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\begin{document}
\section{Equivalence Principals: The Weak, The Strong, and The Grand}
\subsection{Inertial Reference Frames}
Einstein's theory of gravitation, also called general relativity, is more
than just a theory of gravity. It is a complete theory of physical
interactions in the presence of gravitational fields. The basis for this
complete theory is the Principal of Equivalence. The principal can be stated
in several ways and applied weakly or strongly. It is well-tested
experimentally and has never failed. Nevertheless, most physicists are quite
certain that it must fail at some level because it is clearly incompatible
with quantum theory.
Einstein started where Newton did, with Galileo's Principal of Inertia.
Newton states that principal as his first law because it is the essential
test that a reference frame must pass before we use it to state other
physical laws. Only in a frame that passes the first-law test can we expect
other physical laws to be simple. Such a frame is called an \emph{inertial
reference frame}. The first-law test of a reference frame is to isolate an
object from all outside influences, forces, etc., and see if it has zero
acceleration.
\begin{quotation}
\bigskip
\emph{An inertial reference frame is one in which isolated objects do not
accelerate}.
\end{quotation}
\bigskip
\[
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\]
There are several problems with this definition: (1) How do we know that an
object is `isolated'? (2) What is an `object'? (3) How does this definition
apply to quantum objects such as electrons? Einstein focused on the first
problem. An object is isolated if we have done everything possible to shield
it from outside influences: Nothing touches it and we put it in a Faraday
cage to shield it against electromagnetic fields. Einstein's essential
insight was that there is no way to shield against gravitational fields.
Thus, we have no choice except to regard a freely falling object as
isolated. The definition of an inertial reference frame is then modified
slightly.
\bigskip
\emph{An inertial reference frame is one in which freely falling objects do
not accelerate.}
\bigskip
The modified definition works because of another contribution from Galileo,
the Universality of Free Fall. All objects falling in a gravitational field
at a given point have the same acceleration. Thus, it is always possible to
find a reference frame in which all freely falling objects have zero
acceleration. That would be the reference frame defined by a laboratory that
is itself in free fall.
\[
\FRAME{itbpF}{1.2471in}{1.8801in}{0in}{}{}{fall.gif}{\special{language
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\]%
If we set up laboratory (Minkowski) coordinates in such a reference frame,
the speed of an object becomes just the time-slope of the object's
world-line so that Galileo's statement is just that non-interacting
worldlines are \emph{straight}.
\[
\FRAME{itbpF}{0.4419in}{0.71in}{0in}{}{}{strline.gif}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
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"1";cropright "1";cropbottom "0";filename 'graphics/strline.gif';file-properties "XNPEU";}}
\]%
In terms of differential geometry, the world-line of a non-interacting
particle has a tangent-vector field $u$ that is as straight as possible.
Thus, under parallel transport along the curve, the tangent vector does not
turn. In terms of the laboratory connection $\partial $ that we discussed
earlier, the equation of motion of a freely falling object in the laboratory
will be
\[
\partial _{u}u=ku.
\]
\subsection{Clocks}
Take another concept from Special Relativity. The curve parameter along the
worldline of a particle can be thought of as describing a regular sequence
of events that happen inside the particle --- the ticks of an internal
clock. If the clock keeps good time, then its ticks remain evenly spaced
under parallel transport along the worldline --- the \emph{internal clock
time is an affine parameter}.%
\[
\FRAME{itbpF}{0.4419in}{0.7126in}{0in}{}{}{tick.gif}{\special{language
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\]%
The worldline with internal or ``proper'' time as a parameter along it then
obeys the equation%
\[
\partial _{u}u=0
\]%
for a geodesic with an affine parameter. The assumption that the laboratory
connection $\partial $ coincides with the physical connection $D$ somewhere
in the laboratory then means that the particle worldline must obey the
coordinate-independent equation%
\[
D_{u}u=0
\]%
and we conclude that \emph{freely falling objects follow geodesics of the
physical tangent-space connection}.
One important technical point to notice about the inertial frames defined by
the equivalence principal is that they are \emph{local}. In a real
laboratory in free-fall near the earth, a space station, for example, there
will be \emph{only one place} where objects actually have zero acceleration
relative to the lab, namely at the center of mass of the laboratory.
\[
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\]%
Objects placed at other points in the lab will experience tidal forces.
Objects closer to the earth than the station's center of mass will be
attracted to the earth more strongly than the station as a whole and will
accelerate downward relative to the station. Objects farther from the earth
than the station center of mass will accelerate upward. This observation
coincides with our mathematical description, which says that the laboratory
connection only needs to agree with the physical connection at one point in
the lab. In this case, the lab connection agrees with the physical
connection only at the center of mass of the space station.
\subsection{The Weakest Equivalence Principal}
Now consider the geodesic equation in an arbitrary coordinate patch:%
\[
\ddot{x}^{\mu }+\dot{x}^{\alpha }\dot{x}^{\beta }\Gamma ^{\mu }{}_{\alpha
\beta }=0
\]%
Einstein wanted to recover Galileo's description of falling objects.
