The energy-momentum tensor
Various functionality to deal with the stress-energy tensor in special relativity.
perfectfluid(rho,p,u=0) dust(rho,u=0) photongas(rho,u=0) transform_dd(TT, B) transform_ud(TT, B) transform_uu(TT, B) raise(TT) lower(TT)
TT |
A second-rank tensor with indices either downstairs-downstairs, downstairs-upstairs, or upstairs-upstairs |
B |
A boost matrix |
rho,p,u |
Density, pressure, and four-velocity of the dust |
Function perfectfluid() returns the stress-energy tensor, with
two upstairs indices, for a perfect fluid with the conditions
specified. No checking for physical reasonableness (eg the weak
energy condition) is performed: caveat emptor!
Function dust() is a (trivial) function that returns the
stress-energy tensor of a zero-pressure perfect fluid (that is, dust).
Function photongas() returns the stress-energy tensor of a
photon gas. They are here for discoverability reasons; both are
special cases of a perfect fluid.
Functions transform_dd() et seq transform a second-rank tensor
using the Lorentz transform. The letters “u” or “d”
denote the indices of the tensor being upstairs (contravariant) or
downstairs (covariant). The stress-energy tensor is usually written
with two upstairs indices, so use transform_uu() to transform
it.
Function lower() lowers both indices of a tensor with two
upstairs indices. Function raise() raises two downstairs
indices. These two functions have identical R idiom but do not return
identical values if c!=1.
Robin K. S. Hankin
perfectfluid(10,1) u <- as.3vel(c(0.4,0.4,0.2)) ## In the following, LHS is stationary dust and RHS is dust moving at ## velocity 'u', but transformed to a frame also moving at velocity 'u': LHS <- dust(1) RHS <- transform_uu(dust(1,u),boost(u)) max(abs(LHS-RHS)) # should be small ## In the following, negative sign needed because active/passive ## difference: LHS <- dust(1,u) RHS <- transform_uu(dust(1),boost(-u)) max(abs(LHS-RHS)) # should be small ## Now test behaviour when c!=1: sol(299792458) perfectfluid(1.225,101325) # air at STP LHS <- transform_uu(perfectfluid(1.225,101325),boost(as.3vel(c(1000,0,0)))) RHS <- perfectfluid(1.225,101325) LHS-RHS # should be small sol(10) u <- as.3vel(4:6) LHS <- photongas(1,u) RHS <- transform_uu(photongas(1),boost(-u)) LHS-RHS # should be small B1 <- boost(r3vel(1)) %*% boost(r3vel(1)) B2 <- boost(r3vel(1)) %*% boost(r3vel(1)) LHS <- transform_uu(transform_uu(dust(1),B1),B2) RHS <- transform_uu(dust(1),B2 %*% B1) # note order LHS-RHS # should be small ## remember to re-set c: sol(1)
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