
%
% GR Exam 2013, Toby Wiseman
%
% Corrected and revised after comments from external examiners.
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\examtitle{General Relativity}
% for long examtitles, supply an optional argument as a short title, i.e.
% \examtitle[Physics 101]{Physics 101 A course with a very very long coursetitle}
\exammonth{May 2013}
\examtime{Monday 20${}^{th}$ May 2013: 14:00 to 16:00}
\examlevel{For 4th-Year Physics Students}
\examid{PT4.2} % for footer; to be supplied by Exam office
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\begin{document}

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%The paper consists of two sections: A and B \\
%Section A contains one question, comprising small parts. [20 marks total] \\
%Section B will contain four questions on selected parts of the course. [15 marks each] \\
%Candidates are required to:	\\
The paper consists of \emph{two} sections: A and B \\
Section A contains \emph{one} question [40 marks total]. \\
Section B contains \emph{four} questions [30 marks each]. \\

Candidates are required to:	\\
Answer {\bf ALL} parts of Section A and {\bf TWO  QUESTIONS} from Section B.\\
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\hspace{1cm}

\vspace{2cm}

\noindent
{\bf Conventions:} \\

\noindent
We use conventions as in lectures. In particular we take $(-,+,+,+)$ signature. 

\vspace{1cm}

\noindent
{\bf You may find the following formulae useful:} \\

\noindent
The Christoffel symbol is defined as,
\begin{eqnarray*}
\Gamma^\mu_{~~\alpha\beta} \equiv \frac{1}{2} g^{\mu\nu} \left( \partial_\alpha g_{\nu \beta} + \partial_\beta g_{\alpha \nu} - \partial_\nu g_{\alpha \beta}  \right)
\end{eqnarray*}

\noindent
The covariant derivative of a vector field is,
\begin{eqnarray*}
\nabla_\mu v^\nu \equiv \partial_\mu v^\nu + \Gamma^\nu_{~~\mu\alpha} v^\alpha
\end{eqnarray*}
and for a covector field is,
\begin{eqnarray*}
\nabla_\mu w_\nu \equiv \partial_\mu w_\nu - \Gamma^\alpha_{~~\mu\nu} w_\alpha
\end{eqnarray*}

%\noindent
%The Riemann tensor is defined as,
%\begin{eqnarray*}
%R_{\alpha\beta\mu}^{~~~~\delta} & = & \partial_\beta \Gamma^{\delta}_{~~\alpha\mu}  -   \partial_\alpha \Gamma^{\delta}_{~~\beta\mu} + \Gamma^{\nu}_{~~\alpha\mu} \Gamma^{\delta}_{~~\beta\nu} - \Gamma^{\nu}_{~~\beta\mu} \Gamma^{\delta}_{~~\alpha\nu} 
%\end{eqnarray*}

\noindent
For a Lagrangian of a curve $x^\mu(\lambda)$ of the form,
\begin{eqnarray*}
L = \int d\lambda \, \mathcal{L}( x^\mu , \frac{dx^\mu}{d\lambda} )
\end{eqnarray*}
the Euler-Lagrange equations are, 
\begin{eqnarray*}
\frac{d}{d \lambda} \left( \frac{ \partial \mathcal{L} }{ \partial ( \frac{dx^\mu}{d\lambda} ) } \right) =  \frac{ \partial \mathcal{L} }{ \partial x^\mu }
\end{eqnarray*}





\newpage

\hspace{1cm}
\vspace{5cm}

\centerline{\bf \Large Section A}

\vspace{1cm}

\noindent
\centerline{Answer all of section A.}

\newpage



  \section{A}              % Optional
 
  \begin{questions}       % questions will be numbered automatically
  
 %
% Section A  Question 1
%
  
  \begin{question} % 1st question
  
This question concerns the covariant derivative.
  
