sagemathgh-37970: Fix sign error in the Kerr metric of the manifold c…

…atalog

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In the Kerr spacetime constructed from the manifold catalog (i.e.
returned by  `manifolds.Kerr`), there was an error in the sign of the
g_{03} component of the metric tensor with respect to Boyer-Lindquist
coordinates. This is fixed here.

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### ⌛ Dependencies

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URL: sagemath#37970
Reported by: Eric Gourgoulhon
Reviewer(s): Travis Scrimshaw

build/pkgs/configure/checksums.ini

@@ -1,4 +1,4 @@
 tarball=configure-VERSION.tar.gz
-sha1=599182ff950764df31446f7bf8350499717dc76e
-md5=06e0643f4948ad4b4c87d2530d026a1f
-cksum=4286155814
+sha1=fa7d8e54b6563001256eac25159c14cdb3619004
+md5=15cf5e8fb1b18afc819922d8218721a8
+cksum=2753177283

build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-e77b3df44ade2f51f750de90ad255f772a933776
+8eaeb3e77110a5e6b866f7cded0a3e4953fd6c3f

src/sage/manifolds/catalog.py

@@ -127,14 +127,19 @@ def Kerr(m=1, a=0, coordinates="BL", names=None):
         4-dimensional Lorentzian manifold M
         sage: K.atlas()
         [Chart (M, (t, r, th, ph))]
+
+    The Kerr metric in Boyer-Lindquist coordinates (cf. :wikipedia:`Kerr_metric`)::
+
         sage: K.metric().display()
         g = (2*m*r/(a^2*cos(th)^2 + r^2) - 1) dt⊗dt
-         + 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dt⊗dph
+         - 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dt⊗dph
          + (a^2*cos(th)^2 + r^2)/(a^2 - 2*m*r + r^2) dr⊗dr
          + (a^2*cos(th)^2 + r^2) dth⊗dth
-         + 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dph⊗dt
+         - 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dph⊗dt
          + (2*a^2*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) + a^2 + r^2)*sin(th)^2 dph⊗dph
 
+    The Schwarzschild spacetime with the mass parameter set to 1::
+
         sage: K.<t, r, th, ph> = manifolds.Kerr()
         sage: K
         4-dimensional Lorentzian manifold M
@@ -144,6 +149,9 @@ def Kerr(m=1, a=0, coordinates="BL", names=None):
         sage: K.default_chart().coord_range()
         t: (-oo, +oo); r: (0, +oo); th: (0, pi); ph: [-pi, pi] (periodic)
 
+
+    The Kerr spacetime in Kerr coordinates::
+
         sage: m, a = var('m, a')
         sage: K.<t, r, th, ph> = manifolds.Kerr(m, a, coordinates="Kerr")
         sage: K
@@ -205,7 +213,7 @@ def Kerr(m=1, a=0, coordinates="BL", names=None):
         g[0, 0], g[1, 1], g[2, 2], g[3, 3] = -(1-2*m*r/rho**2), \
             rho**2/(r**2-2*m*r+a**2), rho**2, \
             (r**2+a**2+2*m*r*a**2/rho**2*sin(th)**2)*sin(th)**2
-        g[0, 3] = 2*m*r*a*sin(th)**2/rho**2
+        g[0, 3] = -2*m*r*a*sin(th)**2/rho**2
         return M
 
     raise NotImplementedError("coordinates system not implemented, see help"