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<Input><Text-field style="Title" layout="Title"><Font encoding="UTF-8">G\303\266del's Universe</Font></Text-field><Text-field style="Text" layout="Title">Frank Wang, <Font style="Hyperlink">fwang@lagcc.cuny.edu</Font></Text-field><Text-field style="Text" layout="Title" alignment="left"><Font encoding="UTF-8">In 1949, Kurt G\303\266del proposed a solution to Einstein's field equations that exhibited a rotation of matter (with a nonvanishing cosmological constant).  This solution allows the existence of closed timelike curves, which implies the possibility of time travel.  In this worksheet, we demonstrate the use of </Font><Font style="Maple Input" executable="false">DifferentialGeometry</Font> and <Font style="Maple Input" executable="false">Tensor</Font><Font encoding="UTF-8"> packages to calculate the Ricci tensor, Ricci scalar, and Einstein tensor.  We also apply the calculus of variations to calculate the null geodesics (trajectories of light rays) in G\303\266del's universe.  The notation is based on the presentation in the monograph of Stephen Hawking and George Ellis.  </Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">To honor Albert Einstein on the occasion of his 70th birthday, Kurt G\303\266del presented &quot;An example of a new type of cosmological solution of Einstein's field equations of gravitation&quot; (published in </Font><Font italic="true">Reviews of Modern Physics</Font><Font encoding="UTF-8"> in 1949).  G\303\266del's model admits the existence of closed timelike curves, implying that it is possible to go on a journey into one's own past.  Because G\303\266del's model yields no red shift for distant objects, it is considered unphysical and not discussed in most textbooks on general relativity.  In the text </Font><Font italic="true">Large scale structure of space-time</Font><Font encoding="UTF-8"> by Stephen Hawking and George Ellis, there is a terse treatment of G\303\266del's universe, see below.  </Font></Text-field>
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opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="centred" bullet="none" spacebelow="0" pagebreak-before="false"><Font opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">Source: Hawking and Ellis (Ref. 1), page 168.</Font></Text-field><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="centred" bullet="none" spacebelow="0" pagebreak-before="false"></Text-field>
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<Group labelreference="L266" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">We use <Font style="Maple Input" executable="false">DifferentialGeomety</Font> and <Font style="Maple Input" executable="false">Tensor</Font> packages to calculate the Ricci tensor, Ricci scalar, Einstein tensor, and energy-momentum tensor.  </Text-field>
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<Group labelreference="L209" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
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<Group labelreference="L208" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(DifferentialGeometry): with(Tensor):</Text-field>
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<Group hide-input="false" labelreference="L210" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Preferences(&quot;ShowFramePrompt&quot;, false);</Text-field>
</Input>
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<Group labelreference="L267" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Set up the frame:</Text-field>
</Input>
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<Group hide-input="false" labelreference="L207" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">DGsetup([t, x, y, z], M);</Text-field>
</Input>
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<Group labelreference="L268" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Define the metric according to the convention of Hawking and Ellis:</Text-field>
</Input>
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<Group hide-input="false" labelreference="L206" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">g := evalDG(-dt &amp;t dt + dx &amp;t dx - 1/2*exp(2*sqrt(2)*omega*x)*dy &amp;t dy + dz &amp;t dz - exp(sqrt(2)*omega*x)*dt &amp;t dy - exp(sqrt(2)*omega*x)*dy &amp;t dt);</Text-field>
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<Group labelreference="L269" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Calculate the Ricci tensor, Ricci scalar, and Einstein tensor:</Text-field>
</Input>
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<Group hide-input="false" labelreference="L205" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">RicciTensor(g);</Text-field>
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<Group hide-input="false" labelreference="L212" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">RicciScalar(g);</Text-field>
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<Group hide-input="false" labelreference="L211" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">EinsteinTensor(g);</Text-field>
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<Group labelreference="L276" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Double check that Einstein tensor can be constructed from Ricci tensor and Ricci scalar:</Text-field>
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<Group hide-input="false" labelreference="L274" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">evalDG(RicciTensor(g) - 1/2*RicciScalar(g)*g);</Text-field>
</Input>
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<Group hide-input="false" labelreference="L275" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">RaiseLowerIndices(InverseMetric(g), %, [1, 2]);</Text-field>
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<Group labelreference="L270" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Calculate the energy-momentum tensor:</Text-field>
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<Group hide-input="false" labelreference="L218" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">T := evalDG(rho*D_t &amp;t D_t);</Text-field>
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<Group labelreference="L271" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Left-hand side of Einstein's field equations, with a cosmological constant term:</Text-field>