		\begin{subquestions}
		 
\subquestion
\Q{State how the components of a $(1, 0)$ tensor $v^\mu$ and a $(0,1)$ tensor $w_\mu$  transform under a coordinate transformation $x \to x'$.}

\A{



}

			 \qmarks{4}

\subquestion		 
\Q{Use your previous answer to show that $v^\mu w_\mu$ transforms as a scalar under a coordinate transformation $x \to x'$.}

\A{


}

			 \qmarks{4}
			 
			 
	 
\subquestion
\Q{Show that $\partial_\mu w_\nu$, the partial derivative of a covector field $w_\mu$,  does \emph{not} transform as a tensor.  }

\A{


}

			 \qmarks{4}			 



\subquestion\Q{

Under a coordinate transformation the Christoffel symbol transforms as;
\begin{eqnarray*}
\Gamma'^{\mu'}_{~~\alpha'\beta'} = \Gamma^{\mu}_{~~\alpha\beta} \frac{\partial x'^{\mu'}}{\partial x^\mu}  \frac{\partial x^\alpha}{\partial x'^{\alpha'}} \frac{\partial x^\beta}{\partial x'^{\beta'}} - \left( \frac{\partial^2 x'^{\mu'}}{\partial x^\alpha \partial x^\beta} \right) \frac{\partial x^\alpha}{\partial x'^{\alpha'}} \frac{\partial x^\beta}{\partial x'^{\beta'}}
\end{eqnarray*}
Show that this is not a tensor transformation.

}

\A{
}

 \qmarks{4}			 



\subquestion\Q{

Use this to show that the covariant derivative of a covector field $w_\mu$, defined as $\nabla_\mu w_\nu = \partial_\mu w_\nu - \Gamma^\alpha_{~~\mu \nu} w_\alpha$, does transform as a tensor.

}

\A{

So from the previous part we have,
\begin{eqnarray*}
\partial_{\mu'} v'^{\nu'} 
&=&  \frac{\partial x^{\mu}}{\partial x'^{\mu'}}  \frac{ \partial x'^{\nu'} }{ \partial x^\nu } \left( \partial_\mu v^{\nu} \right) + v^{\nu}  \frac{\partial x^{\mu}}{\partial x'^{\mu'}}  \left(  \frac{ \partial^2 x'^{\nu'} }{ \partial x^\mu \partial x^\nu } \right)
\end{eqnarray*}

