Please review the concepts and formulas of relativistic kinematics and relativistic dynamics.
A contravariant 4-vector is a set of 4 quantities which transform under a Lorentz transformation like (ct,r) = (x0,x1,x2,x3). (A0,A1,A2,A3) is a contravariant 4-vector if A'
a = (¶x'a/¶xb)Ab. The repeated index b is summed over.A covariant 4-vector is a set of 4 quantities which transform under a Lorentz transformation like (ct,-r) = (x0,x1,x2,x3). (A0,A1,A2,A3) is a covariant 4-vector if A'a = (¶xb/¶x'a)Ab.
If the primed coordinate system moves with velocity v = bci with respect to the unprimed one, then
.
b = v/c, b = v/c, g = (1 - b2)-1/2.
A contravariant tensor of second rank is a set of 16 quantities which transform under a Lorentz transformation according to F'
ab = (¶x'a/¶xg)(¶x'b/¶xd)Fgd.A covariant tensor of second rank transforms under a Lorentz transformation according to G'ab = (¶x
g/¶x'a)(¶xd/¶x'b)Ggd and a mixed tensor transforms according to H'ab = (¶x'a/¶xg)(¶xd/¶x'b)Hgd. Special tensors: dab = (¶xa/¶xb) is the Kroneker delta extended to 4 indices.
is the metric tensor.
The dot product between two contravariant 4-vectors is defined as A×B = g
abAaBb = AbBb = AaBa. It is invariant under a Lorentz transformation, it is a Lorentz scalar.
xm = (ct,r),
um = (gc,gv) = 4-vector velocity,
pm = (gmc,gmv) = (E/c,p) = (p0,p) = 4-vector momentum,
(¶/¶x
m) = ¶m = (¶/¶x0, -Ñ) = 4-dimensional gradient,jm = (cr,j) = 4-vector current,
Am = (f/c,A) = 4-vector potential, (Si units), Am = (f,A) (Gaussian units)
The divergence of a 4-vector ¶mAm = ¶A0/¶x0 - Ñ×A is a Lorentz scalar.
Examples:
is the D’Alambertian.
is the statement of charge
conservation.
(SI),
(Gaussian),
is the Lorentz condition.
(SI units),
(Gaussian units),
is the inhomogeneous wave equation for the potentials.
All contravariant 4-vectors Am transform as A'0 =
g(A0 - b×A), A'||= g(A|| - bA0), A'^ = A^.The antisymmetric field strength tensor is defined through Fab = ¶aA
b - ¶bAa. It is a second rank contravariant tensor.
(SI units),
(Gaussian),
The transformation F'ab = (¶x'a/¶xg)(¶x'b/¶xd)Fgd yields
E'|| = E||, B'|| = B||, (SI and Gaussian units),
E'^ =
g(E + v´B)^, (SI units), E'^ = g(E + (v/c)´B)^, (Gaussian units),B'^ = g(B - (v/c2)´E)^, (SI units), B'^ = g(B - (v/c)´E)^, (Gaussian units).
Here || and ^ refer to the direction of the relative velocity.
For a charge distribution moving with velocity v with respect to an observer, we have in the frame K’ of the observer
E'|| = E||, B'|| = B|| = 0, (SI and Gaussian units),
E'^ = gE^, B'^ = g(v/c2)´E, or B'^ = (v/c2)´E', (SI units).
E'^ = gE^, B'^ = g(v/c)´E, or B'^ = (v/c)´E', (Gaussian units).
Note: The observer is moving with velocity -v with respect to the rest frame of the charge distribution.
E2 - c2B2 (SI units) or E2 - B2 (Gaussian units) and (E×B)2 are invariant under a Lorentz transformation.