Field Strength Tensor Math - G
The field strength tensor in Gaussian units is:
F^(alpha eta )=[[0,-E_(x),-E_(y),-E_(z)],[+E_(x),0,-B_(z),+B_(y)],[+E_(y),+B_(z),0,-B_(x)],[+E_(z),-B_(y),+B_(x),0]]
The metric tensor is:
g^(alpha eta )=g_(alpha eta )=[[+1,0,0,0],[0,-1,0,0],[0,0,-1,0],[0,0,0,-1]]
The four-dimensional Levi-Civita symbol is:
epsi lon^(alpha eta gamma delta )={[+1 even permutation of 0123],[-1 odd permutation of 0123],[0 otherwise ]}
Explicitly calculate F_(alpha eta )=g_(alpha gamma )F^(gamma delta )g_(delta eta ).
Explicitly calculate F^(alpha eta )=(1)/(2)epsi lon^(alpha eta gamma delta )F_(gamma delta ).
Explicitly calculate del_(alpha )F^(alpha eta )=(4pi )/(c)J^(eta ), where J^(alpha )=(c
ho ,vec(J)) is the 4-current density.
Which equations are these in vector notation?
Explicitly calculate del_(alpha )F^(alpha eta )=0. Which equations are these in vector notation?
Explicitly calculate (dp^(alpha ))/(d au )=(q)/(c)F^(alpha eta )U_(eta ), where p^(alpha )=((E)/(c),vec(p)) is the 4-momentum and
U^(alpha )=gamma _(u)(c,vec(u)) is the 4 -velocity. Which equations are these in vector notation?
Explicitly calculate F^(alpha eta )F_(alpha eta ). Since this is a Lorentz scalar, it is the same in all inertial
reference frames. What is its form in vector notation?
Explicitly calculate F^(alpha eta )F_(alpha eta ). Since this is a Lorentz scalar, it is the same in all inertial
reference frames. What is its form in vector notation?
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Field Strength Tensor Math - G
[24-2024-03-15-PHY-841-SR-06-covariant-EOM, 11.9] The field strength tensor in Gaus- sian units is: 0 -Er Ey -Ez +Er 0 -Bz +By (3) +Ey +Bz 0 -Bx +Ez -By +Br 0
The metric tensor is:
+1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
(4)
903
The four-dimensional Levi-Civita symbol is:
+1 even permutation of 0123 -1 odd permutation of 0123 0 otherwise
(5)
1. Explicitly calculate F = ga~F~8 gs3
2. Explicitly calculate F=8F8
3. Explicitly calculate .Fa = 4 J, where J = (cp, J) is the 4-current density. Which equations are these in vector notation?
4. Explicitly calculate 0.F = 0. Which equations are these in vector notation?
U = Yu(c, u) is the 4-velocity. Which equations are these in vector notation?
6. Explicitly calculate F Fa. Since this is a Lorentz scalar, it is the same in all inertial reference frames. What is its form in vector notation?
7. Explicitly calculate Fa Fog. Since this is a Lorentz scalar, it is the same in all inertial reference frames. What is its form in vector notation?