The second term in the covariant derivative transforms as;
\begin{eqnarray*}
\Gamma'^{\nu'}_{~~\mu' \alpha'} v'^{\alpha'}
&=& \left( \Gamma^{\nu}_{~~\mu\alpha} \frac{\partial x'^{\nu'}}{\partial x^\nu}  \frac{\partial x^\mu}{\partial x'^{\mu'}} \frac{\partial x^\alpha}{\partial x'^{\alpha'}} - \left( \frac{\partial^2 x'^{\nu'}}{\partial x^\mu \partial x^\alpha} \right) \frac{\partial x^\mu}{\partial x'^{\mu'}} \frac{\partial x^\alpha}{\partial x'^{\alpha'}} \right)  \left( \frac{ \partial x'^{\alpha'} }{ \partial x^\beta }  v^{\beta} \right)
\end{eqnarray*}
Recalling that,
\begin{eqnarray*}
\frac{\partial x^\alpha}{\partial x'^{\alpha'}}  \frac{ \partial x'^{\alpha'} }{ \partial x^\beta } = \delta^\alpha_\beta
\end{eqnarray*}
by the chain rule then we obtain,
\begin{eqnarray*}
\Gamma'^{\nu'}_{~~\mu' \alpha'} v'^{\alpha'}
&=& v^{\alpha}  \Gamma^{\nu}_{~~\mu\alpha} \frac{\partial x'^{\nu'}}{\partial x^\nu}  \frac{\partial x^\mu}{\partial x'^{\mu'}}  - v^{\alpha} \left( \frac{\partial^2 x'^{\nu'}}{\partial x^\mu \partial x^\alpha} \right) \frac{\partial x^\mu}{\partial x'^{\mu'}} 
\end{eqnarray*}
Putting this together we have,
\begin{eqnarray*}
\nabla_{\mu'} v'^{\nu'}  &=& \partial_{\mu'} v'^{\nu'} + \Gamma'^{\nu'}_{~~\mu' \alpha'} v'^{\alpha'} \nonumber \\
&=&  \frac{\partial x^{\mu}}{\partial x'^{\mu'}}  \frac{ \partial x'^{\nu'} }{ \partial x^\nu } \left( \partial_\mu v^{\nu} \right) + v^{\nu}  \frac{\partial x^{\mu}}{\partial x'^{\mu'}}  \left(  \frac{ \partial^2 x'^{\nu'} }{ \partial x^\mu \partial x^\nu } \right) + v^{\alpha}  \Gamma^{\nu}_{~~\mu\alpha} \frac{\partial x'^{\nu'}}{\partial x^\nu}  \frac{\partial x^\mu}{\partial x'^{\mu'}}  - v^{\alpha} \left( \frac{\partial^2 x'^{\nu'}}{\partial x^\mu \partial x^\alpha} \right) \frac{\partial x^\mu}{\partial x'^{\mu'}} \nonumber \\
& = &  \frac{\partial x^{\mu}}{\partial x'^{\mu'}}  \frac{ \partial x'^{\nu'} }{ \partial x^\nu } \left( \partial_\mu v^{\nu} + v^{\alpha}  \Gamma^{\nu}_{~~\mu\alpha}\right) \nonumber \\
& = &  \frac{\partial x^{\mu}}{\partial x'^{\mu'}}  \frac{ \partial x'^{\nu'} }{ \partial x^\nu } \nabla_\mu v^\nu
\end{eqnarray*}
which shows that the covariant derivative $ \nabla_\mu v^\nu$ indeed transforms as a $(1,1)$ tensor.
}


			 \qmarks{4}	

			 
\subquestion
\Q{ Given a vector $v^\mu$ then $v^\mu w_\mu$ is a coordinate scalar. Use this fact together with the definition 
of the covariant derivative in the previous part and also 
  the Liebnitz  property (product rule) of covariant derivatives to deduce that the covariant derivative of a vector field is,
\begin{eqnarray*}
\nabla_\mu v^\nu = \partial_\mu v^\nu + \Gamma^\nu_{~~\mu \alpha} v^\alpha
\end{eqnarray*}
}

\A{
The covariant derivative of a scalar is simply the usual partial derivative. Therefore we have,
\begin{eqnarray*}
\nabla_\alpha \left( u_\mu v^\mu \right) = \partial_\alpha \left( u_\mu v^\mu \right) = u_\mu \partial_\alpha v^\mu  + v^\mu \partial_\alpha u_\mu
\end{eqnarray*}
However by the product rule for the covariant derivative we also have,
\begin{eqnarray*}
\nabla_\alpha \left( u_\mu v^\mu \right) = u_\mu \nabla_\alpha v^\mu  + v^\mu \nabla_\alpha u_\mu
\end{eqnarray*}
Hence equating these we obtain,
\begin{eqnarray*}
u_\mu \nabla_\alpha v^\mu  + v^\mu \nabla_\alpha u_\mu = u_\mu \partial_\alpha v^\mu  + v^\mu \partial_\alpha u_\mu
\end{eqnarray*}
Rearranging and using the expression for the covariant derivative of a vector we find;
\begin{eqnarray*}
v^\mu \nabla_\alpha u_\mu &=& u_\mu \partial_\alpha v^\mu  + v^\mu \partial_\alpha u_\mu - u_\mu \nabla_\alpha v^\mu  \nonumber \\
& = & u_\mu \partial_\alpha v^\mu  + v^\mu \partial_\alpha u_\mu - u_\mu \left( \partial_\alpha v^\mu + \Gamma^\mu_{~~\alpha\beta} v^\beta \right)  \nonumber \\
& = & v^\mu \partial_\alpha u_\mu - v^\beta \Gamma^\mu_{~~\alpha\beta}  u_\mu \nonumber \\
& = & v^\mu \partial_\alpha u_\mu - v^\mu \Gamma^\beta_{~~\alpha\mu}  u_\beta \nonumber \\
& = & v^\mu \left( \partial_\alpha u_\mu - \Gamma^\beta_{~~\alpha\mu}  u_\beta \right)
\end{eqnarray*}
Thus we obtain,
\begin{eqnarray*}
v^\mu \nabla_\alpha u_\mu &=& v^\mu \left( \partial_\alpha u_\mu - \Gamma^\beta_{~~\alpha\mu}  u_\beta \right)
\end{eqnarray*}
and since this holds for any $v^\mu$, then we find,
\begin{eqnarray*}
 \nabla_\alpha u_\mu &=&  \partial_\alpha u_\mu - \Gamma^\beta_{~~\alpha\mu}  u_\beta 
\end{eqnarray*}
as we were required to show.

