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TENSORS OF RELATIVISTIC ELECTRODYNAMICS WITH MAPLE

Alexei V. Tikhonenko

General Physics Department,
State Technical University of Nuclear Power Engineering,
Obninsk, Russia

futureprint@obninsk.com

This work develops algorithms of construction, calculation and transformation of physical vectors and tensors in relativistic electrodynamics with MAPLE.

1. Lorentz transformations

>	restart;

>	with(tensor):with(linalg):

Consider transformations of vectors and tensors with Lorentz transformations in four-dimensional pseudo-Euclidean space-time. We will use "tensor" and "linalg" packages for operations with vectors and tensors in four-dimensional space-time.

The first necessary step is creation of the metric tensor of four-dimensional pseudo-Euclidean space-time:

>	coord:=[t,x,y,z]; g11:=1: g22:=-1: g33:=-1: g44:=-1: g:=create([-1,-1],array(1..4,1..4,symmetric,sparse,[(1,1)=g11(r,theta),(2,2)=g22(r,theta),(3,3)=g33(r,theta),(4,4)=g44(r,theta)]));

(1.1)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi(

(1.1)

>	g_matrix:=matrix([[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1]]); detg:=det(g_matrix);

(1.2)

>	`tensor/simp`:= proc(a) simplify(a); end: g_contr := invert(g, 'det_g');

(1.3)

Lorentz transformations (for two reference frames) are determined as

>	dim:=4; Coord_1:=[t,x,y,z]; Coord_2:=[t,x,y,z];

(1.4)

>	*CHANG:=[t=gamma(t-betax),x=gamma(x-betat) ,y=y,z=z];*

(1.5)

>	Jacobian(Coord_2,CHANG,tr,TR); op(tr); op(TR);

(1.6)

where

and V is relative velocity of reference frames in the line of x-axis.

Components of covariant vector

>	VECT:=create([-1],array([V[1],V[2],V[3], V[4]]));

(1.7)

will transform with Lorentz transformations as

VECT_trans:=create([-1],array([V_trans[1], V_trans[2],V_trans[3],V_trans[4]]));
get_compts(VECT_trans)=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma));
get_compts(VECT_trans)[1]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[1];
get_compts(VECT_trans)[2]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[2];
get_compts(VECT_trans)[3]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[3];
get_compts(VECT_trans)[4]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[4];

(1.8)

Components of covariant tensor

>	TENS:=create([-1,-1],array(1..4,1..4,sparse,[[T[1,1],T[1,2],T[1,3],T[1,4]],[T[2,1],T[2,2],T[2,3],T[2,4]],[T[3,1],T[3,2],T[3,3],T[3,4]],[T[4,1],T[4,2],T[4,3],T[4,4]]]));

(1.9)

will transform with Lorentz transformations as

TENS_trans:=create([-1,-1],array(1..4,1..4,sparse,[[T_trans[1,1],T_trans[1,2],T_trans[1,3],T_trans[1,4]],[T_trans[2,1],T_trans[2,2],T_trans[2,3],T_trans[2,4]],[T_trans[3,1],T_trans[3,2],T_trans[3,3],T_trans[3,4]],[T_trans[4,1],T_trans[4,2],T_trans[4,3],T_trans[4,4]]]));
get_compts(TENS_trans)=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma));
get_compts(TENS_trans)[1,1]=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma))[1,1];
get_compts(TENS_trans)[1,2]=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma))[1,2];
get_compts(TENS_trans)[2,4]=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma))[2,4];

(1.10)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-msub(Typesetting:-mi(

(1.10)

Components of the symmetrical covariant tensor

>	TENS_sym:=create([-1,-1],array(1..4,1..4,sparse,[[T[1,1],T[1,2],T[1,3],T[1,4]],[T[1,2],T[2,2],T[2,3],T[2,4]],[T[1,3],T[2,3], T[3,3],T[3,4]],[T[1,4],T[2,4],T[3,4],T[4,4]]]));

(1.11)

will transform with Lorentz transformations as

TENS_sym_trans:=create([-1,-1],array(1..4,1..4,sparse,[[T_trans[1,1],T_trans[1,2],T_trans[1,3],T_trans[1,4]],[T_trans[2,1],T_trans[2,2],T_trans[2,3],T_trans[2,4]],[T_trans[3,1],T_trans[3,2],T_trans[3,3],T_trans[3,4]],[T_trans[4,1],T_trans[4,2],T_trans[4,3],T_trans[4,4]]]));
get_compts(TENS_trans)=get_compts(collect(transform( TENS_sym,CHANG,tr,TR), [gamma,beta]));

(1.12)

TENS_sym_trans:=collect(simplify(subs (gamma =1/sqrt(1-beta^2),matrix([[(T[1,1]-2*beta*T [1,2]+ beta^2*T[2,2])*gamma^2, (beta^2*T[1,2]+(-T[1,1]-T[2,2])*beta+T[1,2])*gamma^2,(T[1,3]-beta*T[2,3])*gamma,(T[1,4]-beta*T[2,4])*gamma],[(beta^2* T[1,2]+(-T[1,1]-T[2,2])*beta +T[1,2])*gamma^2,(beta^2 *T[1,1]-2*beta*T[1,2] +T[2,2])*gamma^2,(-beta*T[1,3] +T[2,3])* gamma, -beta*T[1,4]+T[2,4]*gamma],[(T[1, 3]-beta* T[2,3])*gamma,(-beta*T[1,3] +T[2, 3])*gamma, T[3,3], T[3,4]], [(T[1,4]-beta*T[2,4])*gamma, (-beta*T[1,4]+T[2, 4])*gamma, T[3,4], T[4,4]]]))),beta) assuming (-beta^2+1)>0;

