Metrics

.gitignore

@@ -66,6 +66,9 @@ pip-wheel-metadata
 # VS Code
 .vscode
 
+# Emacs - projectile
+TAGS
+
 # default pytest cache directory
 */.pytest_cache
 .pytest_cache/

docs/Makefile

@@ -0,0 +1,177 @@
+# Makefile for Sphinx documentation
+#
+
+# You can set these variables from the command line.
+SPHINXOPTS    =
+SPHINXBUILD   = sphinx-build
+PAPER         =
+BUILDDIR      = build
+
+# User-friendly check for sphinx-build
+ifeq ($(shell which $(SPHINXBUILD) >/dev/null 2>&1; echo $$?), 1)
+$(error The '$(SPHINXBUILD)' command was not found. Make sure you have Sphinx installed, then set the SPHINXBUILD environment variable to point to the full path of the '$(SPHINXBUILD)' executable. Alternatively you can add the directory with the executable to your PATH. If you don't have Sphinx installed, grab it from http://sphinx-doc.org/)
+endif
+
+# Internal variables.
+PAPEROPT_a4     = -D latex_paper_size=a4
+PAPEROPT_letter = -D latex_paper_size=letter
+ALLSPHINXOPTS   = -d $(BUILDDIR)/doctrees $(PAPEROPT_$(PAPER)) $(SPHINXOPTS) source
+# the i18n builder cannot share the environment and doctrees with the others
+I18NSPHINXOPTS  = $(PAPEROPT_$(PAPER)) $(SPHINXOPTS) source
+
+.PHONY: help clean html dirhtml singlehtml pickle json htmlhelp qthelp devhelp epub latex latexpdf text man changes linkcheck doctest gettext
+
+help:
+	@echo "Please use \`make <target>' where <target> is one of"
+	@echo "  html       to make standalone HTML files"
+	@echo "  dirhtml    to make HTML files named index.html in directories"
+	@echo "  singlehtml to make a single large HTML file"
+	@echo "  pickle     to make pickle files"
+	@echo "  json       to make JSON files"
+	@echo "  htmlhelp   to make HTML files and a HTML help project"
+	@echo "  qthelp     to make HTML files and a qthelp project"
+	@echo "  devhelp    to make HTML files and a Devhelp project"
+	@echo "  epub       to make an epub"
+	@echo "  latex      to make LaTeX files, you can set PAPER=a4 or PAPER=letter"
+	@echo "  latexpdf   to make LaTeX files and run them through pdflatex"
+	@echo "  latexpdfja to make LaTeX files and run them through platex/dvipdfmx"
+	@echo "  text       to make text files"
+	@echo "  man        to make manual pages"
+	@echo "  texinfo    to make Texinfo files"
+	@echo "  info       to make Texinfo files and run them through makeinfo"
+	@echo "  gettext    to make PO message catalogs"
+	@echo "  changes    to make an overview of all changed/added/deprecated items"
+	@echo "  xml        to make Docutils-native XML files"
+	@echo "  pseudoxml  to make pseudoxml-XML files for display purposes"
+	@echo "  linkcheck  to check all external links for integrity"
+	@echo "  doctest    to run all doctests embedded in the documentation (if enabled)"
+
+clean:
+	rm -rf $(BUILDDIR)/*
+
+html:
+	$(SPHINXBUILD) -b html $(ALLSPHINXOPTS) $(BUILDDIR)/html
+	@echo
+	@echo "Build finished. The HTML pages are in $(BUILDDIR)/html."
+
+dirhtml:
+	$(SPHINXBUILD) -b dirhtml $(ALLSPHINXOPTS) $(BUILDDIR)/dirhtml
+	@echo
+	@echo "Build finished. The HTML pages are in $(BUILDDIR)/dirhtml."
+
+singlehtml:
+	$(SPHINXBUILD) -b singlehtml $(ALLSPHINXOPTS) $(BUILDDIR)/singlehtml
+	@echo
+	@echo "Build finished. The HTML page is in $(BUILDDIR)/singlehtml."
+
+pickle:
+	$(SPHINXBUILD) -b pickle $(ALLSPHINXOPTS) $(BUILDDIR)/pickle
+	@echo
+	@echo "Build finished; now you can process the pickle files."
+
+json:
+	$(SPHINXBUILD) -b json $(ALLSPHINXOPTS) $(BUILDDIR)/json
+	@echo
+	@echo "Build finished; now you can process the JSON files."
+
+htmlhelp:
+	$(SPHINXBUILD) -b htmlhelp $(ALLSPHINXOPTS) $(BUILDDIR)/htmlhelp
