Sent this a few days ago, has not posted yet:

I have been working with the Euler (perfect fluid) tensor lately, and had a
question about the formulation of this tensor in relation to the spacetime
metric signature.

For a spacelike signature -+++, this tensor is T^uv=(u+p)U^uU^v + g_uv p,
where u is dust density, p is pressure density, and U^u is the velocity four
vector, and U^uU_u=-1 on account of signature -+++.

The trace T=T^u_u of the above is then T = -u-p+4p = -u+3p. If we have an
equation of state u=p ("stiff matter"), then T=2p=2u, and then the above
tensor becomes T^uv=T U^uU^v + (1/2) g_uv T

For a timelike signature +---, I believe (please help me here) that we must
write T^uv=(u+p)U^uU^v - g_uv p, that is, the g_uv p term now has a minus
sign and U^uU_u=+1 on account of signature -+++.

The Kronecker delta is the same no matter what, so the trace is now T=u+p-4p
= u-3p. For the same equation of state u=p for stiff matter, then T=-2p=-2u
and the tensor then becomes T^uv=-T U^uU^v + (1/2) g_uv T.

Since the dust density p0, that means T0 for the energy tensor where the
metric has a timelike signature +---, and it seems odd to me to have a trace
density 0.

Am I missing something here?

Maybe I should use T^uv=-(u+p)U^uU^v + g_uv p for the timelike signature
+---? Then T=-u-p+4p = -u+3p just as in the spacelike case -+++, then, for
stiff matter u=p, the trace is T=2p=2u.

Then, the final tensor becomes T^uv=-TU^uU^v + (1/2) g_uv T.

What is the right answer / approach here?

Thanks.

Jay.
_____________________________
Jay R. Yablon
Email:

In article ,
Jay R. Yablon wrote:

I have been working with the Euler (perfect fluid) tensor lately, and had a
question about the formulation of this tensor in relation to the spacetime
metric signature.

For a spacelike signature -+++, this tensor is T^uv=(u+p)U^uU^v + g_uv p,
where u is dust density, p is pressure density, and U^u is the velocity four
vector, and U^uU_u=-1 on account of signature -+++.

The trace T=T^u_u of the above is then T = -u-p+4p = -u+3p. If we have an
equation of state u=p ("stiff matter"), then T=2p=2u, and then the above
tensor becomes T^uv=T U^uU^v + (1/2) g_uv T

For a timelike signature +---, I believe (please help me here) that we must
write T^uv=(u+p)U^uU^v - g_uv p, that is, the g_uv p term now has a minus
sign and U^uU_u=+1 on account of signature -+++.

I'm a bit too lazy to work through this in detail, so all I can say is this.

If you work out a manifestly coordinate-independent formula like

T^uv=(u+p)U^uU^v + g_uv p

using some signature, and then you whimsically decide to change your
conventions regarding the signature, the formula will still be true
without any changes.

Proof: if this weren't true, particle physicists would have *killed off*
people working on general relativity long ago, because they would never
agree about the basic formulas of physics. The only way people can
tolerate the coexistence of two conventions, +--- and -+++, is that the
basic formulas are right either way.

So, I think your error was changing your equation to

T^uv=(u+p)U^uU^v - g_uv p

when you switched from +--- to -+++. The change in g_uv should do
precisely the right thing, with no extra fiddling.

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John Baez says...

If you work out a manifestly coordinate-independent formula like

T^uv=(u+p)U^uU^v + g_uv p

using some signature, and then you whimsically decide to change your
conventions regarding the signature, the formula will still be true
without any changes.

Yes, but something that's interesting is that when we deal
with Clifford algebras, the algebra for (+---) spacetime is
not the same as the algebra for (-+++) spacetime. I don't
think that that difference would allow us to say that we are
*really* one signature instead of another, though.

--
Daryl McCullough
Ithaca, NY

In article ,
(Daryl McCullough) wrote:

John Baez says...

If you work out a manifestly coordinate-independent formula like

T^uv=(u+p)U^uU^v + g_uv p

using some signature, and then you whimsically decide to change your
conventions regarding the signature, the formula will still be true
without any changes.

Yes, but something that's interesting is that when we deal
with Clifford algebras, the algebra for (+---) spacetime is
not the same as the algebra for (-+++) spacetime. I don't
think that that difference would allow us to say that we are
*really* one signature instead of another, though.

This just ends up being the fact that there are many (8, I think)
different choices for the extending the usual Spin(1,3) - SO(1,3) cover
to a cover of all of O(1,3). Two of these choices come from the two
Clifford algebras Cl(1,3) and Cl(3,1), but thought of as just a choice
of cover, it has nothing to do with the signature.

Aaron

So how does one account for the fact that this tensor is written as
T^uv=(u+p)U^uU^v + g_uv p for -+++, see, e.g., Stehpani, "Exact
Solutions . . ." equation (5.9), and as T^uv=(u+p)U^uU^v - g_uv p for
+---, see, e.g., Einstein, ". . . The General Theory" (1916), equation
(58) (where u is used in place of u+p)?

What I find confusing are the trace relations: U^uU_u=-1 for -+++ while
U^uU_u=+1 for +---. Yet, g^us g_vs = lambda^u_v no matter what the
signature. Therefore, the trace of T^uv=(u+p)U^uU^v + g_uv p for -+++
is T = -u-p+4p = -u+3p while the trace of the same T^uv=(u+p)U^uU^v +
g_uv p for +--- is +u+p+4p=u+5p. But, the trace of T^uv=(u+p)U^uU^v -
g_uv for +--- is u+p-4p = u-3p which seems to make more sense because
u-3p = -(-u+3p). The u+p+4p=u+5p versus the u+p-4p = u-3p in the trace
equation is what I am trying to sort out.

Jay.

So how does one account for the fact that this tensor is written as
T^uv=(u+p)U^uU^v + g_uv p for -+++, see, e.g., Stehpani, "Exact
Solutions . . ." equation (5.9), and as T^uv=(u+p)U^uU^v - g_uv p for
+---, see, e.g., Einstein, ". . . The General Theory" (1916), equation
(58) (where u is used in place of u+p)?

What I find confusing are the trace relations: U^uU_u=-1 for -+++ while
U^uU_u=+1 for +---. Yet, g^us g_vs = lambda^u_v no matter what the
signature. Therefore, the trace of T^uv=(u+p)U^uU^v + g_uv p for -+++
is T = -u-p+4p = -u+3p while the trace of the same T^uv=(u+p)U^uU^v +
g_uv p for +--- is +u+p+4p=u+5p. But, the trace of T^uv=(u+p)U^uU^v -
g_uv for +--- is u+p-4p = u-3p which seems to make more sense because
u-3p = -(-u+3p). The u+p+4p=u+5p versus the u+p-4p = u-3p in the trace
equation is what I am trying to sort out.

Jay.