(1.13)

Components of the asymmetrical covariant tensor

>	TENS_as:=create([-1,-1],array(1..4,1..4 ,sparse ,[[0,T[1,2],T[1,3],T[1,4]],[-T[1, 2],0,T[2,3], T[2,4]],[-T[1,3],-T[2,3],0, T[3,4]],[-T[1,4],-T[2,4],-T[3,4],0]]));

(1.14)

will transform with Lorentz transformations as

TENS_as_trans:=create([-1,-1],array(1..4,1..4,sparse,[[0,T_trans[1,2],T_trans[1,3],T_trans[1,4]],[-T_trans[1,2],0,T_trans[2, 3],T_trans[2,4]],[-T_trans[1,3],-T_trans [2,3],0,T_trans[3,4]],[-T_trans[1,4],-T_trans[2,4],-T_trans[3,4],0]]));
get_compts(TENS_trans)=get_compts(simplify(collect(transform(TENS_as,CHANG,tr,TR),[gamma,beta])));

(1.15)

TENS_as_trans:=simplify(subs(gamma=1/sqrt (1-beta^2),matrix([[0, -T[1,2]*(-1+beta^2) *gamma^2,-(-T[1,3]+beta*T[2,3])*gamma,-(-T[1,4]+beta*T[2,4] )*gamma], [T[1,2]*(-1+ beta^2)*gamma^2, 0,-(beta* T[1,3]-T[2,3])* gamma, -(beta*T[1,4]-T[2,4])*gamma], [(-T[1,3]+beta*T[2,3])*gamma, (beta*T[1,3]-T[2,3] )*gamma, 0, T[3,4]], [(-T[1,4]+beta *T[2,4])*gamma, (beta*T[1,4]-T[2,4]) *gamma,-T[3,4], 0]]))) assuming (-beta^2+1)>0;

(1.16)

Typesetting:-mrow(Typesetting:-mi(

2. 4-velocity ? 4-acceleration

Consider 4-radius-vector of particle moving in four-dimensional space-time:

>	R(t):=create([-1],array([x[1](t),x[2](t),x[3] (t),x[4](t)]));

(2.1)

Consider 4-velocity of particle in four-dimensional space-time:

V(t):=create([-1],array([v[1](t),v[2](t), v[3](t),v[4](t)]));
V(t):=create([-1],array([1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t)^2)/c^2)/c*diff(x[1](t),t),1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t)^2)/c^2)/c*diff(x[2](t),t),1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t)^2)/c^2)/c*diff(x[3](t),t),1/sqrt(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)/ c*diff(x[4](t),t)]));
V(t):=create([-1],array([1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t)^2)/c^2),1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t)^2)/c^2)/c*v[2](t),1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t) ^2)/c^2)/c*v[3](t),1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t)^2)/c^2)/c*v[4](t)]));

(2.2)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi(

(2.2)

>	V[1](t):=get_compts(V(t))[1]; V[2](t):=get_compts(V(t))[2]; V[3](t):=get_compts(V(t))[3]; V[4](t):=get_compts(V(t))[4];

(2.3)

Square of 4-velocity is

>	V_sqw=simplify(subs(beta=sqrt(v[2](t)^2+ v[3](t)^2 +v[4](t)^2)/c,prod(prod(V(t),V(t)) ,g_contr,[1,1],[2,2])));

(2.4)

Consider 4-acceleration of particle in four-dimensional space-time:

Accl[1](t):=1/sqrt(1-(v[2](t)^2+v[3](t)^2 +v[4](t)^2)/c^2)/c*diff(V[1](t),t);
Accl[2](t):=1/sqrt(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)/c*diff(V[2](t),t);
Accl[3](t):=1/sqrt(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)/c*diff(V[3](t),t);
Accl[4](t):=1/sqrt(1-(v[2](t)^2+v[3](t)^2+v [4](t)^2)/c^2)/c*diff(V[4](t),t);

`:=`(Accl[1](t), `+`(`*`(`/`(1, 2), `*`(`/`(`*`(`+`(`*`(2, `*`(v[2](t), `*`(diff(v[2](t), t)))), `*`(2, `*`(v[3](t), `*`(diff(v[3](t), t)))), `*`(2, `*`(v[4](t), `*`(diff(v[4](t), t)))))), `*`(`^`(`+`...

(2.5)

`:=`(Accl[2](t), `/`(`*`(`+`(`*`(`/`(1, 2), `*`(`/`(`*`(v[2](t), `*`(`+`(`*`(2, `*`(v[2](t), `*`(diff(v[2](t), t)))), `*`(2, `*`(v[3](t), `*`(diff(v[3](t), t)))), `*`(2, `*`(v[4](t), `*`(diff(v[4](t),...

(2.5)

`:=`(Accl[3](t), `/`(`*`(`+`(`*`(`/`(1, 2), `*`(`/`(`*`(v[3](t), `*`(`+`(`*`(2, `*`(v[2](t), `*`(diff(v[2](t), t)))), `*`(2, `*`(v[3](t), `*`(diff(v[3](t), t)))), `*`(2, `*`(v[4](t), `*`(diff(v[4](t),...