}

 \qmarks{4}
 
				 
		\end{subquestions}
	
\end{question}

%\end{questions}

\newpage

\hspace{1cm}
\vspace{5cm}

\centerline{\bf \Large Section B}

\vspace{1cm}

\noindent
\centerline{Answer 2 out of the 4 questions in the following section.}

\newpage

  \section{B}              % Optional
 
%  \begin{questions}       % questions will be numbered automatically
  

%
% Section B  Question 1
%
  
	\begin{question}   
	
	This question concerns the Newtonian spacetime, which we write using coordinates $x^\mu = (t, x^i)$ with $i = 1,2,3$ as,
	\begin{eqnarray}
ds^2 = \left( \eta_{\mu\nu} - 2 \epsilon \Phi(x) \delta_{\mu\nu} \right) dx^\mu dx^\nu \nonumber
\end{eqnarray}
where $\epsilon \Phi$ is the Newtonian potential, and we are interested in the Newtonian limit $\epsilon \to 0$ so that $|\epsilon \Phi| \ll 1$ .

	
	\begin{subquestions}


\subquestion
\Q{
State the stress tensor for a perfect fluid in a general spacetime in terms of its energy density $\rho$, pressure $P$ and local 4-velocity. 
}

\A{

}

\qmarks{1}


\subquestion
\Q{
In the limit $\epsilon \to 0$ the components of the Ricci tensor to leading order in $\epsilon$ are;
\begin{eqnarray*}
R_{tt} &=& \epsilon \, \delta_{ij} \partial_i \partial_j \Phi  \nonumber \\
R_{ti} &= &  0 \nonumber \\
R_{ij} & = & \epsilon \, \delta_{ij} \, \left( \delta_{ab} \partial_a \partial_b \Phi \right) 
\end{eqnarray*}
Use these to compute the components of the stress tensor that satisfies the Einstein equations for this spacetime. Show that this is the stress tensor for a dust fluid (ie. fluid with zero pressure), and determine the 4-velocity and energy density of this dust in terms of the Newtonian potential $\epsilon \Phi$.

}

\A{

}

\qmarks{1}



\subquestion
\Q{
By calculation, show that to leading order in $\epsilon$,
\begin{eqnarray}
\Gamma^i_{~~tt} = + \epsilon \partial_i  \Phi  \nonumber
\end{eqnarray}
Show that a non-accelerated particle that is slowly moving obeys (to leading order in $\epsilon \to 0$),
\begin{eqnarray}
\frac{d^2 x^i}{d t^2} = - \partial_i \left( \epsilon \Phi \right) \nonumber
\end{eqnarray}
}

