The stress energy tensor is subject to Bianchi identity for reasons that were obvious to Einstein and obscure to almost everyone else.
This is true even in special relativity and has direct antecedents in 19th century physics.
It is the tensor formulation of conservation principles that are global in special relativity and local in general relativity.
<a href=a.a68>This program</a> and <a href=yield>its yield</a> describes an infinite cylindrical region of space with positive or negative curvature and reports the Einstein tensor G<sub>αβ</sub> which describe the necessary stress energy tensor to produce that shape of space-time.
For positive curvature and r > tr:
<table><tr><td>G<sub>αβ</sub> = <td><table><tr><td>1<td>0<td>0<td>0
<tr><td>0<td>0<td>0<td>0<tr><td>0<td>0<td>0<td>0
<tr><td>0<td>0<td>0<td>−1</table><td>= T<sub>αβ</sub></table>
which indeed describes tension within the infinite cylinder along the z-axis, but a positive mass density too.
Space is locally flat and empty outside the cylinder!
Globally there is an angular deficit about the cylinder that might be measured by anyone traveling around the cylinder.
<p>It remains to discover whether motion along the string changes the situation.
It should be shown that stitching the curved and flat parts of the space satisfies the essential Einstein equations; do the Bianchi identities hold at the boundary?
<p>We consider a Lorentz boost in the z direction with rapidity ψ:
<table><tr><td>A<sup>α</sup><sub>β</sub> = <td><table><tr><td>cosh ψ<td>0<td>0<td>sinh ψ
<tr><td>0<td>1<td>0<td>0<tr><td>0<td>0<td>1<td>0
<tr><td>sinh ψ<td>0<td>0<td>cosh ψ</table><td>.</table>
<p>It acts on the stress energy tensor thus:
<br>T'<sub>βδ</sub> = A<sup>α</sup><sub>β</sub>A<sup>γ</sup><sub>δ</sub>T<sub>αγ</sub>.
<br>OK I give up. I can’t do this in my head but <a href=boost.a68>this program</a> reports that the transformed T (tp) is identical to the original (t).
This test would have failed if the logic of the Einstein tensor G had not forced me to include the T<sub>00</sub> term.
<p>The continuous symmetries of this space time are thus generated by<ul>
<li>the boost(ψ),
<li>z' = z+c,
<li>φ' = φ+d,
<li>t' = t+e</ul>
z' = −z, t' = −t and φ' = −φ provide discrete symmetries.
<p>It would be good to show that the angle deficit is constant as the thread radius decreases but total curvature within a thread cross-section remains the same.
It would be good to include the 8π factor.

<h3>Embedding</h3>
I should make clearer that a 2D space like cross-section of this space-time, perpendicular to the thread, can be isometrically embedded in 3D Euclidean space where it appears as part unit sphere and part cone.
The two parts join at a circle of tangency between cone and sphere.
<h3>Gravitational Field</h3>
The gravitational field of the string has an unusual shape.
A simply connected region that is disjoint from the string is flat; there are no “gravitational forces” inside.
Two test particles emerging from such a region, traveling parallel with the same velocity, passing on opposite sides of the string, will find themselves approaching as if there had briefly been a mass between them—a crease in space.