(2.5)

`:=`(Accl[4](t), `/`(`*`(`+`(`*`(`/`(1, 2), `*`(`/`(`*`(v[4](t), `*`(`+`(`*`(2, `*`(v[2](t), `*`(diff(v[2](t), t)))), `*`(2, `*`(v[3](t), `*`(diff(v[3](t), t)))), `*`(2, `*`(v[4](t), `*`(diff(v[4](t),...

(2.5)

>	Accl:=create([-1],array([Accl[1](t),Accl[2](t),Accl[3](t),Accl[4](t)]));

(2.6)

The scalar product of 4-velocity ? 4-acceleration is equal to

>	prod(prod(V(t),Accl),g_contr,[1,1],[2,2]);

(2.7)

This value means orthogonality of 4-velocity ? 4-acceleration.

Another form of 4-acceleration is:

Accl[1]:=1/((1-beta^2)^2)/(c^3)*(v[2] (t)*w[2](t) +v[3](t)*w[3](t)+v[4](t)* w[4](t));
Accl[2]:=1/(1-beta^2)^2/c^4*v[2](t)*(v[2](t) *w[2](t)+v[3](t)*w[3](t)+v[4](t)*w[4](t))+1/(1-beta^2)/c^2*w[2](t);
Accl[3]:=1/(1-beta^2)^2/c^4*v[3](t)*(v[2](t) *w[2](t)+v[3](t)*w[3](t)+v[4](t)*w[4](t))+1/(1-beta^2)/c^2*w[3](t);
Accl[4]:=1/(1-beta^2)^2/c^4*v[4](t)*(v[2](t)* w[2](t)+v[3](t)*w[3](t)+v[4](t)*w[4](t))+1/(1-beta^2)/c^2*w[4](t);
Accl:=create([-1],array([Accl[1],Accl[2], Accl[3],Accl[4]]));

(2.8)

3. energy-momentum vector

Consider 4-velocity

V[1](t):=1/((1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)^(1/2));
V[2](t):=1/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)^(1/2)/c*v[2](t);
V[3](t):=1/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)^(1/2)/c*v[3](t);
V[4](t):=1/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)^(1/2)/c*v[4](t);
V(t):=create([-1],array([V[1](t),V[2](t), V[3](t),V[4](t)]));

(3.1)

and define 4-energy-momentum vector as

>	P[1](t):=cmV[1](t); P[2](t):=cmV[2](t); P[3](t):=cmV[3](t); P[4](t):=cmV[4](t); P(t):=create([-1],array([P[1](t),P[2](t),P[3](t), P[4](t)]));

(3.2)

Square of 4-energy-momentum vector is:

>	P_sqw=prod(prod(P(t),P(t)),g_contr,[1,1], [2,2]);

(3.3)

As another form of 4-energy-momentum vector is

>	P_:=create([-1],array([E/c,p[2],p[3], p[4]]));

(3.4)

we have

>	prod(prod(P_,P_),g_contr,[1,1],[2,2])= prod(prod(P(t),P(t)),g_contr,[1,1],[2,2]);

(3.5)

4-energy-momentum vectorwill transform with Lorentz transformations as

>	P:=create([-1],array([E/c,p[2],p[3],p[4]])); P_trans_:=create([-1],array([E_trans/c,p_trans[2], p_trans[3],p_trans[4]])); P_trans:=collect(transform(P,CHANG,tr,TR),gamma);

(3.6)

>	get_compts(P_trans_)[1]=get_compts (P_trans)[1]; get_compts(P_trans_)[2]=get_compts (P_trans)[2]; get_compts(P_trans_)[3]=get_compts (P_trans)[3]; get_compts(P_trans_)[4]=get_compts (P_trans)[4];

(3.7)

Make sure that square of 4-energy-momentum vector is invariant relatively to Lorentz transformations:

>	P_sqw:=get_compts(prod(prod(P,P),g_contr, [1,1],[2,2]));

(3.8)

>	P_trans_sqw:=get_compts(prod(prod(P_trans,P_trans),g_contr,[1,1],[2,2])); P_trans_sqw:=simplify(subs(gamma=1/sqrt (1-beta^2),(gamma^2E^2+gamma^2beta^2* p[2]^2c^2-gamma^2beta^2E^2-gamma^2 p[2]^2c^2-p[3]^2c^2-p[4]^2*c^2)/c^2));

(3.9)

and

Differentiate 4-energy-momentum vector by interval:

P[1](t):=c*m/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2) /c^2)^(1/2):
P[2](t):=m/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2) /c^2)^(1/2)*v[2](t):
P[3](t):=m/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2) /c^2)^(1/2)*v[3](t):
P[4](t):=m/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2) /c^2)^(1/2)*v[4](t):
Dif_P(t):=create([-1],array([1/sqrt(1-(v[2](t)^ 2+v[3](t)^2+v[4](t)^2)/c^2)/c *diff(P[1](t),t),1/sqrt(1-(v[2](t)^2+ v[3](t)^2+v[4](t)^2)/c^2)/c* diff(P[2] (t),t),1/sqrt(1-(v[2](t)^2+v[3](t) ^2+v[4](t)^2)/c^2 )/c*diff(P[3](t), t),1/sqrt(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)/c* diff(P[4](t),t)]));

(3.10)

>	DP[1]:=get_compts(Dif_P(t))[1]; DP[2]:=get_compts(Dif_P(t))[2]; DP[3]:=get_compts(Dif_P(t))[3]; DP[4]:=get_compts(Dif_P(t))[4];

`:=`(DP[1], `+`(`*`(`/`(1, 2), `*`(`/`(`*`(m, `*`(`+`(`*`(2, `*`(v[2](t), `*`(diff(v[2](t), t)))), `*`(2, `*`(v[3](t), `*`(diff(v[3](t), t)))), `*`(2, `*`(v[4](t), `*`(diff(v[4](t), t))))))), `*`(`^`(...