Consider an object that falls from rest in the coordinate system that has
been chosen. At the instant that the object is dropped, we have%
\[
\dot{x}^{a}=0;\qquad a=1,2,3
\]%
and the geodesic equations become just%
\[
\begin{array}{c}
\ddot{x}^{0}+\dot{x}^{0}\dot{x}^{0}\Gamma ^{0}{}_{00}=0 \\
\ddot{x}^{1}+\dot{x}^{0}\dot{x}^{0}\Gamma ^{1}{}_{00}=0 \\
\ddot{x}^{2}+\dot{x}^{0}\dot{x}^{0}\Gamma ^{2}{}_{00}=0 \\
\ddot{x}^{3}+\dot{x}^{0}\dot{x}^{0}\Gamma ^{3}{}_{00}=0%
\end{array}%
\]%
or, equivalently, with $x^{0}=t,\quad x^{1}=x,\quad x^{2}=y,\quad x^{3}=z$
and with the curve parameter denoted by $\lambda $%
\[
\begin{array}{c}
\frac{d^{2}x}{d\lambda ^{2}}+\left( \frac{dt}{d\lambda }\right) ^{2}\Gamma
^{1}{}_{00}=0\smallskip \\
\frac{d^{2}y}{d\lambda ^{2}}+\left( \frac{dt}{d\lambda }\right) ^{2}\Gamma
^{2}{}_{00}=0\smallskip \\
\frac{d^{2}z}{d\lambda ^{2}}+\left( \frac{dt}{d\lambda }\right) ^{2}\Gamma
^{3}{}_{00}=0%
\end{array}%
\]%
Use the fact that this object is initially at rest in these coordinates so
that $\frac{dx}{d\lambda }=\frac{dy}{d\lambda }=\frac{dz}{d\lambda }=0$ and
convert the derivatives to local coordinate time $t$ by dividing by the
factor $\left( \frac{dt}{d\lambda }\right) ^{2}.$%
\[
\begin{array}{c}
\frac{d^{2}x}{dt^{2}}+\Gamma ^{1}{}_{00}=0\smallskip \\
\frac{d^{2}y}{dt^{2}}+\Gamma ^{2}{}_{00}=0\smallskip \\
\frac{d^{2}z}{dt^{2}}+\Gamma ^{3}{}_{00}=0\smallskip%
\end{array}%
\]%
In a local frame we find that the acceleration of an object at rest is given
by the vector%
\[
\vec{a}=\vec{\Gamma},\qquad \vec{\Gamma}=-\Gamma ^{1}{}_{00}\partial
_{x}-\Gamma ^{2}{}_{00}\partial _{y}-\Gamma ^{3}{}_{00}\partial _{z}.
\]%
The acceleration of the object is determined entirely by the connection
coefficients of the geometry and not at all by any properties of the object
(such as its mass). This statement agrees with what Galileo observed.
If we watch the object fall for a while, it will gain velocity. The geodesic
equation will then exhibit some velocity-dependent terms such as $\frac{dx}{%
dt}\Gamma ^{2}{}_{01}$. However, these velocity-dependent terms will be
negligible until speeds reach a substantial fraction of the speed of light.
For objects with ordinary velocities, the velocity-independent acceleration $%
\vec{\Gamma}$ dominates, which agrees with Galileo's observation that all
objects (in a particular place such as the surface of the Earth) fall with
the same acceleration.
Thus, the combination of Galileo, special relativity, and the tools of
differential geometry leads to theories in which \emph{freely falling
objects follow geodesics of an affine connection on a four dimensional
space-time manifold}. This is a statement of what I choose to call the
``Weakest'' Equivalence Principal. It is not very restrictive because even
Newton's Theory of Gravitation can be cast into a form that is consistent
with it --- just define an affine connection that does the same job as
Newton's gravitational potential without producing any other effects.
\subsection{The Weak Equivalence Principal: Clocks and Light}
In order for a theory to reduce to Special Relativity, we need the idea of a
local Lorentz Frame. Given the ``weakest equivalence principle'', we have an
affine connection that governs the motion of particles and we can always
find coordinates near any event that will make particle paths appear
straight and unaccelerated. These particle paths have affine parameters
along them that we can interpret as ``proper time'' if we want to. What is
missing, however, is any account of how to compare clocks moving along
different paths.
In all of the following material, I will be using units in which the speed
of light is equal to one. These $c=1$ units emphasize the symmetry between
space and time and might be realized, for example, by using the second as
the unit of time and the light-second ($3\times 10^{8}m$) as the unit of
distance.
In Special Relativity, proper time $\tau $ is expressed in terms of the
standard Minkowski coordinates by
\[
d\tau ^{2}=-dt^{2}+dx^{2}+dy^{2}+dz^{2},
\]%
which is shorthand for the relation:%
\[
1=-\left( \frac{dt}{d\tau }\right) ^{2}+\left( \frac{dx}{d\tau }\right)
^{2}+\left( \frac{dy}{d\tau }\right) ^{2}+\left( \frac{dz}{d\tau }\right)
^{2}
\]%
or simply,
\[
u\cdot u=-1
\]%
Suppose we insist that we can always find local coordinates such that this
expression takes this form? Converting the expression back to general
coordinates, it means that there is a second-rank tensor field $g_{\mu \nu }$
such that the proper time parameter obeys%
\[
d\tau ^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }\qquad \text{or}\qquad 1=g_{\mu \nu
}\frac{dx^{\mu }}{d\tau }\frac{dx^{\nu }}{d\tau }.
\]
Thus, we are reduced to asserting the existence of a space-time \emph{metric
tensor} field and insisting that good (affine parameter) clocks read proper
time.
If $\tau $ is an affine parameter, then the tangent vector to a geodesic
with this parameter is
\[
u=\frac{d}{d\tau }=\frac{dx^{\mu }}{d\tau }\frac{\partial }{\partial x^{\mu }%
}=\frac{dx^{\mu }}{d\tau }\partial _{\mu }
\]%
so that there is a constraint that must be preserved all along the geodesic:
\[
u\bullet u=-1\qquad g\left( u,u\right) =-1.
\]%
Now evaluate the derivative of this constraint along the geodesic.%
\begin{eqnarray*}
D_{u}\left( g\left( u,u\right) \right) &=&0 \\
\left( D_{u}g\right) \left( u,u\right) +g\left( D_{u}u,u\right) +g\left(
u,D_{u}u\right) &=&0 \\
\left( D_{u}g\right) \left( u,u\right) &=&0.
\end{eqnarray*}%
The covariant derivative of the metric tensor $g$ is a third rank tensor $Q$
that will be called the \emph{metricity tensor} and is defined by%
\[
Q\left( u,v,w\right) =\left( D_{w}g\right) \left( u,v\right) .