\A{

}

\qmarks{1}




\subquestion
\Q{
Briefly state the physical significance of the results in part ii) and part iii) in understanding how General Relativity is compatible with Newton's law of gravity.
}

\A{

}

\qmarks{1}




	\end{subquestions}
	
\end{question}


%
% Section B  Question 2
%
  
	\begin{question}   
	
	This question concerns Schwarschild metric, which we write using coordinates $x^\mu = (t, r, \theta, \phi)$ as,
	\begin{eqnarray}
ds^2 = - \left( 1 - \frac{2 M}{r} \right) dt^2 + \left(1 - \frac{2 M}{r}\right)^{-1} dr^2 + r^2 \left( d\theta^2 + \sin^2{\theta} d\phi^2 \right) \nonumber
\end{eqnarray}
	
	\begin{subquestions}


\subquestion
\Q{
Consider a timelike geodesic $x^\mu(\tau) = \left( T(\tau), R(\tau), \Theta(\tau), \Phi(\tau) \right)$ in the Schwarzschild metric where $\tau$ is proper time. Write a Lagrangian that we may vary to determine the geodesic.
}

\A{

}

\qmarks{1}


\subquestion
\Q{

By varying the Lagrangian, deduce the Euler-Lagrange equations for $\Theta$ and $\Phi$. Show these are consistent with a geodesic that lies in the plane $\theta = \pi/2$. We now restrict our attention to such geodesics. Show then that,
\begin{eqnarray}
R^2 \frac{d\Phi}{d\tau} = J
\end{eqnarray}
where $J$ is a constant.
}

\A{

}

\qmarks{1}



\subquestion
\Q{

Further deduce the equations that govern $T$ and $R$. Show that,
\begin{eqnarray}
\left( 1 - \frac{2 M}{r} \right) \frac{dT}{d\tau}  & = & E \nonumber
\end{eqnarray}
were $E$ is a constant. Hence show the equation governing the radial motion in the plane $\theta = \pi/2$ looks like that of a  one dimensional unit mass particle in a potential $V(R)$ with energy $E$ so,
\begin{eqnarray}
E = \frac{1}{2} \left(  \frac{dR}{d\tau} \right)^2 + V(R) 
\end{eqnarray}
where you should determine the potential $V(R)$.
}

\A{

}

\qmarks{1}



\subquestion
\Q{

Consider a circular orbit, so that $R$ is constant. How is the radius $R$ related to the constants $E$ and $J$? If the orbit is perturbed a little so that it is no longer circular, show that the orbit does not close to an ellipse as in Newtonian gravity, but instead the perihelion precesses. 

}

\A{

}

\qmarks{1}


	\end{subquestions}
	
\end{question}



%
% Section B  Question 3
%
  
  \begin{question} % 1st question
  
  This question concerns particle motion in curved spacetimes.
  
		\begin{subquestions}
		 
\subquestion
\Q{
Consider a particle following a timelike curve $x^\mu(\tau)$ in a general spacetime, where $\tau$ is the particle's proper time. The 4-velocity $v^\mu = dx^\mu/d\tau$. Give the expression for the 4-acceleration $a^\mu$ in terms of $v^\mu$ and its covariant derivative.

}


\A{}

\qmarks{1}

\subquestion
\Q{
Show that for the case of Minkowski spacetime in Minkowski coordinates $x^\mu = (t, x^i)$ so that $ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu$ then this reduces to the Special Relativity result,
\begin{eqnarray}
a^\mu = \frac{d^2 x^\mu}{d \tau^2}
\end{eqnarray}

}


\A{}

\qmarks{1}


\subquestion
\Q{
By carefully varying the action,
\begin{eqnarray}
L = \int d\tau \left( g_{\mu\nu} \frac{dx^\mu}{d\tau}  \frac{dx^\nu}{d\tau} \right)
\end{eqnarray}
show that the Euler-Lagrange equations derived from it reproduce the geodesic equation,
\begin{eqnarray}
v^\mu \nabla_\mu v^\nu = 0
\end{eqnarray}