(3.11)

`:=`(DP[2], `/`(`*`(`+`(`*`(`/`(1, 2), `*`(`/`(`*`(m, `*`(v[2](t), `*`(`+`(`*`(2, `*`(v[2](t), `*`(diff(v[2](t), t)))), `*`(2, `*`(v[3](t), `*`(diff(v[3](t), t)))), `*`(2, `*`(v[4](t), `*`(diff(v[4](t...

(3.11)

`:=`(DP[3], `/`(`*`(`+`(`*`(`/`(1, 2), `*`(`/`(`*`(m, `*`(v[3](t), `*`(`+`(`*`(2, `*`(v[2](t), `*`(diff(v[2](t), t)))), `*`(2, `*`(v[3](t), `*`(diff(v[3](t), t)))), `*`(2, `*`(v[4](t), `*`(diff(v[4](t...

(3.11)

`:=`(DP[4], `/`(`*`(`+`(`*`(`/`(1, 2), `*`(`/`(`*`(m, `*`(v[4](t), `*`(`+`(`*`(2, `*`(v[2](t), `*`(diff(v[2](t), t)))), `*`(2, `*`(v[3](t), `*`(diff(v[3](t), t)))), `*`(2, `*`(v[4](t), `*`(diff(v[4](t...

(3.11)

These formulas can be rewritten as:

DP[1]:=m/(1-beta^2)^2/c^2*(v[2](t)*w[2] (t)+v[3] (t)*w[3](t)+v[4](t)*w[4](t));
DP[2]:=m/(1-beta^2)^2/c^3*v[2](t)*(v[2] (t)*w[2](t)+v[3](t)*w[3](t)+v[4](t)* w[4](t))+1/(1-beta^2)/c*w[2](t);
DP[3]:=m/(1-beta^2)^2/c^3*v[3](t)*(v[2](t)*w[2] (t)+v[3](t)*w[3](t)+v[4] (t)*w[4](t))+1/(1-beta^2)/c*w[3](t);
DP[4]:=m/(1-beta^2)^2/c^3*v[4](t)*(v[2](t)*w[2] (t)+v[3](t)*w[3](t)+v[4] (t)*w[4](t))+1/(1-beta^2)/c*w[4](t);

(3.12)

Therefore, we have that derivatives of components of 4-energy-momentum vector differ from accel-eration components by the factor m/c.

Now we denote derivatives of components of 4-energy-momentum vector by

f[1]:=m/(1-beta^2)^2/c^3*(v[2]*w[2]+v[3]*w[3]+v[4]*w[4]);
f[2]:=m/(1-beta^2)^2/c^4*v[2]*(v[2]*w[2]+v[3]*w[3]+v[4]*w[4])+m/(1-beta^2)/c^2* w[2];
f[3]:=m/(1-beta^2)^2/c^4*v[3]*(v[2]*w[2]+v[3]*w[3]+v[4]*w[4])+m/(1-beta^2) /c^2*w[3];
f[4]:=m/(1-beta^2)^2/c^4*v[4]*(v[2]*w[2]+v[3]*w[3]+v[4]*w[4])+m/(1-beta^2 )/c^2*w[4];

(3.13)

whence

and 4-force vector is:

>	F:=create([-1],array([f[1],f[2],f[3],f[4]]));

(3.14)

>	*F:=create([-1],array([(v[2]f[2]+v[3]f[3]+v[4]f[4])/c,f[2],f[3],f[4]])):**

4. Tensor of electromagnetic field

Consider covariant 4-tensor of electromagnetic field:

>	T[1,2]:=E[1]; T[1,3]:=E[2]; T[1,4]:=E[3]; T[2,3]:=-H[3]; T[2,4]:=H[2]; T[3,4]:=-H[1]; F:=create([-1,-1],array(1..4,1..4,sparse,[[0,T[1,2],T[1,3],T[1,4]],[-T[1,2],0,T[2,3],T[2,4]],[-T[1,3],-T[2,3],0,T[3,4]],[-T[1,4],-T[2,4],-T[3,4],0]]));

(4.1)

Contravariant tensor of electromagnetic field is:

>	F_contr:=raise(g_contr,raise(g_contr,F,1),2);

(4.2)

Dual tensor to covariant tensor of electromagnetic field is:

>	Levi_Civita (detg,4,cov_LC,con_LC); F_star:=dual(con_LC,F,[1,2]);

(4.3)

Dual tensor to contravariant tensor of electromagnetic field is:

>	F_star_cov:=lower(g,lower(g,F_star,1),2);

(4.4)

Calculate invariants of electromagnetic field:

>	INV[1]:=get_compts(prod(F,F_contr,[1,1], [2,2]));

(4.5)

>	INV[2]:=get_compts(prod(F,F_star,[1,1], [2,2]));

(4.6)

Components of tensor of electromagnetic field will transform with Lorentz transformations as

