eigenmath/lisp/example2.tex

\parindent=0pt
{\tt ;\ Page\ references\ are\ for\ the\ book\ "Gravitation."}

{\tt ;\ generic\ metric}

{\tt (setq\ gdd\ (sum}

{\tt \ \ (product\ (g00)\ (tensor\ x0\ x0))}

{\tt \ \ (product\ (g11)\ (tensor\ x1\ x1))}

{\tt \ \ (product\ (g22)\ (tensor\ x2\ x2))}

{\tt \ \ (product\ (g33)\ (tensor\ x3\ x3))}

{\tt ))}

{\tt (gr)\ ;\ compute\ g,\ guu,\ GAMUDD,\ RUDDD,\ RDD,\ R,\ GDD,\ GUD\ and\ GUU}

{\tt ;\ generic\ vectors}

{\tt (setq\ u\ (sum}

{\tt \ \ (product\ (u0)\ (tensor\ x0))}

{\tt \ \ (product\ (u1)\ (tensor\ x1))}

{\tt \ \ (product\ (u2)\ (tensor\ x2))}

{\tt \ \ (product\ (u3)\ (tensor\ x3))}

{\tt ))}

{\tt (setq\ v\ (sum}

{\tt \ \ (product\ (v0)\ (tensor\ x0))}

{\tt \ \ (product\ (v1)\ (tensor\ x1))}

{\tt \ \ (product\ (v2)\ (tensor\ x2))}

{\tt \ \ (product\ (v3)\ (tensor\ x3))}

{\tt ))}

{\tt (setq\ w\ (sum}

{\tt \ \ (product\ (w0)\ (tensor\ x0))}

{\tt \ \ (product\ (w1)\ (tensor\ x1))}

{\tt \ \ (product\ (w2)\ (tensor\ x2))}

{\tt \ \ (product\ (w3)\ (tensor\ x3))}

{\tt ))}

$$V_{[\mu\nu\lambda]}={1\over3!}
(V_{\mu\nu\lambda}
+V_{\nu\lambda\mu}
+V_{\lambda\mu\nu}
-V_{\nu\mu\lambda}
-V_{\mu\lambda\nu}
-V_{\lambda\nu\mu}
)$$
{\tt ;\ how\ to\ antisymmetrize\ three\ indices\ (p.\ 86)}

{\tt (setq\ V\ (sum}

{\tt \ \ (product\ 1/6\ V)}

{\tt \ \ (product\ 1/6\ (transpose12\ (transpose23\ V)))\ ;\ nu\ lambda\ mu\ ->\ mu\ nu\ lambda}

{\tt \ \ (product\ 1/6\ (transpose23\ (transpose12\ V)))\ ;\ lambda\ mu\ nu\ ->\ mu\ nu\ lambda}

{\tt \ \ (product\ -1/6\ (transpose12\ V))\ \ \ \ \ \ \ \ \ \ \ \ \ \ ;\ nu\ mu\ lambda\ ->\ mu\ nu\ lambda}

{\tt \ \ (product\ -1/6\ (transpose23\ V))\ \ \ \ \ \ \ \ \ \ \ \ \ \ ;\ mu\ lambda\ nu\ ->\ mu\ nu\ lambda}

{\tt \ \ (product\ -1/6\ (transpose13\ V))\ \ \ \ \ \ \ \ \ \ \ \ \ \ ;\ lambda\ nu\ mu\ ->\ mu\ nu\ lambda}

{\tt ))}

$$[{\bf u},{\bf v}]=
(u^\beta{v^\alpha}_{,\beta}-v^\beta{u^\alpha}_{,\beta}){\bf e}_\alpha$$
{\tt ;\ commutator\ (p.\ 206)}

{\tt (define\ commutator\ (sum}

{\tt \ \ (contract13\ (product\ +1\ arg1\ (gradient\ arg2)))}

{\tt \ \ (contract13\ (product\ -1\ arg2\ (gradient\ arg1)))}

{\tt ))}

$${\Gamma^\alpha}_{\mu\nu}=\hbox{$1\over2$}g^{\alpha\beta}
(g_{\beta\mu,\nu}+g_{\beta\nu,\mu}-g_{\mu\nu,\beta})$$
{\tt (print\ "connection\ coefficients\ (p.\ 210)")}

{\tt (setq\ temp\ (gradient\ gdd))}

{\tt (setq\ Gamma\ (contract23\ (product\ 1/2\ guu\ (sum}

{\tt \ \ temp}

{\tt \ \ (transpose23\ temp)}

{\tt \ \ (product\ -1\ (transpose12\ (transpose23\ temp)))}

{\tt ))))}

{\tt ;\ check}

{\tt (print\ (equal\ Gamma\ GAMUDD))}

$${T^\alpha}_{;\gamma}={T^\alpha}_{,\gamma}+
T^\mu{\Gamma^\alpha}_{\mu\gamma}$$
{\tt ;\ covariant\ derivative\ of\ a\ vector\ (p.\ 211)}