\]%
For the $u\cdot u=-1$ constraint to be preserved along a geodesic, the
metricity must satisfy the condition%
\[
Q\left( u,u,u\right) =0
\]%
or, in components,%
\[
u^{\alpha }Q_{\alpha \beta \delta }u^{\beta }u^{\delta }=0.
\]%
If the constraint is preserved along any geodesic, then the condition must
hold for arbitrary four-velocity components and the metricity tensor is
restricted to have a zero symmetric part:%
\[
Q_{\left( \alpha \beta \delta \right) }=0.
\]%
This same restriction guarantees that timelike geodesics stay timelike, and
lightlike geodesics stay lightlike.
There is still a little room for a non-zero metricity. To get rid of it,
assert the possibility of carrying a Lorentz frame along with a freely
falling particle. Thus, one has a set of local basis vectors $e_{\mu }\left(
\tau \right) $ for each value of the proper time and they are carried along
so that $\nabla _{u}e_{\mu }=0$. Requiring that this frame should remain a
Lorentz frame means that the dot products, $g\left( e_{\alpha },e_{\beta
}\right) $ should be the appropriate ones and zeros of the Lorentz metric.
But then%
\[
\begin{array}{c}
D_{u}\left( g\left( e_{\alpha },e_{\beta }\right) \right) =\left(
D_{u}g\right) \left( e_{\alpha },e_{\beta }\right) +g\left( \nabla
_{u}e_{\alpha },e_{\beta }\right) +g\left( e_{\alpha },\nabla _{u}e_{\beta
}\right) \\
=Q\left( e_{\alpha },e_{\beta },u\right) =0.%
\end{array}%
\]%
This full set of assertions then says that freely falling particles move
along the geodesics of a \emph{metric-compatible} connection.
Thus we have two equivalent statements of what is usually called the Weak
Equivalence Principle:
\begin{itemize}
\item Freely falling particles move along the geodesics of an affine
connection $D$ on a space-time manifold and freely falling Lorentz frames,
constant with respect to the connection $D,$ exist and remain Lorentz frames
as they fall.
\item Freely falling particles move along the geodesics of an affine
connection that is compatible with a Riemannian metric $g$ on a space-time
manifold.\FRAME{dtbpF}{1.7608in}{1.3595in}{0pt}{}{}{wep.gif}{\special%
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\end{itemize}
As we have seen there is some room for weaker versions of the weak
equivalence principle. One can squeeze in a little metricity, for example,
without messing up the behavior of clocks and light too much. Ordinarily,
however, we try the simplest approach first and let the metricity be zero.
However, it is useful to keep open the possibility that Nature has chosen
the simplest approach only in the low energy limit of gravitational theory.
\subsection{The Strong Equivalence Principle}
The weak equivalence principle states that the connection coefficients are
the only variables that affect the trajectories of freely falling objects.
When these coefficients are zero, objects behave as if they are not in a
gravitational field. Thus, you cannot distinguish between a small freely
falling reference frame in a gravitational field and a frame at rest in the
absence of any gravitational fields by watching the behavior of freely
moving objects. The \emph{strong equivalence principle} extends these
statements by requiring \emph{all gravitational effects on matter} to be
mediated by the connection coefficients. When these coefficients are zero,
all physical processes behave as if they are not in a gravitational field.
Thus, you cannot distinguish between a small freely falling reference frame
in a gravitational field and a frame at rest in the absence of any
gravitational fields \emph{by any local observations}.
The restriction to a ``small freely falling reference frame'' and ``local
observations'' is a modern modification of what Einstein actually said.
Einstein stated the equivalence principle in terms of any observations in a
\emph{uniform} gravitational field. Thus, he said that no experiment can
distinguish between being in a uniform gravitational field and being
uniformly accelerated in the absence of a gravitational field. That form of
the principle has largely been discarded because there is no such thing as a
uniform gravitational field. Saying that a field is nearly uniform is the
same as saying that observations are restricted to a small enough region for
the non-uniformity to be ignored.
The restriction either to uniform gravitational fields or to sufficiently
local observations is important. Without it, the equivalence principle is
obviously incorrect. If you are in a spaceship that is either floating in
intergalactic space or falling toward the Earth, you can tell which
situation applies by looking out the window. That, however, is a non-local
observation. If your spaceship is rather large, you can tell whether or not
you are in a strong gravitational field without looking out the window.%
\[
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\]%
Just watch the behavior of a cloud of particles in the ship. If the cloud
changes from a spherical shape to an ellipsoidal shape, then you know that
you are falling toward some object that lies along the long axis of the
ellipsoid. This observation works by using tidal forces that are
proportional to the gradients of the gravitational field. A reference frame
in which such gradients have observable effects is too large for the
equivalence principle to apply. Yet another way to tell where you are is to
watch a charged particle inside the ship. If you are floating in
intergalactic space, the charged particle will just stay where it is. If you
are really accelerating toward the earth, the charged particle will radiate
electromagnetic waves that will react back on it so as to reduce its
acceleration. The resulting back-reaction forces will cause the charged
particle to rise toward the part of the ship that is farthest from the
Earth. However, the charged particle itself is a non-local object with an
electric field stretching to infinity in all directions. Its behavior is
governed by boundary conditions on the electromagnetic field at infinity. To
perform a local experiment with charged particles, the spaceship must be
shielded by a conducting shell. In that case, there is no radiation-reaction
force and the equivalence principle still works.
\subsection{Strong Equivalence as a Correspondence Rule}
The strong equivalence principle says that all gravitational interactions
are mediated by the connection coefficients. The geometrical or
coordinate-independent object that one builds from the connection
coefficients is the covariant derivative. Whenever the connection
coefficients are zero --- in flat, gravity-free Minkowski space-time or in a
freely falling frame --- covariant derivatives become ordinary derivatives.
Thus, the strong equivalence principal can be stated as a rule for passing
from the Special Relativity version of a theory to the version that includes
gravitational interactions: Replace ordinary derivatives everywhere by
covariant derivatives.
\[
\frac{\partial }{\partial x^{\alpha }}\rightarrow \nabla _{e_{\alpha }}
\]%
We have seen this \emph{Minimal Coupling Rule} before in our discussion of
field theory construction.