>	dim:=4: Coord_1:=[t,x,y,z]; Coord_2:=[t,x,y,z]; CHANG:=[t=gamma(t-betax),x=gamma(x-betat),y=y,z=z]; Jacobian(Coord_2,CHANG,tr,TR);op(tr):op(TR);

(4.7)

>	F_trans:=transform(F,CHANG,tr,TR);

(4.8)

matrix([[0,E_trans[1],E_trans[2],E_trans[3]],[-E_trans[1],0,-H_trans[3],H_trans[2]],[-E_trans[2],H[3],0,-H_trans[1]],[-E_trans[3],-H_trans[2],H_trans[1],0]])=simplify(subs(gamma=1/sqrt(1-beta^2),matrix([[0,-gamma^2*beta^2*E[1]+gamma^2*E[1],gamma*E[2]+gamma*beta*H[3],gamma*E[3]-gamma*beta*H[2]],[-gamma^2*E[1]+gamma^2*beta^2*E[1],0,-gamma*beta*E[2]-gamma*H[3],-gamma*beta*E[3]+gamma*H[2]],[-gamma*E[2] -gamma*beta*H[3],gamma*beta*E[2]+gamma*H [3],0,-H[1]],[-gamma*E[3]+gamma*beta*H[2], gamma*beta*E[3]-gamma*H[2],H[1],0]])));

(4.9)

5. Energy-momentum tensor of electromagnetic field

Consider covariant 4-tensor of electromagnetic field:

>	T[1,2]:=E[1]; T[1,3]:=E[2]; T[1,4]:=E[3]; T[2,3]:=-H[3]; T[2,4]:=H[2]; T[3,4]:=-H[1]; F:=create([-1,-1],array(1..4,1..4,sparse, [[0,T[1,2],T[1,3],T[1,4]],[-T[1,2],0,T[2, 3],T[2,4]],[-T[1,3],-T[2,3],0,T[3,4]],[-T[1,4],-T[2,4],-T[3,4],0]]));

(5.1)

Mixed and contravariant 4-tensors of electromagnetic field are:

>	F_:=raise(g_contr,F,1); F_contr:=raise(g_contr,raise(g_contr,F,1),2);

(5.2)

Calculate miscellaneous functions:

>	T1:=prod(F_contr,F_,[2,2]); T2:=prod(F,F_contr,[1,1],[2,2]);

(5.3)

>	*T2:=-2E[1]^2-2E[2]^2-2E[3]^2+2H[3]^2+2H[2]^2+2H[1]^2;*

(5.4)

So contravariant energy-momentum tensor of electromagnetic field is

T[1,1]:=simplify(1/4/Pi*(-get_compts(T1) [1,1]+ 1/4*get_compts(g_contr)[1,1]*T2));
T[1,2]:=simplify(1/4/Pi*(-get_compts(T1) [1,2]+ 1/4*get_compts(g_contr)[1,2]*T2));
T[1,3]:=simplify(1/4/Pi*(-get_compts(T1) [1,3]+1/4*get_compts(g_contr)[1,3]*T2));
T[1,4]:=simplify(1/4/Pi*(-get_compts(T1) [1,4]+1/4*get_compts(g_contr)[1,4]*T2));
T[2,2]:=simplify(1/4/Pi*(-get_compts(T1) [2,2]+1/4*get_compts(g_contr)[2,2]*T2));
T[2,3]:=simplify(1/4/Pi*(-get_compts(T1) [2,3] +1/4*get_compts(g_contr)[2,3]*T2));
T[2,4]:=simplify(1/4/Pi*(-get_compts(T1) [2,4]+ 1/4*get_compts(g_contr)[2,4]*T2));
T[3,3]:=simplify(1/4/Pi*(-get_compts(T1) [3,3]+ 1/4*get_compts(g_contr)[3,3]*T2));
T[3,4]:=simplify(1/4/Pi*(-get_compts(T1) [3,4]+ 1/4*get_compts(g_contr)[3,4]*T2));
T[4,4]:=simplify(1/4/Pi*(-get_compts(T1) [4,4]+ 1/4*get_compts(g_contr)[4,4]*T2));

(5.5)

>	T_EP_contr:=create([1,1],array(1..4,1..4,sparse,[[T[1,1],T[1,2],T[1,3],T[1,4]],[T[1,2],T[2,2],T[2,3],T[2,4]],[T[1,3],T[2,3],T[3,3],T[3,4]],[T[1,4],T[2,4],T[3,4],T[4,4]]]));

(5.6)

or in standart denotes

w=1/8*(E[1]^2+E[2]^2+E[3]^2+H[3]^2+H[2]^2+H[1]^2)/Pi;
T[1,1]:=w;
S[1]=c*1/4*1/Pi*(E[2]*H[3]-E[3]*H[2]);
T[1,2]:=S[1]/c;
S[2]=-1/4*(E[1]*H[3]-E[3]*H[1])/Pi;
T[1,3]:=S[2]/c;
S[3]=1/4*1/Pi*(E[1]*H[2]-E[2]*H[1]);
T[1,4]:=S[3]/c;
sigma[2,2]=-1/8*(E[1]^2-H[3]^2-H[2]^2-E[2]^2-E[3]^2+H[1]^2)/Pi;
T[2,2]:=sigma[2,2];
sigma[2,3]=-1/4*(E[1]*E[2]+H[2]*H[1])/Pi;
T[2,3]:=sigma[2,3];
sigma[2,4]=-1/4*(E[1]*E[3]+H[3]*H[1])/Pi;
T[2,4]:=sigma[2,4];
sigma[3,3]=1/8*(-E[2]^2+H[3]^2+H[1] ^2+E[1]^2+E[3]^2-H[2]^2)/Pi;
T[3,3]:=sigma[3,3];
sigma[3,4]=-1/4*(E[2]*E[3]+H[3]*H[2])/Pi;
T[3,4]:=sigma[3,4];
sigma[4,4]=1/8*(-E[3]^2+H[2]^2+H[1] ^2+E[1]^2 +E[2]^2-H[3]^2)/Pi;
T[4,4]:=sigma[4,4];

