Question: Can a boosted frame remove unwanted terms from a transformed metric?

[Mentz114]

A certain metric gives an Einstein tensor that has the form below. The coordinate labelling is
$x^0=t,\ x^1=r,\ x^2=\theta,\ x^3=\phi$
$$G_{\mu\nu}= \left[ \begin{array}{cccc}<br /> A & B & 0 & 0\\<br /> B & p1 & 0 & 0\\<br /> 0 & 0 & p2 & 0\\<br /> 0 & 0 & 0 & p3<br /> \end{array} \right]$$
where $A,B,C,p1,p2,p3$ are functions of t and r. A transformation $\Lambda$ so $\Lambda^\mu_\rho\ \Lambda^\nu_\sigma\ G_{\mu\nu}$ is diagonal is easily found,
$$\Lambda^\mu_\rho=\left[ \begin{array}{cccc}<br /> 1 & -\frac{B}{p1} & 0 & 0\\<br /> 0 & 1 & 0 & 0\\<br /> 0 & 0 & 1 & 0\\<br /> 0 & 0 & 0 & 1<br /> \end{array} \right]$$
This seems to be transforming $t$ into $T=t-h\ r$ where $h=B/p1$. This can be used to give the differential transformation
$$dT=dt -hdr-rdh=dt-hdr-r(\partial_t h\ dt + \partial_r h\ dr)$$
so we can find $dt^2$ and substitute into the original metric to get a transformed one written in coordinates $T,r,\theta,\phi$.

Question: will the Einstein tensor obtained from the transformed metric be $\Lambda^\mu_\rho\ \Lambda^\nu_\sigma\ G_{\mu\nu}$ ?

I think it will be but I haven't convinced myself.

 

[bcrowell]

Question: will the Einstein tensor obtained from the transformed metric be $\Lambda^\mu_\rho\ \Lambda^\nu_\sigma\ G_{\mu\nu}$ ?

Isn't this true regardless of the specifics of the problem, just because that's how a rank-2 tensor transforms?

 

[Mentz114]

Isn't this true regardless of the specifics of the problem, just because that's how a rank-2 tensor transforms?

That is what I think - but I have some doubts.

It would be true if $\Lambda$ were a frame field, i.e. a transformation from the coordinate basis to a frame basis - but it's not.

In the untransformed Einstein tensor, the terms I've called $p1$ etc do come out as isotropic pressure, i.e. $p1=Pg_{11}$, with the same P in all three. So if the off-diagonal terms were absent, it might be a static perfect fluid with pressure. Can one change an unphysical metric to a physical one with a coordinate transformation ? It seems too easy.

 

[PAllen]

I could be all wet here, but it looks like your lambda is impossible as a coordinate transform. I think the (0,0) component being 1 is saying you must have T=t + f(r). Your solution would then be possible if B and p1 depended only on r. But you've said they depend on r and t. Contradiction.

 

[Mentz114]

I could be all wet here, but it looks like your lambda is impossible as a coordinate transform. I think the (0,0) component being 1 is saying you must have T=t + f(r). Your solution would then be possible if B and p1 depended only on r. But you've said they depend on r and t. Contradiction.

I think you're right. It's a bust.

But I have found a boosted frame that removes the unwanted terms, which is what I should have done in the first place.