\subsection{The Scalar Field Problem}
The minimal coupling rule fails in situations where we are not really sure
about the physical interactions involved. A notorious example is the
massless scalar field. In Special Relativity, it obeys the Klein-Gordon
equation%
\[
\eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\Phi =0
\]%
where $\eta ^{\mu \nu }$ is the Minkowski metric. Minimal coupling suggests
that the equation in a theory coupled to gravity should be%
\[
g^{\mu \nu }\nabla _{\mu }\nabla _{\nu }\Phi =0.
\]%
Since the metric tensor is symmetric, there is no factor ordering problem
here and this looks like a good theory. However, recall that Maxwell's
equations of electromagnetism are invariant under conformal transformations
--- multiply the metric tensor and each field component by various powers of
a single function on space-time and get new solutions from old. The minimal
coupling scalar field equation is not conformally invariant but we can make
it so by adding a curvature term to obtain%
\[
g^{\mu \nu }\nabla _{\mu }\nabla _{\nu }\Phi +\frac{1}{6}R\Phi =0.
\]%
This is another, perfectly good generalization of the Klein-Gordon equation
because the curvature term will vanish in Minkowski space-time. We do not
actually know of any fundamental massless scalar field. The one massless
field that we know really well, the electromagnetic field, has conformal
symmetry. Which equation should we use? The equation without the curvature
term seems simpler when written down but the equation with the curvature
term has simpler properties.
\subsection{The Grand Equivalence Principals}
The Strong Equivalence Principal, like the Weak Equivalence Principal is
strictly local. It applies to first derivatives at just one point $P_{0}$ in
a given laboratory (geometrically, in a given coordinate patch or chart on
the manifold). If we extend the principal to \emph{second derivatives} at
the point $P_{0}$ we get what I will call the Grand Equivalence Principals,
the Grand Weak Equivalence Principal and the Grand Strong Equivalence
Principal or GWEP and GSEP. Unlike the WEP and SEP which have never been
shown to fail, the GWEP and GSEP do not always work. The difficulty is
basically the factor ordering problem that is familiar from quantum theory.
Directional derivatives that use the laboratory connection $\partial $
always commute with one another, so the order in which they are taken is not
important. When we replace the lab connection with the physical connection,
the order in which the derivatives are taken becomes important. Another way
to say the same thing is to note that the laboratory connection is flat, so,
as we saw in the scalar field problem discussed above, there is no way to
tell whether or not a given physical law should include terms that are
proportional to the curvature. In general, the GWEP and GSEP provide good
initial guesses at the global forms of second-order physical laws, but these
guesses must often be corrected by adding terms that are proportional to the
space-time curvature tensor.
\section{Stress Energy Tensor}
\subsection{Definition and interpretation of the tensor}
We will need to understand the stress-energy tensor shortly, so I will use
it as an example of minimal coupling.
In Special Relativity, the stress-energy tensor takes the form%
\[
\left[ T\right] =\left[
\begin{array}{cccc}
T^{00} & T^{01} & T^{02} & T^{03} \\
T^{10} & T^{11} & T^{12} & T^{13} \\
T^{20} & T^{21} & T^{22} & T^{23} \\
T^{30} & T^{31} & T^{32} & T^{33}%
\end{array}%
\right] =\left[
\begin{array}{cccc}
\rho & j^{1} & j^{2} & j^{3} \\
j^{1} & S^{11} & S^{12} & S^{13} \\
j^{2} & S^{21} & S^{22} & S^{23} \\
j^{3} & S^{31} & S^{32} & S^{33}%
\end{array}%
\right]
\]%
and represents the four-momentum density seen by any observer. For example,
if the observer's four-velocity is $u^{\mu }$ then the observer will see the
four-momentum density%
\[
\frac{dp^{\mu }}{dV}=-T^{\mu }{}_{\nu }u^{\nu },
\]%
where $dV$ stands for a three-dimensional volume element. In particular,
this density is to be integrated over a constant-time surface in the
observer's frame. In terms of components, the energy density becomes the
ratio of the energy $dp^{0}$ contained within a given small region to the
volume $dV$\ of that region.%
\[
\begin{array}{c}
\frac{dp^{0}}{dV}%
=-T^{0}{}_{0}u^{0}-T^{0}{}_{1}u^{1}-T^{0}{}_{2}u^{2}-T^{0}{}_{3}u^{3} \\
=T^{00}u^{0}-T^{01}u^{1}-T^{02}u^{2}-T^{03}u^{3}%
\end{array}%
\]%
or, in terms of more familiar symbols,%
\[
\frac{dE}{dV}=\rho \frac{dt}{d\tau }-\vec{j}\cdot \vec{u}
\]%
If the observer is at rest in the frame chosen to represent the
stress-energy tensor, then $\vec{u}=0$ and $t=\tau $ and the expression
becomes just%
\[
\frac{dE}{dV}=\rho .
\]%
Thus, the energy density is identified as $\rho $. The spatial momentum
density seen by the moving observer is%
\[
\begin{array}{c}
\frac{dp^{i}}{dV}%
=-T^{i}{}_{0}u^{0}-T^{i}{}_{1}u^{1}-T^{i}{}_{2}u^{2}-T^{i}{}_{3}u^{3} \\
=T^{i0}u^{0}-T^{i1}u^{1}-T^{i2}u^{2}-T^{i3}u^{3}%
\end{array}%
\]%
or%
\[
\frac{dp^{i}}{dV}=j^{i}\frac{dt}{d\tau }-S^{ij}u_{j}
\]%
If the observer is at rest in this frame, , then $\vec{u}=0$ and $t=\tau $
and the expression becomes just%
\[
\frac{dp^{i}}{dV}=j^{i}
\]%
so that the terms $j^{i}$ represent the \emph{spatial momentum densities}.
