Fixed some tensor index mistakes in alcubierre metric

book/knowledge-library/alcubierre-metric-intro.ipynb

@@ -167,7 +167,7 @@
    "id": "09e2456a",
    "metadata": {},
    "source": [
-    "Now, we can calculate the expansion and contraction of space resultant from the metric - the magnitude of which is termed the **York Time**. The trace of the extrinsic curvature tensor $K_{ij}$ is given by:"
+    "Now, we can calculate the expansion and contraction of space resultant from the metric - the magnitude of which is termed the **York Time**. The trace (contraction) of the extrinsic curvature tensor $K_{ij}$ is given by:"
    ]
   },
   {
@@ -176,7 +176,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "K^i{}_j = \\partial_j X^i\n",
+    "K = K^i{}_i = \\partial_i X^i\n",
     "$$"
    ]
   },
@@ -194,7 +194,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "K^i{}_j = v_s \\frac{x_s}{r_s} \\frac{df(r_s)}{dr_s}\n",
+    "K = v_s \\frac{x_s}{r_s} \\frac{df(r_s)}{dr_s}\n",
     "$$"
    ]
   },
@@ -334,7 +334,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "T_{00} = -\\frac{1}{8\\pi} \\frac{(v_s)^2 (y^2 + z^2)}{4 (r_s)^2} \\left(\\frac{df(r_s)}{dr_s}\\right)^2\n",
+    "T_{00} = \\frac{1}{8\\pi} G^{00} = -\\frac{1}{8\\pi} \\frac{(v_s)^2 (y^2 + z^2)}{4 (r_s)^2} \\left(\\frac{df(r_s)}{dr_s}\\right)^2\n",
     "$$"
    ]
   },
@@ -344,7 +344,7 @@
    "metadata": {},
    "source": [
     "```{note}\n",
-    "Here $G = c = 1$, which is the standard when using the ADM formalism.\n",
+    "Here $G = c = 1$, which is the standard when using the ADM formalism. Thus $\\frac{c^4}{8\\pi G} \\Rightarrow \\frac{1}{8\\pi}$.\n",
     "```"
    ]
   },