Let us define 4 vector by 4 co-ordinates (x1,x2,x3,x4) where (x1,x2,x3) are space components (like x,y,z) and x4 is related to time as x4=ict.Express the following equations in tensor notation.
(i)The continuity equation: div J+(del*rho/del t)=0
(ii)The wave equation laplacian ø-(1/c^2) [del^2 ø/del t^2]=0
(iii)What will be the value of j4 instead of above?
my attempts:Please tell me if I am going through the right way.
div(V)+d(pho)/dt=
dJ1/dx1+dJ2/dx2+dJ3/dx3+ic*d(Rho)/(ic*dt)=
sum(dJi/dxi,i=1..4)= di (Ji)
Where J is the quadrivector (Jx,Jy,jz,icRho)
So the charge conservation equation is just the divergence of the 4vector.
Along the same line, the generalisation of the Laplacian is:
Laplacian=sum[(d/dxi)^2,i=1..3]=dii
D'Alembertian= sum[(d/dxi)^2,i=1..3]+(d/d(ict))^2
= sum[(d/dxi)^2,i=1..4]
=dii
So a wave equation looks like a generalization of a Poisson equation to a 4 dimensional space.
[Laplacian-1/c^2(d/dt)](phi)=dii(phi)