## Question about Algebraic classification and Compatibility of Energy Momentum Tensors

unquoted text! -P.H.]

I am studying Stephani et al.'s "Exact Solutions of Einstein's Field
the Energy tensors discussed in Chapter 5, which are still new to me.

Because of the Einstein equations, the Ricci R_uv tensor takes on the
same algebraic type as the energy tensor T_uv to which it is linked.
This, by implication, seems to have consequences regarding the
"compatibility" of various energy tensors with one another.
Specifically:

The non-null Maxwell tensor T^uv (Maxwell) is of the Segre type
[(11),(1,1)] / Plebanski type [2S-2T]. It is also a function of the
metric tensor g_uv and the field strength tensor F^uv, that is,
T^uv(Maxwell) = T^uv(g_uv, F_uv)

The Perfect fluid tensor T^uv(Euler) is of the Segre Type [(111),1] /
Plebanski type [3S-T].

My question is about compatibility between energy tensors. Suppose one
wanted to somehow understand whether there exists an electrodynamic
basis underlying a perfect fluid. That is, suppose one wanted to
explore the possibility that T^uv(Euler) = T^uv'(g_uv, F_uv), similarly
to the Maxwell tensor (I have used a prime ' to denote that the
combination of fields T^uv'(g_uv, F_uv) may be different from that in
T^uv(g_uv, F_uv) = T^uv(Maxwell)).

At some juncture, one would need to set T^uv(Euler) = T^uv'(g_uv, F_uv).
In this situation, would it be necessary for T^uv(Euler) to be of the
same algebraic type as T^uv'(g_uv, F_uv)? That is, would one need to
find a T^uv'(g_uv, F_uv) in the Segre Type [(111),1] / Plebanski type
[3S-T] class rather than in the [(11),(1,1)] / Plebanski type [2S-2T]
class for this to be valid? If one tried to set a Segre Type [(111),1]
/ Plebanski type [3S-T] tensor equal to a [(11),(1,1)] / Plebanski type
[2S-2T] class tensor, would this be an apples to oranges equation that
is effectively trying to set the Ricci tensor to two different algebraic
type simultaneously? (It is to be understood that in all cases these
are energy tensors which would also be set to the R^uv-(1/2)g^uvR and so
would have vanishing divergence and establish a second order non-linear
differential equation for the metric tensor).

More generally, how precisely must the algebraic classes of two tensors
match if one is to equate them in a valid manner to one another as well
as to R^uv-(1/2)g^uvR?

Finally, if one were to take T^uv(Maxwell) and augment it with a further
term "S g^uv" where S is a scalar *field* (not a cosmological constant),
that is, if one were to migrate T^uv(Maxwell) --> T^uv(Maxwell) + S
g^uv, which would add a non-zero trace, what would that do to the class
of T^uv(Maxwell)? What class would T^uv(Maxwell) + S g^uv fall into?
Would it by chance move from the Segre type [(11),(1,1)] / Plebanski
type [2S-2T] into the [(111),1] / Plebanski type [3S-T] class of a
perfect fluid tensor? If not, what class would T^uv(Maxwell) + S g^uv
fall into?

Thanks.

Jay.
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com

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