\subsection{Mass-energy conservation}
Now consider the following Lorentz-invariant system of equations:%
\[
T^{\mu \nu }{}_{,\nu }=0
\]%
or, equivalently%
\[
\frac{\partial T^{\mu \nu }}{\partial x^{\nu }}=0.
\]%
The $\mu =0$ equation is familiar when we look at it in detail:%
\[
T^{00}{}_{,0}+T^{0i}{}_{,i}=0
\]%
or, naming the components of the stress-energy,%
\[
\rho _{,0}+j^{i}{}_{,i}=0.
\]%
The equation looks more familiar still in the form%
\[
\dot{\rho}+\nabla \cdot \vec{j}=0
\]%
and is just the mass-energy conservation law.
\subsection{The relativistic generalization of F = ma}
Earlier, it was noted that there is no such thing as a relativistic force
vector because of the constraint $p\cdot p=-c^{2}$ on the four-momentum.
Thus, there is no straightforward generalization of Newton's Third Law in
relativity. Now, however, we can see what takes its place.
If the mass-conservation law works in a reference frame with $e_{0}=u$, then
the $\mu =0$ equation discussed above becomes%
\[
u_{\mu }T^{\mu \nu }{},_{\nu }=0.
\]%
If this equation is to hold for any $u$ so that mass-conservation works for
everyone, then we need
\[
T^{i\nu }{}_{,\nu }=0
\]%
for $i=1,2,3$ \ or%
\[
T^{i0}{}_{,0}+T^{in}{}_{,n}=0
\]%
which becomes%
\[
\dot{j}^{i}=-S^{in}{}_{,n}
\]%
or%
\[
\frac{d}{dt}\frac{dp^{i}}{dV}=-\nabla _{n}S^{in}
\]%
The left-hand side
\[
\frac{d}{dt}\frac{dp^{i}}{dV}=\frac{dp^{i}/dt}{dV}
\]%
is a \emph{force-density} and the right-hand side can be interpreted as the
net force on a volume element if $S$ is the stress tensor. The derivatives
come from taking the stresses on opposite sides of the volume element.
There is still no force vector in general relativity, but the relativistic
generalization of the mass-energy conservation law%
\[
T^{\mu \nu }{}_{,\nu }=0
\]%
also tells us how to use the stress-tensor components as potentials for
finding the force-density that acts on a system. Thus, this law is also the
relativistic generalization of $F=ma$.
\subsection{Perfect fluid as an example}
In a perfect fluid, if one sits in a reference frame that moves with the
fluid, one should see the exact same properties in all directions. A perfect
fluid is said to be \emph{isotropic}. For such a fluid, the stress tensor
must be of the form%
\[
\left[ S\right] =\left[
\begin{array}{ccc}
p & 0 & 0 \\
0 & p & 0 \\
0 & 0 & p%
\end{array}%
\right]
\]%
and the force per unit volume on the fluid will be just the negative
gradient of $p:$%
\[
\frac{d}{dt}\frac{dp^{i}}{dV}=-\nabla _{i}p.
\]%
This result becomes familiar if we identify the function $p$ as the \emph{%
pressure} of the fluid.
\subsection{Minimal Coupling}
The principal of minimal coupling lets us move this entire discussion into
the context of a theory of gravity by replacing the conservation laws%
\[
T^{\mu \nu }{}_{,\nu }=0\qquad \rightarrow \qquad T^{\mu \nu }{}_{;\nu }=0
\]
or, if you prefer%
\[
\partial _\nu T^{\mu \nu }=0\qquad \rightarrow \qquad \nabla _\nu T^{\mu \nu
}=0.
\]
The name ``minimal coupling'' refers to the fact that we make the simplest
and most local possible modifications to the SR equations in order to get a
coordinate-independent version of the law. Because Minkowski space-time is
flat, we could also add terms that are constructed from the space-time
curvature because these would be zero in Special Relativity. Thus, a
non-minimal generalization of the stress-energy conservation law might be%
\[
\nabla _{\nu }T^{\mu \nu }+\nabla _{\nu }T^{\alpha \beta }R^{\mu }{}_{\alpha
\nu \beta }=0
\]%
We do not put such extra terms in because we do not need them to account for
any known physical effects.
There are times when the minimal coupling rule fails to tell us what to do.
The curvature tensor is essentially the commutator of covariant derivatives.
If a theory contains second derivatives, then taking these derivatives in
different orders leads to minimal coupling theories that differ by curvature
terms. Thus, there is a \emph{factor-ordering problem} in this
correspondence rule just as there is in the rules for passing from a
classical to a quantized theory.
\section{Gravitational Field Equations from The GWEP.}
\subsection{The dust cloud}
Suppose that a cloud of dust particles floats near the center of mass of a
space station. This happens to be a doughnut-shaped space station, so there
is only vacuum between the particles. Also suppose that initially, the
particles are not moving. Adopt a spherical coordinate system around the
station center of mass and label particles by angle coordinates and their
distance $s$ from the center of mass. Define the spatial separation vector $%
\vec{w}=\frac{\partial }{\partial s}$. The position vector of a particle at
distance $\delta s$ from the center is then given by
\[
\delta \vec{x}=\delta s\vec{w}
\]%
$.$
\[
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"1";cropright "1";cropbottom "0";filename 'graphics/dustball.gif';file-properties "XNPEU";}}
\]%
The spatial velocity of each particle is $\vec{v}\left( s,\theta ,\varphi
\right) $. The velocity of that particle relative to the center is
\[
\delta \vec{v}=\delta s\frac{\partial \vec{v}}{\partial s}=\delta sD_{\vec{w}%
}\vec{v}
\]%
The acceleration of the dust particle relative to the center is then the
rate of change of the relative velocity or
\[
\delta \vec{a}=\frac{d}{dt}\delta \vec{v}.