(5.7)

>	T_EP_contr:=create([1,1],array(1..4,1..4,sparse,[[T[1,1],T[1,2],T[1,3],T[1,4]],[T[1,2],T[2,2],T[2,3],T[2,4]],[T[1,3],T[2,3],T[3,3],T[3,4]],[T[1,4],T[2,4],T[3,4],T[4,4]]]));

(5.8)

Covariant energy-momentum tensor of electromagnetic field is

>	T_EP:=lower(g,lower(g,T_EP_contr,1),2);

(5.9)

Components of energy-momentum tensor will transform with Lorentz transformations as

>	dim:=4: Coord_1:=[t,x,y,z]; Coord_2:=[t,x,y,z]; CHANG:=[t=gamma(t-betax),x=gamma(x-betat),y=y,z=z]; Jacobian(Coord_2,CHANG,tr,TR);op(tr):op(TR);

(5.10)

>	T_EP_trans:=collect(transform(T_EP,CHANG, tr, TR),[gamma,beta,c]);

(5.11)

T_EP_trans[1,1]=get_compts(T_EP_trans)[1,1];
T_EP_trans[1,2]=get_compts(T_EP_trans)[1,2];
T_EP_trans[1,3]=get_compts(T_EP_trans)[1,3];
T_EP_trans[1,4]=get_compts(T_EP_trans)[1,4];
T_EP_trans[2,2]=get_compts(T_EP_trans)[2,2];
T_EP_trans[2,3]=get_compts(T_EP_trans)[2,3];
T_EP_trans[2,4]=get_compts(T_EP_trans)[2,4];
T_EP_trans[3,3]=get_compts(T_EP_trans)[3,3];
T_EP_trans[3,4]=get_compts(T_EP_trans)[3,4];
T_EP_trans[4,4]=get_compts(T_EP_trans)[4,4];

(5.12)

So energy density (w), components of energy flux density vector (S) and components of Maxwell stress tensor () will transform as

w_trans=(sigma[2,2]*beta^2+2*S[1]/c*beta+ w)*gamma^2;
S_trans[1]=(-S[1]/c*beta^2+(-w-sigma[2,2]) *beta-S[1]/c)*gamma^2;
S_trans[2]=(-sigma[2,3]*beta-S[2]/c)*gamma;
S_trans[3]=(-sigma[2,4]*beta-S[3]/c)*gamma;
sigma_trans[2,2]=(w*beta^2+2*S[1]/c*beta+sigma[2,2])*gamma^2;
sigma_trans[2,3]=(S[2]/c*beta+sigma[2,3])*gamma;
sigma_trans[2,4]=(S[3]/c*beta+sigma[2,4])*gamma;
sigma_trans[3,3]=sigma[3,3];
sigma_trans[3,4]=sigma[3,4];
sigma_trans[4,4]=sigma[4,4];

(5.13)

6. Energy-momentum tensor of macroscopic particles

Consider energy-momentum tensor of macroscopic particles:

>	w:=epsilon; S[1]:=0; S[2]:=0; S[3]:=0; sigma[2,2]:=p; sigma[2,3]:=0; sigma[2,4]:=0; sigma[3,3]:=p; sigma[3,4]:=0; sigma[4,4]:=p;

(6.1)

Components of energy-momentum tensor of macroscopic particleswill transform with Lorentz transformations as

(6.2)

or in standart denotes

w_trans=(sigma[2,2]*beta^2+2*S[1]/c*beta+w)*gamma^2;
S_trans[2]=(-S[1]/c*beta^2+(-w-sigma[2,2]) *beta-S[1]/c)*gamma^2;
S_trans[3]=(-sigma[2,3]*beta-S[2]/c)*gamma;
S_trans[4]=(-sigma[2,4]*beta-S[3]/c)*gamma;
sigma_trans[2,2]=(w*beta^2+2*S[1]/c*beta+sigma[2,2])*gamma^2;
sigma_trans[2,3]=(S[2]/c*beta+sigma[2,3])*gamma;
sigma_trans[2,4]=(S[3]/c*beta+sigma[2,4])*gamma;
sigma_trans[3,3]=sigma[3,3];
sigma_trans[3,4]=sigma[3,4];
sigma_trans[4,4]=sigma[4,4];

(6.3)

7. Energy-momentum tensor of plane electromagnetic wave

Consider plane electromagnetic wave with filed vectors:

>	H[1]:=0; E[1]:=0; H[2]:=-E[3]; H[3]:=E[2];

(7.1)