+	@echo
+	@echo "Build finished; now you can run HTML Help Workshop with the" \
+	      ".hhp project file in $(BUILDDIR)/htmlhelp."
+
+qthelp:
+	$(SPHINXBUILD) -b qthelp $(ALLSPHINXOPTS) $(BUILDDIR)/qthelp
+	@echo
+	@echo "Build finished; now you can run "qcollectiongenerator" with the" \
+	      ".qhcp project file in $(BUILDDIR)/qthelp, like this:"
+	@echo "# qcollectiongenerator $(BUILDDIR)/qthelp/einsteinpy.qhcp"
+	@echo "To view the help file:"
+	@echo "# assistant -collectionFile $(BUILDDIR)/qthelp/einsteinpy.qhc"
+
+devhelp:
+	$(SPHINXBUILD) -b devhelp $(ALLSPHINXOPTS) $(BUILDDIR)/devhelp
+	@echo
+	@echo "Build finished."
+	@echo "To view the help file:"
+	@echo "# mkdir -p $$HOME/.local/share/devhelp/einsteinpy"
+	@echo "# ln -s $(BUILDDIR)/devhelp $$HOME/.local/share/devhelp/einsteinpy"
+	@echo "# devhelp"
+
+epub:
+	$(SPHINXBUILD) -b epub $(ALLSPHINXOPTS) $(BUILDDIR)/epub
+	@echo
+	@echo "Build finished. The epub file is in $(BUILDDIR)/epub."
+
+latex:
+	$(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex
+	@echo
+	@echo "Build finished; the LaTeX files are in $(BUILDDIR)/latex."
+	@echo "Run \`make' in that directory to run these through (pdf)latex" \
+	      "(use \`make latexpdf' here to do that automatically)."
+
+latexpdf:
+	$(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex
+	@echo "Running LaTeX files through pdflatex..."
+	$(MAKE) -C $(BUILDDIR)/latex all-pdf
+	@echo "pdflatex finished; the PDF files are in $(BUILDDIR)/latex."
+
+latexpdfja:
+	$(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex
+	@echo "Running LaTeX files through platex and dvipdfmx..."
+	$(MAKE) -C $(BUILDDIR)/latex all-pdf-ja
+	@echo "pdflatex finished; the PDF files are in $(BUILDDIR)/latex."
+
+text:
+	$(SPHINXBUILD) -b text $(ALLSPHINXOPTS) $(BUILDDIR)/text
+	@echo
+	@echo "Build finished. The text files are in $(BUILDDIR)/text."
+
+man:
+	$(SPHINXBUILD) -b man $(ALLSPHINXOPTS) $(BUILDDIR)/man
+	@echo
+	@echo "Build finished. The manual pages are in $(BUILDDIR)/man."
+
+texinfo:
+	$(SPHINXBUILD) -b texinfo $(ALLSPHINXOPTS) $(BUILDDIR)/texinfo
+	@echo
+	@echo "Build finished. The Texinfo files are in $(BUILDDIR)/texinfo."
+	@echo "Run \`make' in that directory to run these through makeinfo" \
+	      "(use \`make info' here to do that automatically)."
+
+info:
+	$(SPHINXBUILD) -b texinfo $(ALLSPHINXOPTS) $(BUILDDIR)/texinfo
+	@echo "Running Texinfo files through makeinfo..."
+	make -C $(BUILDDIR)/texinfo info
+	@echo "makeinfo finished; the Info files are in $(BUILDDIR)/texinfo."
+
+gettext:
+	$(SPHINXBUILD) -b gettext $(I18NSPHINXOPTS) $(BUILDDIR)/locale
+	@echo
+	@echo "Build finished. The message catalogs are in $(BUILDDIR)/locale."
+
+changes:
+	$(SPHINXBUILD) -b changes $(ALLSPHINXOPTS) $(BUILDDIR)/changes
+	@echo
+	@echo "The overview file is in $(BUILDDIR)/changes."
+
+linkcheck:
+	$(SPHINXBUILD) -b linkcheck $(ALLSPHINXOPTS) $(BUILDDIR)/linkcheck
+	@echo
+	@echo "Link check complete; look for any errors in the above output " \
+	      "or in $(BUILDDIR)/linkcheck/output.txt."
+
+doctest:
+	$(SPHINXBUILD) -b doctest $(ALLSPHINXOPTS) $(BUILDDIR)/doctest
+	@echo "Testing of doctests in the sources finished, look at the " \
+	      "results in $(BUILDDIR)/doctest/output.txt."
+
+xml:
+	$(SPHINXBUILD) -b xml $(ALLSPHINXOPTS) $(BUILDDIR)/xml
+	@echo
+	@echo "Build finished. The XML files are in $(BUILDDIR)/xml."
+
+pseudoxml:
+	$(SPHINXBUILD) -b pseudoxml $(ALLSPHINXOPTS) $(BUILDDIR)/pseudoxml
+	@echo
+	@echo "Build finished. The pseudo-XML files are in $(BUILDDIR)/pseudoxml."