}


\A{}

\qmarks{1}

\subquestion
\Q{
Consider now a particle coupled to a scalar field $\phi(x)$ in a general spacetime, with Lagrangian,
\begin{eqnarray}
L = \int d\tau \left( g_{\mu\nu}(x) \frac{dx^\mu}{d\tau}  \frac{dx^\nu}{d\tau} + \phi(x) \right)
\end{eqnarray}
By varying the action, show that it experiences a 4-acceleration,
\begin{eqnarray}
a_\mu = \partial_\mu \phi(x)
\end{eqnarray}

}


\A{}

\qmarks{1}



\subquestion
\Q{
Consider now a particle with charge $q$ coupled to an electromagnetic field $A_\mu(x)$ in a general spacetime. Its Lagrangian is modified to,
\begin{eqnarray}
L = \int d\tau \left( g_{\mu\nu}(x) \frac{dx^\mu}{d\tau}  \frac{dx^\nu}{d\tau} + q \, A_\mu(x) \frac{dx^\mu}{d\tau} \right)
\end{eqnarray}
Compute the 4-acceleration of the particle due to the electromagnetic field.

}


\A{}

\qmarks{1}


	
\end{subquestions}

	
\end{question}



%
% Section B  Question 4
%
  
	\begin{question}   
	
	This question concerns Killing vectors.
	
	\begin{subquestions}


\subquestion
\Q{
The Lie derivative of a $(0,2)$ tensor $A_{\mu\nu}$ with respect to a vector field $w^\mu$ is defined to be,
\begin{eqnarray}
\mathrm(Lie)(w,A)_{\mu\nu} = w^\alpha \partial_\alpha A_{\mu\nu} + A_{\mu\alpha} \partial_\nu w^\alpha + A_{\alpha\nu} \partial_\mu w^\alpha \nonumber
\end{eqnarray}
Suppose we consider the Lie derivative of the metric $g_{\mu\nu}$. Show that this can also be written in terms of the covariant derivative as,
\begin{eqnarray}
\mathrm(Lie)(w,g)_{\mu\nu} = \nabla_{\mu} w_\nu + \nabla_\nu w_\mu \nonumber
\end{eqnarray}
If this vanishes, we say $w^\mu$ is a Killing vector field.
}

\A{

}

\qmarks{1}



\subquestion
\Q{
Consider a timelike particle with velocity $v^\mu = dx^\mu/d\tau$ for proper time $\tau$. Suppose it follows a geodesic in a spacetime with a Killing vector field $w^\mu$. Show that the quantity,
\begin{eqnarray}
\phi = - w^\mu v_\mu \nonumber
\end{eqnarray}
is constant along the particles trajectory.
}

\A{

}

\qmarks{1}



\subquestion
\Q{
Consider the spacetime with coordinates $x^\mu = (t, x^i)$ 
\begin{eqnarray}
\label{eq:static}
ds^2 = - N(x) dt^2 + g_{ij}(x) dx^i dx^j 
\end{eqnarray}
where $N$ and $g_{ij}$ only depend on the spatial coordinates $x^i$ and not time $t$. Show that there is a Killing vector $w^\mu$ for this spacetime and explicitly check that $\mathrm{Lie}(w,g) = 0$. Write down the conserved quantity $\phi$ for a non-accelerated particle's motion. Is this the energy of the particle as measured by observers sitting at constant spatial position?
}

\A{

}

\qmarks{1}


\subquestion
\Q{
In the spacetime in equation~(\ref{eq:static}) above write down a Lagrangian that may be varied to deduce geodesic motion in the spacetime. Show using the Euler-Lagrange equations that the quantity $\phi$ is indeed conserved. 
}

\A{

}

\qmarks{1}



	\end{subquestions}
	
\end{question}



	
	
 \end{questions}

\end{document}