\]
\subsection{The Newtonian tidal force matrix}
We analyzed this situation in an example earlier. Newton's Theory of Gravity
would say that there is a gravitational potential $\phi $ and the relative
acceleration in the direction of a unit vector $\hat{n}$ is given by
\[
\hat{n}\cdot \frac{d}{dt}D_{\vec{w}}\vec{v}=-\text{Hess}\phi \left( \hat{n},%
\vec{w}\right)
\]%
or,
\[
\hat{n}\cdot \delta \vec{a}=-\text{Hess}\phi \left( \hat{n},\delta \vec{x}%
\right) .
\]%
This result can be expressed in terms of the Tidal Force Matrix $\left[ A%
\right] $ in the form
\[
\delta \vec{a}=\left[ A\right] \delta \vec{x}
\]%
where, for any two unit vectors $\hat{n},\hat{m}$%
\[
\hat{n}\cdot \left[ A\right] \hat{m}=-\text{Hess}\phi \left( \hat{n},\hat{m}%
\right) .
\]%
Newton's theory says that the gravitational potential $\phi $ obeys a
Poisson equation
\[
\nabla ^{2}\phi =4\pi G\rho
\]%
where $\rho $ is the mass density (of the dust cloud in this case). But
\[
\nabla ^{2}\phi =\text{Tr}\left[ A\right]
\]%
so we get a constraint on the motion of the dust particles:
\[
\text{Tr}\left[ A\right] =-4\pi G\rho .
\]
\subsection{Geodesic deviation tidal force matrix}
The Weak Equivalence Principal tells us that each dust particle should
follow a geodesic. The result of the previous analysis was the geodesic
deviation equation,
\[
\frac{d}{dt}\left( \frac{du}{ds}\right) =\mathcal{R}\left( u,w\right) u
\]
where $u$ is the four-velocity and $w=\vec{w}$ is the same spatial
separation vector as in the Newtonian analysis. Notice that the identity
\[
u\cdot u=-c^{2}
\]
means that $\frac{du}{ds}$ is perpendicular to the four-velocity $u$ so
that, in the reference frame of the space station, this is a purely spatial
vector. Multiplying this vector by the radial separation $\delta s$ gives
the relative velocity vector
\[
\delta \vec{v}=\delta s\frac{du}{ds}
\]
and the relative acceleration vector is then
\[
\delta \vec{a}=\frac{d}{dt}\left( \delta s\frac{du}{ds}\right) =\mathcal{R}%
\left( u,\delta sw\right) u
\]
or
\[
\delta \vec{a}=\mathcal{R}\left( u,\delta \vec{x}\right) u.
\]
The Tidal Force Matrix can now be identified from
\[
\left[ A\right] \delta \vec{x}=\mathcal{R}\left( u,\delta \vec{x}\right) u.
\]
Because Newton's Theory of Gravitation should apply to these slowly moving
dust particles, we expect the corresponding constraint on the trace of $%
\left[ A\right] .$ Thus, we expect
\[
\text{Tr}\left[ A\right] =\omega ^{a}\cdot \left[ A\right] e_{a}=\omega
^{a}\cdot \mathcal{R}\left( u,e_{a}\right) u=-4\pi G\rho
\]%
where the index $a$ is summed from one to three and the basis vectors $e_{a}$
are associated with the rest-frame of the space station. The antisymmetry of
the curvature tensor,
\[
\mathcal{R}\left( h,k\right) =-\mathcal{R}\left( k,h\right)
\]%
and the fact that the four-velocity $u$ is just the timelike basis vector $%
e_{0}$ let us extend the sum to all four space-time indexes without changing
anything:
\[
\omega ^{a}\cdot \mathcal{R}\left( u,e_{a}\right) u=\omega ^{\alpha }\cdot
\mathcal{R}\left( u,e_{\alpha }\right) u.
\]%
In this form we are contracting the curvature tensor and have a new tensor,
called the Ricci curvature
\[
\text{Ricci}\left( h,k\right) =\omega ^{\alpha }\cdot \mathcal{R}\left(
e_{\alpha },h\right) k=-\omega ^{\alpha }\cdot \mathcal{R}\left( u,e_{\alpha
}\right)
\]%
and the constraint becomes
\[
\text{Ricci}\left( u,u\right) =4\pi G\rho .
\]
\subsection{The vacuum field equations}
If the dust cloud has negligible density and floats in a vacuum, then we can
take $\rho =0$ and obtain the constraint
\[
\text{Ricci}\left( u,u\right) =0.
\]%
This constraint is supposed to be a law of nature, so it must hold for all
observers and thus, for any values of the vector $u$. The constraint holds
for the vector $u$ if and only if it holds for the vector $U=\lambda u$
where $\lambda $ can be any non-zero (possibly complex) scale factor. Thus,
we can forget the constraint $u\cdot u=-c^{2}$ and insist that the
constraint hold for \emph{any} vector $U$. If the constraint holds for the
vectors $U,V,$ and $U+V$ then we find that
\[
\text{Ricci}\left( U,V\right) +\text{Ricci}\left( V,U\right) =0
\]%
must hold for any vectors $U,V$. In other words, the symmetric part of the
Ricci tensor must vanish. In index notation,
\[
R_{\left( \mu \nu \right) }=\frac{1}{2}\left( \text{Ricci}\left( e_{\mu
},e_{\nu }\right) +\text{Ricci}\left( e_{\nu },e_{\mu }\right) \right)
\]%
and
\[
\text{Ricci}\left( e_{\mu },e_{\nu }\right) =R^{\sigma }{}_{\mu \sigma \nu }
\]%
so that the curvature of empty space-time is constrained by the equations
\[
R_{\left( \mu \nu \right) }=0.