src/riccipy/__init__.py

@@ -1,5 +1,4 @@
 """
-=======
 RicciPy
 =======
 
@@ -8,5 +7,5 @@
 """
 
 __version__ = "0.1.dev0"
-from .metric import Metric, SpacetimeMetric
+from .metric import Metric, SpacetimeMetric, load_metric
 from .tensor import Index, Tensor, expand_tensor, indices

src/riccipy/metric.py

@@ -1,4 +1,5 @@
 from collections import namedtuple
+from pathlib import Path
 
 from sympy import Array, Pow, Rational, S, ones, tensorproduct, zeros
 from sympy.tensor.tensor import TensorIndexType
@@ -256,3 +257,11 @@ def signature(self):
         sig = sign * ones(1, self.dim)
         sig[0] *= -1
         return tuple(sig)
+
+
+def load_metric(symbol, name):
+    base = Path(__file__).parent
+    var = dict(globals(), **locals())
+    with open(base / "metrics" / (name + ".py"), "r") as file:
+        exec(file.read(), var)
+        return SpacetimeMetric(symbol, var["coords"], var["metric"], timelike=False)

src/riccipy/metrics/anti_de_sitter_1.py

@@ -0,0 +1,16 @@
+"""
+Anti-de Sitter space
+Hawking and Ellis (5.9) p131
+"""
+from sympy import cos, diag, sin, sinh, symbols
+
+coords = symbols("t chi theta phi", real=True)
+variables = ()
+functions = ()
+t, ch, th, ph = coords
+metric = diag(
+    -1,
+    cos(t) ** 2,
+    cos(t) ** 2 * sinh(ch) ** 2,
+    cos(t) ** 2 * sinh(ch) ** 2 * sin(th) ** 2,
+)

src/riccipy/metrics/anti_de_sitter_2.py

@@ -0,0 +1,11 @@
+"""
+Static form of Anti-de Sitter space
+Hawking and Ellis (5.9) p131
+"""
+from sympy import cosh, diag, sin, sinh, symbols
+
+coords = symbols("t r theta phi", real=True)
+variables = ()
+functions = ()
+t, r, th, ph = coords
+metric = diag(-cosh(r) ** 2, 1, sinh(r) ** 2, sinh(r) ** 2 * sin(th) ** 2)

src/riccipy/metrics/bayin.py

@@ -0,0 +1,18 @@
+"""
+Bayin's spherically symmetric static perfect fluid solution II
+Bayin, Phys. Rev. D, v18, p2745-2751, (1978)
+"""
+from sympy import Rational, asin, sin, symbols, zeros
+
+coords = symbols("t r theta phi", real=True)
+variables = symbols("w_0 C_0 C_1", constant=True)
+functions = ()
+t, r, th, ph = coords
+w0, C0, C1 = variables
+metric = zeros(4)
+metric[0, 0] = -C0 ** 2 * exp(
+    -asin((-2 * r ** 2 + C1) / (C1 ** 2 + 4 * w0 ** 2) ** (Rational(1, 2)))
+)
+metric[1, 1] = w0 ** 2 / (-r ** 4 + C1 * r ** 2 + w0 ** 2)
+metric[2, 2] = r ** 2
+metric[3, 3] = r ** 2 * sin(th) ** 2

src/riccipy/metrics/beckers_sinzinkayo_demaret_1.py

@@ -0,0 +1,13 @@
+"""
+Beckers, Sinzinkayo, and Demaret solution
+Beckers et al., Phys. Rev. D, v30, p1846, (1984)
+"""
+from sympy import diag, symbols
+
+coords = symbols("t x y z", real=True)
+variables = symbols("k d m", constant=True)
+functions = ()
+t, x, y, z = coords
+k, d, m = variables
+expr = (k * x + d) ** (2 * m)
+metric = diag(-expr, expr, expr, expr)

src/riccipy/metrics/beckers_sinzinkayo_demaret_2.py

@@ -0,0 +1,13 @@
+"""
+Beckers, Sinzinkayo, and Demaret solution
+Beckers et al., Phys. Rev. D, v30, p1846, (1984)
+"""
+from sympy import diag, symbols
+
+coords = symbols("t x y z", real=True)
+variables = symbols("m", constant=True)
+functions = ()
+t, x, y, z = coords
+m = variables
+expr = x ** (2 * m)
+metric = diag(-expr, expr, expr, expr)