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<Group hide-input="false" labelreference="L217" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">l := evalDG(RicciTensor(g) - 1/2*g*RicciScalar(g) + Lambda*g);</Text-field>
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<Group labelreference="L272" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Right-hand side of Einstein's field equations, after changing the contravariant tensor to covariant one:</Text-field>
</Input>
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<Group hide-input="false" labelreference="L216" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">r := evalDG(8*Pi*RaiseLowerIndices(g, T, [1, 2]));</Text-field>
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<Group labelreference="L273" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">The equations needed to be satisfied:</Text-field>
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<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">epr1 := evalDG(l - r);</Text-field>
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<Group hide-input="false" labelreference="L214" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq1 := Hook([D_t, D_t], epr1) = 0;</Text-field>
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<Group hide-input="false" labelreference="L213" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2 := Hook([D_x, D_x], epr1) = 0;</Text-field>
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<Group hide-input="false" labelreference="L220" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq3 := Hook([D_y, D_y], epr1) = 0;</Text-field>
</Input>
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<Group labelreference="L277" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Solve for Lambda and rho:</Text-field>
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<Group hide-input="false" labelreference="L221" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">solve({eq1, eq2, eq3}, {Lambda, rho});</Text-field>
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<Input><Text-field style="Text" layout="Normal">We found that the sign of the cosmological constant negative, corresponding to a positive pressure.  </Text-field>
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<Input><Text-field style="Text" layout="Normal">After a change of coordinates, the line element takes the form shown in the following, which is page 169 of Hawking and Ellis.  </Text-field>
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opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="centred" bullet="none" spacebelow="0" pagebreak-before="false"><Font opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">Source: Hawking and Ellis (Ref. 1), page 169.</Font></Text-field><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="centred" bullet="none" spacebelow="0" pagebreak-before="false"></Text-field>
</Group>
<Group labelreference="L28" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
</Input>
</Group>
<Group labelreference="L278" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Based on the line element, we write the Lagrangian, with lambda as the parameter:</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L29" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">L := 1/omega^2*(-diff(t(lambda), lambda)^2 + diff(r(lambda), lambda)^2 - (sinh(r(lambda))^4 - sinh(r(lambda))^2)*diff(phi(lambda), lambda)^2 + 2*sqrt(2)*sinh(r(lambda))^2*diff(t(lambda),lambda)*diff(phi(lambda), lambda));</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L27" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">omega := 1;</Text-field>
</Input>
</Group>
<Group labelreference="L279" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">The process of finding the Euler-Lagrange equations through substitutions and differentiations has been discussed in Ref. 2 and references therein:</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L32" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">L1 := subs({t(lambda) = var1, diff(t(lambda), lambda) = var2, r(lambda) = var3, diff(r(lambda), lambda) = var4, phi(lambda) = var5, diff(phi(lambda), lambda) = var6}, L);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L35" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr11 := diff(L1, var2);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L34" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr12 := diff(L1, var1);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L33" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr13 := subs({var1 = t(lambda), var2 = diff(t(lambda), lambda), var3 = r(lambda), var4 = diff(r(lambda), lambda), var5 = phi(lambda), var6 = diff(phi(lambda), lambda)}, Epr11);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L31" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr14 := subs({var1 = t(lambda), var2 = diff(t(lambda), lambda), var3 = r(lambda), var4 = diff(r(lambda), lambda), var5 = phi(lambda), var6 = diff(phi(lambda), lambda)}, Epr12);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L36" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr15 := diff(Epr13, lambda);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L30" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq16 := Epr15 - Epr14 = 0;</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L44" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr21 := diff(L1, var4);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L41" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr22 := diff(L1, var3);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L42" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr23 := subs({var1 = t(lambda), var2 = diff(t(lambda), lambda), var3 = r(lambda), var4 = diff(r(lambda), lambda), var5 = phi(lambda), var6 = diff(phi(lambda), lambda)}, Epr21);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L40" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr24 := subs({var1 = t(lambda), var2 = diff(t(lambda), lambda), var3 = r(lambda), var4 = diff(r(lambda), lambda), var5 = phi(lambda), var6 = diff(phi(lambda), lambda)}, Epr22);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L43" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr25 := diff(Epr23, lambda);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L39" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq26 := Epr25 - Epr24 = 0;</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L50" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr31 := diff(L1, var6);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L47" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr32 := diff(L1, var5);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L48" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr33 := subs({var1 = t(lambda), var2 = diff(t(lambda), lambda), var3 = r(lambda), var4 = diff(r(lambda), lambda), var5 = phi(lambda), var6 = diff(phi(lambda), lambda)}, Epr31);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L46" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr34 := subs({var1 = t(lambda), var2 = diff(t(lambda), lambda), var3 = r(lambda), var4 = diff(r(lambda), lambda), var5 = phi(lambda), var6 = diff(phi(lambda), lambda)}, Epr32);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L49" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Epr35 := diff(Epr33, lambda);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L45" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq36 := Epr35 - Epr34 = 0;</Text-field>
</Input>
</Group>
<Group labelreference="L280" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Set the initial conditions for massless particles:</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L52" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">t0 := 0; r0 := 1/100; phi0 := Pi/10; dr0 := 1; dphi0 := 0; dt0 := dr0;</Text-field>
</Input>
</Group>
<Group labelreference="L281" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Numerically solve the differential equations:</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L85" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">soln0 := dsolve({Eq16, Eq26, Eq36, t(0) = t0, D(t)(0) = dt0, r(0) = r0, D(r)(0) = dr0, phi(0) = phi0, D(phi)(0) = dphi0}, {t(lambda), r(lambda), phi(lambda)}, numeric, output = operator); </Text-field>
</Input>
</Group>
<Group labelreference="L38" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(plots):</Text-field>
</Input>
</Group>
<Group labelreference="L319" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Plot the trajectory of light in the xy-plane:</Text-field>
</Input>
</Group>
<Group labelreference="L77" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">polarplot([rhs(soln0(lambda)[4]), rhs(soln0(lambda)[2]), lambda = 0..5]);</Text-field>
</Input>
</Group>
<Group labelreference="L323" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">When the phi-phi component of the metric is zero, calculated below, the light reaches the maximal radial distance:</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L237" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">solve(sinh(r)^4 - sinh(r)^2 = 0, r);</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L238" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">evalf(%);</Text-field>
</Input>
</Group>
<Group labelreference="L285" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Plot and animate the trajectory of light in spacetime:</Text-field>
</Input>
</Group>
<Group labelreference="L232" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">spacecurve([rhs(soln0(lambda)[4])*cos(rhs(soln0(lambda)[2])), rhs(soln0(lambda)[4])*sin(rhs(soln0(lambda)[2])), rhs(soln0(lambda)[6])], lambda = 0..8);</Text-field>
</Input>
</Group>
<Group labelreference="L260" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">animate(spacecurve,[[rhs(soln0(lambda)[4])*cos(rhs(soln0(lambda)[2])), rhs(soln0(lambda)[4])*sin(rhs(soln0(lambda)[2])), rhs(soln0(lambda)[6])], lambda = 0..s, color = red], s=0..8);</Text-field>
</Input>
</Group>
<Group labelreference="L315" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Produce a bunch of light trajectories:</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L115" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to 20 do soln[i] := dsolve({Eq16, Eq26, Eq36, t(0) = t0, D(t)(0) = dt0, r(0) = r0, D(r)(0) = dr0, phi(0) = Pi/10*i, D(phi)(0) = dphi0}, {t(lambda), r(lambda), phi(lambda)}, numeric, output = operator); end do:</Text-field>
</Input>
</Group>
<Group labelreference="L116" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to 20 do p[i] := spacecurve([rhs(soln[i](lambda)[4])*cos(rhs(soln[i](lambda)[2])), rhs(soln[i](lambda)[4])*sin(rhs(soln[i](lambda)[2])), rhs(soln[i](lambda)[6])], lambda = -1/10..5.8, color = red); end do:</Text-field>
</Input>
</Group>
<Group labelreference="L114" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(seq(p[i], i=1..20));</Text-field>
</Input>
</Group>
<Group labelreference="L318" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Create an animation:</Text-field>
</Input>
</Group>
<Group labelreference="L251" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to 20 do ani[i] := animate(spacecurve, [[rhs(soln[i](lambda)[4])*cos(rhs(soln[i](lambda)[2])), rhs(soln[i](lambda)[4])*sin(rhs(soln[i](lambda)[2])), rhs(soln[i](lambda)[6])], lambda = -1/10..s, color = red], s=-1/10..5.8); end do: #It will take a while</Text-field>
</Input>
</Group>
<Group labelreference="L252" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(seq(ani[i], i=1..20));</Text-field>
</Input>
</Group>
<Group labelreference="L325" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">The null geodesics from a point on the axis diverge from the axis initially, reach a caustic at r = ln(1+sqrt(2)), and then reconverge to a point.  </Text-field>
</Input>
</Group>
<Group labelreference="L151" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
<Group labelreference="L246" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" bold="true" layout="Normal"><Font bold="true">References</Font></Text-field><Text-field style="Text" bold="true" layout="Normal"></Text-field><Text-field style="Text" bold="true" layout="Normal"><Font bold="false">1. S. W. Hawking and G. F. R. Ellis, <Font italic="true">The large scale structure of space-time</Font>, Cambridge University Press, Cambridge, 1973, Pp 168 - 170.  </Font></Text-field><Text-field style="Text" bold="true" layout="Normal"><Font bold="false">2. F. Wang, &quot;Relativistic orbits and black holes,&quot; Maple Application Center, 2014.  </Font><Font style="Hyperlink">https://www.maplesoft.com/applications/view.aspx?SID=4508</Font></Text-field><Text-field style="Text" bold="true" layout="Normal"><Font bold="false">2. F. Wang, <Font italic="true">Physics with Maple</Font>, Wiley, New York, 2006.    </Font></Text-field>
</Input>
</Group>
</Worksheet>