So energy-momentum tensor of plane electromagnetic wave is

T_EP_contr:=TABLE([compts=matrix([[1/8*(E[1]^2+E[2]^2+E[3]^2+H[3]^2+H[2]^2+H[1]^2)/Pi,1/4*1/Pi*(E[2]*H[3]-E[3]*H[2]),-1/4*(E[1]*H[3]-E[3]*H[1])/Pi,1/4*1/Pi*(E[1]*H[2]-E[2]*H[1])],[1/4*1/Pi*(E[2]*H[3]-E[3]*H[2]),-1/8*(E[1]^2-H[3]^2-H[2]^2-E[2]^2-E[3]^2+H[1]^2)/Pi,-1/4*(E[1]*E[2]+H[2]*H[1])/Pi,-1/4*(E[1]*E[3]+H[3]*H[1])/Pi],[-1/4*(E[1]*H[3]-E[3]*H[1])/Pi,-1/4*(E[1]*E[2]+H[2]*H[1])/Pi,1/8*(-E[2]^2+H[3]^2+H[1]^2+E[1]^2+E[3]^2-H[2]^2)/Pi,-1/4*(E[2]*E[3]+H[3]*H[2])/Pi],[1/4*1/Pi*(E[1]*H[2]-E[2]*H[1]),-1/4*(E[1]*E[3]+H[3]*H[1])/Pi,-1/4*(E[2]*E[3]+H[3]*H[2])/Pi,1/8*(-E[3]^2+H[2]^2+H[1]^2+E[1]^2+E[2]^2-H[3]^2)/Pi]]), index_char=[1,1]]);

(7.2)

w:=1/4*(E[2]^2+E[3]^2)/Pi; S[2]:=c*w; S[3]:=0; S[4]:=0; sigma[2,2]:=w;
sigma[2,3]:=0; sigma[2,4]:=0; sigma[3,3]:=0; sigma[3,4]:=0; sigma[4,4]:=0; p:=0;
T_EP := TABLE([index_char = [-1, -1], compts = matrix([[w, -S[1]/c, -S[2]/c, -S[3]/c], [-S[1]/c, sigma[2,2], sigma[2,3], sigma[2,4]], [-S[2]/c, sigma[2,3], sigma[3,3], sigma[3,4]], [-S[3]/c, sigma[2,4], sigma[3,4], sigma[4,4]]])]);

(7.3)

8. General Lorentz transformations

Consider general Lorentz transformations.

>	restart;

>	with(tensor):with(linalg):

g11:=1: g22:=-1: g33:=-1: g44:=-1:
g:=create([-1,-1],array(1..4,1..4,symmetric,sparse,[(1,1)=g11(r,theta),(2,2)=g22(r,theta),(3,3)=g33(r,theta),(4,4)=g44(r,theta)]));
g_matrix:=matrix([[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1]]);
detg:=det(g_matrix);
`tensor/simp`:= proc(a) simplify(a); end:
g_contr := invert(g, 'det_g');

(8.1)

>	dim:=4: Coord_1:=[t,x,y,z]; Coord_2:=[t,x,y,z];

(8.2)

gamma=1/sqrt(1-(beta[1]^2+beta[2]^2 +beta [3]^2));
CHANG:=[t=gamma*(t-(beta[1]*x+beta[2]*y+ beta[3]*z)),x=x+(gamma-1)*beta[1]*(beta[1]*x+beta[2]*y+beta[3]*z)/(beta[1]^2+beta [2]^2+beta[3]^2)+gamma*beta[1]*t,y=y+(gamma-1)*beta[2]*(beta[1]*x+beta[2]*y+beta[3]*z)/(beta[1]^2+beta[2]^2+beta[3]^2)+gamma*beta[2]*t,z=z+(gamma-1)*beta[3]*(beta[1]*x+beta[2]*y+beta[3]*z)/(beta[1]^2+beta[2]^2+beta[3]^2)+gamma*beta[3]*t];

(8.3)

>	Jacobian(Coord_2,CHANG,tr,TR); op(tr): op(TR);

(8.4)

Components of covariant vector will transform with Lorentz transformations as

>	VECT:=create([-1],array([V[1],V[2],V[3], V[4]]));

(8.5)

VECT_trans:=create([-1],array([V_trans [1], V_trans[2],V_trans[3],V_trans[4]]));
get_compts(VECT_trans)=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma)):
get_compts(VECT_trans)[1]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[1];
get_compts(VECT_trans)[2]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[2];
get_compts(VECT_trans)[3]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[3];
get_compts(VECT_trans)[4]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[4];

(8.6)

Components of covariant tensor will transform with Lorentz transformations as

>	TENS:=create([-1,-1],array(1..4,1..4,sparse,[[T[1,1],T[1,2],T[1,3],T[1,4]],[T[2,1],T[2,2],T[2,3],T[2,4]],[T[3,1],T[3,2],T[3,3],T[3,4]],[T[4,1],T[4,2],T[4,3],T[4,4]]]));

(8.7)

TENS_trans:=create([-1,-1],array(1..4,1.. 4,sparse,[[T_trans[1,1],T_trans[1,2],T_trans[1,3],T_trans[1,4]],[T_trans[2,1],T_trans[2,2],T_trans[2,3],T_trans[2,4]],[T_trans[3,1],T_trans[3,2],T_trans[3,3],T_trans[3,4]],[T_trans[4,1],T_trans[4,2],T_trans[4,3],T_trans[4,4]]]));
get_compts(TENS_trans)=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma)):

get_compts(TENS_trans)[1,1]=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma))[1,1];
get_compts(TENS_trans)[1,2]=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma))[1,2];
get_compts(TENS_trans)[2,4]=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma))[2,4];