src/riccipy/metrics/bertotti_robinson_1.py

@@ -0,0 +1,14 @@
+"""
+The Bertotti-Robinson solution
+Bertotti, Phys. Rev., v116, p1331, (1959)
+Bertotti, Commun. Math. Phys., v5, p257, (1967)
+Robinson, Commun. Math. Phys., v9, p161, (1968)
+Stephani (10.16) p120
+"""
+from sympy import diag, symbols
+
+coords = symbols("t x y z", real=True)
+variables = ()
+functions = ()
+t, x, y, z = coords
+metric = diag(-(1 + z ** 2), 1 - y ** 2, 1 / (1 - y ** 2), 1 / (1 + z ** 2))

src/riccipy/metrics/bertotti_robinson_2.py

@@ -0,0 +1,15 @@
+"""
+The Bertotti-Robinson solution
+Bertotti, Phys. Rev., v116, p1331, (1959)
+Lovelock, Commun. Math. Phys., v5, p257, (1967)
+Dolan, Commun. Math. Phys., v9, p161, (1968)
+Stephani (10.18) p121
+"""
+from sympy import diag, symbols
+
+coords = symbols("t r theta phi", real=True)
+variables = symbols("q", constant=True)
+functions = ()
+t, r, th, ph = coords
+q = variables
+metric = diag(-q ** 2 / r ** 2, q ** 2 / r ** 2, q ** 2, q ** 2 * sin(th) ** 2)

src/riccipy/metrics/bertotti_robinson_3.py

@@ -0,0 +1,15 @@
+"""
+The Bertotti-Robinson solution
+Bertotti, Phys. Rev., v116, p1331, (1959)
+Lovelock, Commun. Math. Phys., v5, p257, (1967)
+Dolan, Commun. Math. Phys., v9, p161, (1968)
+Stephani (10.19) p120
+"""
+from sympy import diag, sin, sinh, symbols
+
+coords = symbols("t x theta phi", real=True)
+variables = symbols("k", constant=True)
+functions = ()
+t, x, th, ph = coords
+k = variables
+metric = diag(-k ** 2 * sinh(x) ** 2, k ** 2, k ** 2, k ** 2 * sin(th) ** 2)

src/riccipy/metrics/bertotti_robinson_4.py

@@ -0,0 +1,15 @@
+"""
+The Bertotti-Robinson solution
+Bertotti, Phys. Rev., v116, p1331, (1959)
+Lovelock, Commun. Math. Phys., v5, p257, (1967)
+Dolan, Commun. Math. Phys., v9, p161, (1968)
+Stephani (32.95) p372
+"""
+from sympy import cos, diag, symbols
+
+coords = symbols("t z x y", real=True)
+variables = symbols("omega", constant=True)
+functions = ()
+t, x, y, z = coords
+om = variables
+metric = diag(-1, 1, cos(om * x) ** 2, cos(om * t) ** 2)

src/riccipy/metrics/bianchi_1.py

@@ -0,0 +1,11 @@
+"""
+General metric from Bianchi II automorphisms
+"""
+from sympy import Function, diag, exp, symbols
+
+coords = symbols("t x y z", real=True)
+variables = ()
+functions = symbols("alpha", cls=Function)
+t, x, y, z = coords
+al = functions
+metric = diag(1, exp(-2 * al(t)), exp(al(t)), exp(al(t)))

src/riccipy/metrics/bianchi_2.py

@@ -0,0 +1,11 @@
+"""
+General metric from Bianchi IV automorphisms
+"""
+from sympy import Function, diag, exp, symbols
+
+coords = symbols("t x y z", real=True)
+variables = ()
+functions = symbols("alpha beta gamma", cls=Function)
+t, x, y, z = coords
+al, be, ga = functions
+metric = diag(1, exp(al(t)), exp(be(t)), exp(ga(t)))

src/riccipy/metrics/bondi_1.py

@@ -0,0 +1,13 @@
+"""
+Spherical Bondi metric in advanced (ingoing) coordinates
+Bondi, Proc. Roy. Soc. Lond. A, v282, p303, (1964)
+"""
+from sympy import Function, diag, sin, symbols
+
+coords = symbols("r v theta phi", real=True)
+variables = ()
+functions = symbols("C M", cls=Function)
+r, v, th, ph = coords
+C, M = functions
+metric = diag(0, -C(r, v) ** 2 * (1 - 2 * M(r, v) / r), r ** 2, r ** 2 * sin(th) ** 2)
+metric[0, 1] = metric[1, 0] = C(r, v)