{\tt (define\ covariant-derivative\ (sum}

{\tt \ \ (gradient\ arg)}

{\tt \ \ (contract13\ (product\ arg\ GAMUDD))}

{\tt ))}

$${G^{\mu\nu}}_{;\nu}=0$$
{\tt (print\ "divergence\ of\ einstein\ is\ zero\ (p.\ 222)")}

{\tt ;\ covariant-derivative-of-up-up\ is\ already\ defined\ in\ grlib}

{\tt (setq\ temp\ (covariant-derivative-of-up-up\ GUU))}

{\tt (setq\ temp\ (contract23\ temp))\ ;\ sum\ over\ 2nd\ and\ 3rd\ indices}

{\tt (print\ (equal\ temp\ 0))}

$${R^\alpha}_{\beta\gamma\delta}=
{\partial{\Gamma^\alpha}_{\beta\delta}\over\partial x^\gamma}-
{\partial{\Gamma^\alpha}_{\beta\gamma}\over\partial x^\delta}+
{\Gamma^\alpha}_{\mu\gamma}{\Gamma^\mu}_{\beta\delta}-
{\Gamma^\alpha}_{\mu\delta}{\Gamma^\mu}_{\beta\gamma}$$
{\tt (print\ "computing\ riemann\ tensor\ (p.\ 219)")}

{\tt (setq\ temp1\ (gradient\ GAMUDD))}

{\tt (setq\ temp2\ (contract24\ (product\ GAMUDD\ GAMUDD)))}

{\tt (setq\ riemann\ (sum}

{\tt \ \ (transpose34\ temp1)}

{\tt \ \ (product\ -1\ temp1)}

{\tt \ \ (transpose23\ temp2)}

{\tt \ \ (product\ -1\ (transpose34\ (transpose23\ temp2)))}

{\tt ))}

{\tt ;\ check}

{\tt (print\ (equal\ riemann\ RUDDD))}

$$T_{\alpha\beta;\gamma}=T_{\alpha\beta,\gamma}
-T_{\mu\beta}{\Gamma^\mu}_{\alpha\gamma}
-T_{\alpha\mu}{\Gamma^\mu}_{\beta\gamma}$$
{\tt ;\ p.\ 259}

{\tt (define\ covariant-derivative-of-down-down\ (prog\ (temp)}

{\tt \ \ (setq\ temp\ (product\ arg\ GAMUDD))}

{\tt \ \ (return\ (sum}

{\tt \ \ \ \ (gradient\ arg)}

{\tt \ \ \ \ (product\ -1\ (transpose12\ (contract13\ temp)))}

{\tt \ \ \ \ (product\ -1\ (contract23\ temp))}

{\tt \ \ ))}

{\tt ))}

$${T^\alpha}_{\beta;\gamma}={T^\alpha}_{\beta,\gamma}+
{T^\mu}_\beta{\Gamma^\alpha}_{\mu\gamma}-
{T^\alpha}_\mu{\Gamma^\mu}_{\beta\gamma}$$
{\tt (define\ covariant-derivative-of-up-down\ (prog\ (temp)}

{\tt \ \ (setq\ temp\ (product\ arg\ GAMUDD))}

{\tt \ \ (return\ (sum}

{\tt \ \ \ \ (gradient\ arg)}

{\tt \ \ \ \ (transpose12\ (contract14\ temp))}

{\tt \ \ \ \ (product\ -1\ (contract23\ temp))}

{\tt \ \ ))}

{\tt ))}

$$\nabla_{\bf u}{\bf v}={v^\alpha}_{;\beta}u^{\beta}$$
{\tt (define\ directed-covariant-derivative}

{\tt \ \ (contract23\ (product\ (covariant-derivative\ arg1)\ arg2))}

{\tt )}

$$\nabla_{\bf u}{\bf v}-\nabla_{\bf v}{\bf u}=[{\bf u},{\bf v}]$$
{\tt (print\ "symmetry\ of\ covariant\ derivative\ (p.\ 252)")}

{\tt (setq\ temp1\ (sum}

{\tt \ \ (directed-covariant-derivative\ v\ u)}

{\tt \ \ (product\ -1\ (directed-covariant-derivative\ u\ v))}

{\tt ))}

{\tt (setq\ temp2\ (commutator\ u\ v))}

{\tt (equal\ temp1\ temp2)}

$$\nabla_{\bf u}(f{\bf v})=
f\nabla_{\bf u}{\bf v}+{\bf v}\partial_{\bf u}f$$
{\tt (print\ "covariant\ derivative\ chain\ rule\ (p.\ 252)")}