\]
\subsection{The Coupling to Matter: GWEP Fails}
\subsubsection{First guess at a matter coupling}
The straightforward application of the Grand Weak Equivalence Principal to
the case of a self-gravitating cloud of matter is not obvious. The
constraint
\[
\text{Ricci}\left( u,u\right) =4\pi G\rho
\]%
needs to have a second-rank tensor component on its right-hand side. From
the Special Theory of Relativity we know that the mass-density $\rho $ is
indeed a component of such a tensor, the stress-energy tensor. For a cloud
of dust particles with four-velocity $u$, such as we are considering, the
contravariant stress-energy tensor is just
\[
\check{T}=\rho u\otimes u
\]%
so the covariant tensor is
\[
T=\rho g\left( u\right) \otimes g\left( u\right)
\]%
and
\[
T\left( u,u\right) =\rho \left( u\cdot u\right) \left( u\cdot u\right) =\rho
\]%
so the obvious first guess at the link between matter and space-time
curvature is
\[
\text{Ricci}\left( u,u\right) =4\pi GT\left( u,u\right) .
\]%
For that relation to be true for all observers, we would need a relation of
the form
\[
\text{Ricci}\left( U,V\right) +\text{Ricci}\left( V,U\right) =4\pi G\left(
T\left( U,V\right) +T\left( V,U\right) \right) .
\]%
This relation was actually the first one that Einstein considered.
\subsubsection{Best guess at a matter coupling}
Now try to put matter back into the theory. It is convenient to switch to an
index notation at this point because we will need to use contracted tensors.
Also, I will drop the very general approach of the vacuum case and take both
the torsion and the metricity to be zero in what follows. We want the Ricci
tensor to be equal to some tensor whose $00$ component is proportional to
the mass-density. The first thing that springs to mind is the stress energy
tensor with component $T_{00}=\rho $. However, the components of $T_{\mu \nu
}$ are not all independent and neither are the components of the Ricci
tensor. The stress-energy components satisfy the conservation laws%
\[
\nabla _{\nu }T^{\mu \nu }=0
\]%
while the Ricci tensor obeys the twice contracted Bianchi identities%
\[
\nabla _{\nu }\left( R^{\mu \nu }-\frac{1}{2}Rg^{\mu \nu }\right) =0.
\]%
For these conditions to be consistent, we need%
\[
R^{\mu \nu }-\frac{1}{2}Rg^{\mu \nu }=\kappa T^{\mu \nu }
\]%
for some constant $\kappa $.
The form $B^{\mu \nu }-\frac 12Bg^{\mu \nu }$ is encountered fairly often,
so it is useful to consider its properties. Take the trace of this
expression:%
\[
\begin{array}{c}
g_{\mu \nu }\left( B^{\mu \nu }-\frac 12Bg^{\mu \nu }\right) =B-\frac
12Bg_{\mu \nu }g^{\mu \nu } \\
=B-\frac 12B\delta _\mu ^\mu =B-2B=-B%
\end{array}
\]
The trace has been reversed in sign. Thus, we define%
\[
\bar B^{\mu \nu }=B^{\mu \nu }-\frac 12Bg^{\mu \nu }
\]
and call it the trace-reversed tensor. Obviously reversing the trace twice
gets the tensor back where it started:%
\[
\overline{\bar B}{}^{\mu \nu }=B^{\mu \nu }.
\]
\subsubsection{Matching the Newtonian tidal tensor}
Now return to the proposed field equations:%
\[
\bar R^{\mu \nu }=\kappa T^{\mu \nu }
\]
Trace-reverse both sides of the equation to obtain%
\[
R^{\mu \nu }=\kappa \bar T^{\mu \nu }=\kappa \left( T^{\mu \nu }-\frac
12Tg^{\mu \nu }\right)
\]
so that%
\[
R_{00}=\kappa \left( T_{00}-\frac 12Tg_{00}\right) .
\]
In a local Lorentz frame,
\[
g_{00}=-1
\]
and%
\[
\begin{array}{c}
T=g_{\mu \nu }T^{\mu \nu
}=g_{00}T^{00}+g_{11}T^{11}+g_{22}T^{22}+g_{33}T^{33} \\
=-\rho +S^{11}+S^{22}+S^{33}%
\end{array}
\]
In an isotropic medium, the diagonal components of the stress tensor are all
equal to the pressure. The trace of the stress tensor is therefore $3p$. The
trace of the stress-energy tensor is then%
\[
T=3p-\rho
\]
and the proposed field equation becomes%
\[
\begin{array}{c}
R_{00}=\kappa \left( \rho -\frac 12\left( 3p-\rho \right) \left( -1\right)
\right) \\
=\kappa \left( \rho +\frac 32p-\frac 12\rho \right) \\
=\frac 12\kappa \left( \rho +3p\right)%
\end{array}
\]
For this result to correspond to Newtonian gravity, we need%
\[
\frac{1}{2}\kappa \left( \rho +3p\right) \simeq 4\pi G\rho .
\]%
In the $c=1$ units that we are using, typical values of pressure are far
smaller than typical mass densities, so the correspondence will work if we
choose
\[
\kappa =8\pi G.
\]%
The final result for the Gravitational Field Equations is then%
\[
\bar{R}^{\mu \nu }=8\pi GT^{\mu \nu }.
\]
\subsubsection{A new phenomenon: Pressure attracts}
We have tried to carry over all of the local physics including a local
version of Newton's theory of gravitation. It almost worked, but we were
forced to make one modification in the local physics in order to get a
consistent theory. The local tidal forces act as if the source of the
gravitational potential is not just the mass density $\rho $ as Newton
thought, but the combination $\rho +3p$ of mass density and pressure. For
ordinary objects this turns out to be a negligible modification, so we do
recover Newton's theory of gravitation in those cases where we are sure of
its validity. However, for extremely contracted objects such as collapsing
stars, the pressure terms will become important and will modify the way that
gravity works.
\subsubsection{The physical origin of black holes}
The nature of the modification is easy to see. If pressure creates a
gravitational field in the same way that mass density does, then the
pressure that normally keeps a star from collapsing can, when it becomes
high enough, help to cause a collapse. Beyond a certain level, more pressure
means more gravity which means a faster collapse. Thus, we find the origin
of ``black holes'' embedded in Einstein's field equations from the very
beginning.