(8.8)

For symmetrical tensor we have transformations:

>	TENS_sym:=create([-1,-1],array(1..4,1..4,sparse,[[Tau[1,1],Tau[1,2],Tau[1,3],Tau[1,4]],[Tau[1,2],Tau[2,2],Tau[2,3],Tau[2,4]],[Tau[1,3],Tau[2,3],Tau[3,3],Tau[3,4]],[Tau[1,4],Tau[2,4],Tau[3,4],Tau[4,4]]]));

(8.9)

(8.10)

For asymmetrical tensor we have transformations:

>	TENS_as:=create([-1,-1],array(1..4, 1..4,sparse, [[0,Tau[1,2],Tau[1,3], Tau[1,4]],[-Tau[1,2],0,Tau [2,3],Tau[2, 4]],[-Tau[1,3],-Tau[2,3],0,Tau[3,4]],[-Tau[1,4],-Tau[2,4],-Tau[3,4],0]]));

(8.11)

TENS_as_trans:=create([-1,-1],array(1..4, 1..4, sparse,[[0,T_trans[1,2],T_trans [1,3],T_trans[1,4]],[-T_trans[1,2],0, T_trans[2,3], T_trans[2,4]],[-T_trans[1,3] ,-T_trans[2,3],0, T_trans[3,4]],[-T_trans [1,4],-T_trans[2,4],-T_trans[3,4],0]]));
get_compts(TENS_trans)=get_compts(simplify(collect(transform(TENS_as,CHANG,tr,TR), [gamma,beta]))):

(8.12)

9. Tensor of moment of momentum

Define antisymmetric tensor of moment of momentum. We need to determine 4-radius-vector of particle and 4-energy-momentum vector:

>	R4(t):=create([-1],array([x[1],x[2],x[3], x[4]])); gamma=1/sqrt(1-(beta[1]^2+beta[2]^2+beta[3]^2)); P4(t):=create([-1],array([mcgamma,mgamma v[2],mgammav[3],mgammav[4]]));

(9.1)

Define miscellaneous functions

>	MOM1:=prod(R4(t),P4(t)); MOM2:=prod(P4(t),R4(t));

(9.2)

and create covariant antisymmetric tensor of moment of momentum:

T[1,2]:=collect(get_compts(MOM1)[1,2]-get_compts(MOM2)[1,2],[gamma,m]);
T[1,3]:=collect(get_compts(MOM1)[1,2]-get_compts(MOM2)[1,3],[gamma,m]);
T[1,4]:=collect(get_compts(MOM1)[1,2]-get_compts(MOM2)[1,4],[gamma,m]);
T[2,3]:=collect(get_compts(MOM1)[1,2]-get_compts(MOM2)[2,3],[gamma,m]);
T[2,4]:=collect(get_compts(MOM1)[1,2]-get_compts(MOM2)[2,4],[gamma,m]);
T[3,4]:=collect(get_compts(MOM1)[1,2]-get_compts(MOM2)[3,4],[gamma,m]);

(9.3)

>	MOM:=create([-1,-1],array(1..4,1..4,sparse, [[0,T[1,2],T[1,3],T[1,4]],[-T[1,2],0,T[2,3], T[2,4]],[-T[1,3],-T[2,3],0,T[3,4]],[-T[1,4],-T[2,4],-T[3,4],0]])); get_char(MOM); get_rank(MOM);

(9.4)

Contravariant antisymmetric tensor of moment of momentum is:

>	MOM_contr:=raise(g_contr,raise(g_contr, MOM, 1),2); get_char(MOM_contr); get_rank(MOM_contr);

(9.5)

We can rewrite these relations in standart denotes:

>	R4(t):=create([-1],array([c*t,x,y,z])); P4(t):=create([-1],array([En/c,P[2],P[3],P[4]])); MOM1:=prod(R4(t),P4(t)); MOM2:=prod(P4(t),R4(t));

(9.6)

T[1,2]:=collect(get_compts(MOM1)[1,2]-get_compts (MOM2)[1,2],gamma);
T[1,3]:=collect(get_compts(MOM1)[1,2]-get_compts (MOM2)[1,3],gamma);
T[1,4]:=collect(get_compts(MOM1)[1,2]-get_compts (MOM2)[1,4],gamma);
T[2,3]:=collect(get_compts(MOM1)[1,2]-get_compts (MOM2)[2,3],gamma);
T[2,4]:=collect(get_compts(MOM1)[1,2]-get_compts (MOM2)[2,4],gamma);
T[3,4]:=collect(get_compts(MOM1)[1,2]-get_compts (MOM2)[3,4],gamma);

(9.7)

>	MOM_:=create([-1,-1],array(1..4,1..4, sparse ,[[0,T[1,2],T[1,3],T[1,4]],[-T[1,2] ,0,T[2,3], T[2,4]],[-T[1,3],-T[2,3],0, T[3,4]],[-T[1,4],-T[2,4],-T[3,4],0]])); get_char(MOM_): get_rank(MOM_):

(9.8)

>	MOM_contr:=raise(g_contr,raise(g_contr, MOM_, 1),2); get_char(MOM_contr): get_rank(MOM_contr):

(9.9)

References

1. Landau L.D., Lifshits E.V. Theoretical physics. V. 2. Field theory. - Moscow. Fizmatlit. 2003.

2. Tikhonenko A.V. Calculus of tensors and applications in applied mathematical packets. Obninsk. IATE. 2007

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