{\tt (setq\ temp1\ (directed-covariant-derivative\ (product\ (f)\ v)\ u))}

{\tt (setq\ temp2\ (sum}

{\tt \ \ (product\ (f)\ (directed-covariant-derivative\ v\ u))}

{\tt \ \ (product\ v\ (contract12\ (product\ (gradient\ (f))\ u)))}

{\tt ))}

{\tt (print\ (equal\ temp1\ temp2))}

$$\nabla_{{\bf v}+{\bf w}}{\bf u}=
\nabla_{\bf v}{\bf u}+\nabla_{\bf w}{\bf u}$$
{\tt (print\ "additivity\ of\ covariant\ derivative\ (p.\ 257)")}

{\tt (setq\ temp1\ (directed-covariant-derivative\ u\ (sum\ v\ w)))}

{\tt (setq\ temp2\ (sum}

{\tt \ \ (directed-covariant-derivative\ u\ v)}

{\tt \ \ (directed-covariant-derivative\ u\ w)}

{\tt ))}

{\tt (print\ (equal\ temp1\ temp2))}

$${R^\alpha}_{\beta\gamma\delta}={R^\alpha}_{\beta[\gamma\delta]}$$
{\tt (print\ "riemann\ is\ antisymmetric\ on\ last\ two\ indices\ (p.\ 286)")}

{\tt (setq\ temp\ (sum}

{\tt \ \ (product\ 1/2\ RUDDD)}

{\tt \ \ (product\ -1/2\ (transpose34\ RUDDD))}

{\tt ))}

{\tt (print\ (equal\ RUDDD\ temp))}

$${R^\alpha}_{[\beta\gamma\delta]}=0$$
{\tt (print\ "riemann\ vanishes\ when\ antisymmetrized\ on\ last\ three\ indices\ (p.\ 286)")}

{\tt (setq\ temp\ (sum}

{\tt \ \ (product\ 1/6\ RUDDD)}

{\tt \ \ (product\ 1/6\ (transpose34\ (transpose24\ RUDDD)))}

{\tt \ \ (product\ 1/6\ (transpose34\ (transpose23\ RUDDD)))}

{\tt \ \ (product\ -1/6\ (transpose23\ RUDDD))}

{\tt \ \ (product\ -1/6\ (transpose34\ RUDDD))}

{\tt \ \ (product\ -1/6\ (transpose24\ RUDDD))}

{\tt ))}

{\tt (print\ (equal\ temp\ 0))}

$${G^{\alpha\beta}}_{\gamma\delta}=
\hbox{$1\over2$}\epsilon^{\alpha\beta\mu\nu}
{R_{\mu\nu}}^{\rho\sigma}
\hbox{$1\over2$}\epsilon_{\rho\sigma\gamma\delta}$$
{\tt (print\ "double\ dual\ of\ riemann\ (p.\ 325)")}

{\tt (setq\ temp\ (contract23\ (product\ gdd\ RUDDD)))\ ;\ lower\ 1st\ index}

{\tt (setq\ temp\ (transpose34\ (contract35\ (product\ temp\ guu))))\ ;\ raise\ 3rd\ index}

{\tt (setq\ RDDUU\ (contract45\ (product\ temp\ guu)))\ ;\ raise\ 4th\ index}

{\tt (setq\ temp\ (product\ epsilon\ RDDUU))}

{\tt (setq\ temp\ (contract35\ temp))\ ;\ sum\ over\ mu}

{\tt (setq\ temp\ (contract34\ temp))\ ;\ sum\ over\ nu}

{\tt (setq\ temp\ (product\ temp\ epsilon))}

{\tt (setq\ temp\ (contract35\ temp))\ ;\ sum\ over\ rho}

{\tt (setq\ temp\ (contract34\ temp))\ ;\ sum\ over\ sigma}

{\tt (setq\ GUUDD\ (product\ -1/4\ temp))\ ;\ negative\ due\ to\ levi-civita\ tensor}

{\tt ;\ check}

{\tt (print\ (equal}

{\tt \ \ (contract13\ GUUDD)}

{\tt \ \ GUD}

{\tt ))}

$${B^\mu}_{;\alpha\beta}-{B^\mu}_{;\beta\alpha}=
{R^\mu}_{\nu\beta\alpha}B^\nu$$
{\tt (print\ "noncommutation\ of\ covariant\ derivatives\ (p.\ 389)")}