\section{The Equation of Motion Problem}
\subsection{The worm bites its tail}
A curious logical problem has always haunted Einstein's Theory of General
Relativity. To recap what has been done so far:
\begin{enumerate}
\item Galileo's understanding of free fall was generalized to the statement
that falling objects have world-lines that are geodesics of a space-time
with a metric-compatible connection.
\item Newton's Theory of Universal Gravitation was generalized to Einstein's
Gravitational Field Equations, which determine the space-time metric from
the stress-energy of matter.
\end{enumerate}
We can take the result of step 2 and use it to show that objects built from
`reasonable' matter must follow geodesics. \ Thus, step 1 almost follows
from step 2. That situation, in itself, would merely be curious but not a
problem. Self-consistency is good.
The problem arises when, after careful consideration, we conclude that there
is no precisely `reasonable' matter and the theory cannot be precisely
self-consistent.
\subsection{Dust}
The very simplest sort of `reasonable' matter would be a cloud of dust. Each
dust particle has a world line with a tangent four-velocity vector $u$, so
we have a field of such vectors all through the world-tube occupied by the
cloud. The stress-energy of the matter in the tube is extremely simple. In
the rest-frame of the cloud, the only non-zero component is $T^{00}=\rho $
so that, in an arbitrary frame, the contravariant stress-energy tensor is%
\[
\breve{T}=\rho u\otimes u
\]%
with components%
\[
T^{\mu \nu }=\rho u^{\mu }u^{\nu }\text{.}
\]%
Einstein's equations then become%
\[
\bar{R}^{\mu \nu }=8\pi G\rho u^{\mu }u^{\nu }.
\]%
Now take the covariant divergence of both sides of this equation%
\[
\bar{R}^{\mu \nu }{}_{;\nu }=8\pi G\left( \rho u^{\mu }u^{\nu }\right)
{}_{;\nu }
\]%
The left side of the resulting equation is identically zero, so we conclude
that%
\[
\left( \rho u^{\mu }u^{\nu }\right) {}_{;\nu }=0
\]%
or%
\[
\rho {}_{;\nu }u^{\mu }u^{\nu }+\rho u^{\mu }{}_{;\nu }u^{\nu }+\rho u^{\mu
}u^{\nu }{}_{;\nu }=0
\]%
which can be rewritten as%
\[
u\left( \nabla _{u}\rho +\rho u^{\nu }{}_{;\nu }\right) +\rho \nabla _{u}u=0.
\]%
Because of the identity $u\cdot u=-1$ and the resulting identity $u\cdot
\nabla _{u}u=0$, we find that there are two consequences of the equation:%
\begin{eqnarray*}
\nabla _{u}\rho +\rho u^{\nu }{}_{;\nu } &=&0 \\
\nabla _{u}u &=&0
\end{eqnarray*}%
The first consequence connects the matter density to the divergence of its
world lines while the second is exactly the geodesic equation.
Thus, a cloud of dust particles can be the source of a gravitational field
only if each individual particle follows a geodesic.
\subsection{Real objects}
For objects such as stars and planets, pressure and stress cause each
individual part of the object to be accelerated relative to geodesic motion.
Under suitable assumptions, one can show that there is still a `central'
geodesic that stays entirely within the world-tube swept out by the object
but it becomes a more difficult task. A very large and old body of research
literature is built around the attempt to define a new kind of `geodesic
center of mass' that is elegant and natural and can also be proven to have
geodesic motion.
To see why it is difficult to construct a geodesic center of mass, it is
useful to skip ahead a bit and consider some consequences of Einstein's
Field Equations. These equations support waves, called gravitational waves
and these waves can carry both energy and momentum away from an object. The
1993 Nobel Prize in Physics was awarded to Russell Hulse and Joseph Taylor
for showing that this process really occurs in a binary pulsar system.
{\tiny (http://astrosun.tn.cornell.edu/courses/astro201/psr1913.htm)}. In
particular, a rotating bar will generate a tiny amount of gravitational
radiation. Now imagine a sequence of such bars rotating on a common axis but
with each one tilted slightly relative to the next.%
\[
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With the correct combination of tilt and frequency, such an array would emit
gravitational waves in just one direction and would, therefore, be a
gravitational wave `rocket'. It would not be a very effective propulsion
device but, given time, it would accelerate relative to any proposed
`geodesic center of mass' of the device.
\subsection{Quantum theory problems}
It has always been known that general relativity does not get along very
well with quantum theory. The difficulty can be seen at many levels,
including the level of equations of motion. For a particle to follow a
well-defined geodesic without spreading due to quantum effects, it must be
massive. One simple requirement is that one can locate the particle to
within a given distance $\Delta x$ of a point by shining light on it and
have no more than a specified fraction of the incident light energy
transferred to the particle. The characteristic length for such a Compton
scattering process is the Compton wavelength $\lambda _{c}=\frac{h}{mc}$ of
the particle. Unfortunately, making $m$ very large causes the particle to
become a significant source of the gravitational field and thus not a very
good test particle for measuring the field. It has been argued (By Bryce
DeWitt in 1962) that there is an optimum size and mass for gravitational
test particles. The optimum mass of about $10^{-4}$gm means that if such
things were made of ordinary matter, they would be large enough to see with
the naked eye!
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The smallest distance that can be measured would be the Compton wavelength
of one of these objects, or about $10^{-32}$cm.
The corresponding test-particle measurement problem for quantum
electrodynamics was solved by Bohr and Rosenfeld in 1930. Their solution
depended on the fact that the charge-to-mass ratio $e/m$ of a particle can
be adjusted to be arbitrarily small. The difficulty with general relativity
is that the Compton wavelength depends on the \emph{inertial mass} of an
object while its effect on the structure of space time depends on its
gravitational `charge' or active mass. Because of the equivalence principle,
the gravitational `charge to mass ratio' cannot be adjusted and is always
equal to exactly 1.
\end{document}
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