to show that [Lx,Ly] = Lz
[Ly,Lx] = -Lz
[Lx,Lx] = 0
and I have done this. We then need to comment on the significance of these results, which I'm not sure of. I know in QM you get similar results for commutators of these quantities, and it means that you can't simultaneously know...
"A parcel of air is lifted slowly from the ground, where the temperature is 295K, to an elevation of 5km, and then returned rapidly to the ground. Estimate the air parcel temperature at 5km and after it returns to the groundm explanation any assumptions."
I assumed an adiabatic process both...
We are given a form of Einstein's field equations:
3R'' = -pR
R''R + 2((R')^2) = p(R^2)
where p is a constant and R' = dR/dt
Assuimg that R and R' are both positive, we are asked to show that the general solution is R(t) = A*[(t-ti)^(2/3)]
I'm very confused about this. If we...
I managed to get this. I was differentiating with respect to the wrong coordinate system, which messed up the calculation. I then tried using the chain rule and differentiating with respect to the other coord system and it all fell out.
"Show that if a space time metric admits three linearly independent 4 vector fields with vanishing covariant derivatives then Rabcd = 0"
We can set the three vectors as (1,0,0,0), (0,1,0,0) and (0,0,1,0). Use covariant derivative of vector field X^b is:
d(X^b)/d(x^a) + (Christoffel symbol...
We have a vector X^a (n.b ^ indicates superscript) and covector Aa. We need to show that
X^b (d(Aa)/d(x^b) - d(Ab)/d(x^a))
transforms correctly under an artbitrary smooth change of coords. N.b the derivatives are partial.
By using the transformation rules for the...
This equation gives us (delta (rho))/rho (which I understand is the fractional perturbation in the energy density), at the time of "horizon entry" (which I'm unsure about). Does this mean the time that decoupling occured?
We need to show that using the Schwarzchild metric, an incoming radial spacelike geodesic satisfies r>= 1m/(1 + E^2)
I know that E = (1-2m/r) is constant, and I think that ds squared should be negative for a spacelike geodesic. I try substituting E into the metric and setting ds squared <=0...
"Under what conditions does a geodesic represent a possible wordline for a particle in free fall, parametrized by proper time."
I can only think of one conditions - the fact that the metric ds squared is positive. Is this enough?
I start from the defintion of H, and then plug in that p is the partial derivative of L wrt q dot.
The next stage is a bit iffy. I assume that the Kinetic energy can be assumed to be 1/2 m * ((q dot) squared), where q is the position vector. I argue that any other velocity dependent terms in...
Two equal masses are connected by two massless springs of constant k and nat. length l. The masses are constrained by a frictionless tube on a pivot, (also massless) so that they remain colinear with the pivot. The pivot subtends angle theta with the vertical. The 1st mass...
We have two particles mass m called p1 and p2. P1 is stationary, p2 has energy E. They annihilate to produce to particles of mass 100m. We need to find the min value of E
The Attempt at a Solution
I suspect that the minimum initial energy of...