{\tt (setq\ B\ (sum}

{\tt \ \ (product\ (B0)\ (tensor\ x0))}

{\tt \ \ (product\ (B1)\ (tensor\ x1))}

{\tt \ \ (product\ (B2)\ (tensor\ x2))}

{\tt \ \ (product\ (B3)\ (tensor\ x3))}

{\tt ))}

{\tt (setq\ temp\ (covariant-derivative-of-up-down\ (covariant-derivative\ B)))}

{\tt (setq\ temp1\ (sum\ temp\ (product\ -1\ (transpose23\ temp))))}

{\tt (setq\ temp2\ (transpose23\ (contract25\ (product\ RUDDD\ B))))}

{\tt (print\ (equal\ temp1\ temp2))}

{\tt (print\ "bondi\ metric")}

{\tt ;\ erase\ any\ definitions\ for\ U,\ V,\ beta\ and\ gamma}

{\tt (setq\ U\ (quote\ U))}

{\tt (setq\ V\ (quote\ V))}

{\tt (setq\ beta\ (quote\ beta))}

{\tt (setq\ gamma\ (quote\ gamma))}

{\tt ;\ erase\ any\ definitions\ for\ u,\ r,\ theta\ and\ phi}

{\tt (setq\ u\ (quote\ u))}

{\tt (setq\ r\ (quote\ r))}

{\tt (setq\ theta\ (quote\ theta))}

{\tt (setq\ phi\ (quote\ phi))}

{\tt ;\ new\ coordinate\ system}

{\tt (setq\ x0\ u)}

{\tt (setq\ x1\ r)}

{\tt (setq\ x2\ theta)}

{\tt (setq\ x3\ phi)}

{\tt ;\ U,\ V,\ beta\ and\ gamma\ are\ functions\ of\ u,\ r\ and\ theta}

$$g_{uu}=Ve^{2\beta}/r-U^2r^2e^{2\gamma}$$
{\tt (setq\ g\_uu\ (sum}

{\tt \ \ (product}

{\tt \ \ \ \ (V\ u\ r\ theta)}

{\tt \ \ \ \ (power\ r\ -1)}

{\tt \ \ \ \ (power\ e\ (product\ 2\ (beta\ u\ r\ theta)))}

{\tt \ \ )}

{\tt \ \ (product}

{\tt \ \ \ \ -1}

{\tt \ \ \ \ (power\ (U\ u\ r\ theta)\ 2)}

{\tt \ \ \ \ (power\ r\ 2)}

{\tt \ \ \ \ (power\ e\ (product\ 2\ (gamma\ u\ r\ theta)))}

{\tt \ \ )}

{\tt ))}

$$g_{ur}=2e^{2\beta}$$
{\tt (setq\ g\_ur\ (product\ 2\ (power\ e\ (product\ 2\ (beta\ u\ r\ theta)))))}

$$g_{u\theta}=2Ur^2e^{2\gamma}$$
{\tt (setq\ g\_utheta\ (product}

{\tt \ \ 2}

{\tt \ \ (U\ u\ r\ theta)}

{\tt \ \ (power\ r\ 2)}

{\tt \ \ (power\ e\ (product\ 2\ (gamma\ u\ r\ theta)))}

{\tt ))}

$$g_{\theta\theta}=-r^2e^{2\gamma}$$
{\tt (setq\ g\_thetatheta\ (product}

{\tt \ \ -1}

{\tt \ \ (power\ r\ 2)}

{\tt \ \ (power\ e\ (product\ 2\ (gamma\ u\ r\ theta)))}

{\tt ))}

$$g_{\phi\phi}=-r^2e^{-2\gamma}\sin^2\theta$$
{\tt (setq\ g\_phiphi\ (product}

{\tt \ \ -1}

{\tt \ \ (power\ r\ 2)}

{\tt \ \ (power\ e\ (product\ -2\ (gamma\ u\ r\ theta)))}

{\tt \ \ (power\ (sin\ theta)\ 2)}

{\tt ))}

{\tt ;\ metric\ tensor}

{\tt (setq\ gdd\ (sum}

{\tt \ \ (product\ g\_uu\ (tensor\ u\ u))}

{\tt \ \ (product\ g\_ur\ (tensor\ u\ r))}

{\tt \ \ (product\ g\_ur\ (tensor\ r\ u))}

{\tt \ \ (product\ g\_utheta\ (tensor\ u\ theta))}

{\tt \ \ (product\ g\_utheta\ (tensor\ theta\ u))}

{\tt \ \ (product\ g\_thetatheta\ (tensor\ theta\ theta))}

{\tt \ \ (product\ g\_phiphi\ (tensor\ phi\ phi))}

{\tt ))}

{\tt (gr)\ ;\ compute\ g,\ guu,\ GAMUDD,\ RUDDD,\ RDD,\ R,\ GDD,\ GUD\ and\ GUU}

{\tt ;\ is\ covariant\ derivative\ of\ metric\ zero?}

{\tt (print\ (equal\ (covariant-derivative-of-down-down\ gdd)\ 0))}

{\tt ;\ is\ divergence\ of\ einstein\ zero?}

{\tt (setq\ temp\ (contract23\ (covariant-derivative-of-up-up\ GUU)))}

{\tt (print\ (equal\ temp\ 0))}

\end