Input: How is the function beta(r) determined in the derivation?

Context:
\section{Introduction}
The Schwarzschild solution plays a key role in teaching about general relativity: It describes the simplest version of a black hole. By Birkhoff's theorem, it more generally describes the gravitational field around any spherical mass distribution, such as the Sun in our own Solar system. As one of two particularly simple, yet physically relevant examples of a non-trivial metric (the other being the FLRW spacetime of an expanding universe), it is particularly well-suited for teaching about general techniques of ``reading'' and interpreting a spacetime metric.

Consider undergraduate courses where students are introduced to selected concepts and results from general relativity without exposing them to the full mathematical formalism. Such courses have the advantage of introducing students to one of the two great fundamental theories of 20th century physics early on (the other being quantum mechanics); they also profit from subject matter that meets with considerable interest from students.\cite{Hartle2006} Using the terminology of Christensen and Moore,\cite{Christensen2012} in the ``calculus only'' approach pioneered by Taylor and Wheeler,\cite{Taylor2001,Taylor2018} spacetime metrics are not derived, but taken as given, and the focus is on learning how to interpret a given spacetime metric. Similar presentations can be found in the first part of the ``physics first'' approach exemplified by Hartle's text book,\cite{Hartle2003} where the concepts of the metric and of geodesics are introduced early on, and their physical consequences explored, while the mathematics necessary for the Einstein equations is only introduced at a later stage.

Whenever the approach involves an exploration of simple metrics such as the Schwarzschild solution, but stops short of the formalism required for the full tensorial form of Einstein's equations, access to a simple derivation of the Schwarzschild solution that does not make use of the advanced formalism can be a considerable advantage.

Simplified derivations of the Schwarzschild solution have a long tradition within general relativity education,\cite{Schiff1960,Harwit1973} although specific simplifications have met with criticism.\cite{Rindler1968} This article presents a derivation which requires no deeper knowledge of the formalism of differential geometry beyond an understanding of how to interpret a given spacetime metric $\mathrm{d} s^2$. The derivation avoids the criticism levelled at attempts to derive the Schwarzschild solution from the Einstein equivalence principle in combination with a Newtonian limit,\cite{Gruber1988} relying as it does on a simplified version of the vacuum Einstein equation.

More specifically, I combine the restrictions imposed by the symmetry with the simple form of Einstein's equations formulated by Baez and Bunn.\cite{BaezBunn2005} That same strategy was followed by Kassner in 2017,\cite{Kassner2017} but in this text, I use the ``infalling coordinates'' that are commonly associated with the Gullstrand-Painlev\'e form of the Schwarzschild metric,\cite{Martel2001,Visser2005,HamiltonLisle2008} not the more common Schwarzschild coordinates. That choice simplifies the argument even further. In the end, what is required is no more than the solution of an ordinary differential equation for a single function, which yields to standard methods, to obtain the desired result.

\section{Coordinates adapted to spherical symmetry and staticity}
\label{SymmetriesCoordinates}

Assume that the spacetime we are interested in is spherically symmetric and static. In general relativity, a symmetry amounts to the possibility of being able to choose coordinates that are adapted to the symmetry, at least within a restricted sub-region of the spacetime in question. That the spacetime is static is taken to mean that we can introduce a (non-unique) time coordinate ${t}$ so that our description of spacetime geometry does not depend explicitly on ${t}$, and that space and time are completely separate --- in the coordinates adapted to the symmetry, there are no ``mixed terms'' involving $\mathrm{d} {t}$ times the differential of a space coordinate in the metric. If we use ${t}$ to slice our spacetime into three-dimensional hyperplanes, each corresponding to ``space at time ${t}$,'' then each of those 3-spaces has the same spatial geometry. A mixed term would indicate that those slices of space would need to be shifted relative to another in order to identify corresponding points. The mixed term's absence indicates that in adapted coordinates, there is no need for such an extra shift. In those coordinates, we can talk about the 3-spaces as just ``space,'' without the need for specifying which of the slices we are referring to.

In the case of spherical symmetry, we can introduce spherical coordinates that are adapted to the symmetry: a radial coordinate $r$ and the usual angular coordinates $\vartheta,\varphi$, so that the spherical shell at constant $r$ has the total area $4\pi r^2$. In consequence, the part of our metric involving $\mathrm{d}\vartheta$ and $\mathrm{d}\varphi$ will have the standard form
\begin{equation}
r^2(\mathrm{d}\vartheta^2+\sin^2\theta\mathrm{d}\varphi^2) \equiv r^2\mathrm{d}\Omega^2,
\end{equation}
where the right-hand side defines $\mathrm{d}\Omega^2$, the infinitesimal solid angle corresponding to each particular combination of $\mathrm{d}\vartheta$ and $\mathrm{d}\varphi$.

The radial coordinate slices space into spherical shells, each corresponding to a particular value $r=const.$ The rotations around the origin, which are the symmetry transformations of spherical symmetry, map each of those spherical shells onto itself, and they leave all physical quantities that do not explicitly depend on $\vartheta$ or $\varphi$ invariant.

In what follows, we will use the basic structures introduced in this way --- the slices of simultaneous ${t}$, the radial directions within each slice, the angular coordinates spanning the symmetry--adapted spherical shells of area $4\pi r^2$ --- as auxiliary structures for introducing spacetime coordinates. For now, let us write down the shape that our metric has by simple virtue of the spherical symmetry, the requirement that the spacetime be static, and the adapted coordinates, namely
\begin{equation}
\mathrm{d} s^2 = -c^2F(r) \mathrm{d} {t}^2 + G(r) \mathrm{d} r^2 + r^2\:\mathrm{d}\Omega^2.
\label{StaticForm}
\end{equation}
Students familiar with  ``reading'' a spacetime metric will immediately recognize the sign difference between the parts describing space and describing time that is characteristic for spacetime, and the speed of light $c$ that gives us the correct physical dimensions. That there is no explicit dependence on $\varphi$ and $\vartheta$ in the remaining functions $F$ and $G$ is a direct consequence of spherical symmetry. That the factor in front of $\mathrm{d}\Omega^2$ is $r^2$ is a consequence of our coordinate choice, with spherical angular coordinates so that the area of a spherical surface of constant radius $r$ is $4\pi r^2$. That there is no explicit dependence on ${t}$ is one consequence of the spacetime being static; the absence of the mixed term $\mathrm{d} {t}\cdot \mathrm{d} r$ is another. We are left with two unknown functions $F(r)$ and $G(r)$. In the following, let us call ${t}$ and $r$ the {\em static coordinates}.

Note that, since $G(r)$ is as yet undefined, we have not yet chosen a specific physical meaning for the length measurements associated with our $r$ coordinate. But because of the $\mathrm{d}\Omega^2$ part, it is clear that whatever choice we make, the locally orthogonal lengths $r\cdot\mathrm{d}\vartheta$ and $r\cdot\sin\vartheta\cdot\mathrm{d}\varphi$ will have the same physical interpretation as for the length measurement corresponding to $\mathrm{d} r$.

\section{Infalling observer coordinates}
\label{Sec:InfallingObservers}

Now that we know what the radial directions are, at each moment of time ${t}$, we follow Visser\cite{Visser2005} as well as Hamilton and Lisle\cite{HamiltonLisle2008} in defining a family of radially infalling observers.  Observers in that family are in free fall along the radial direction, starting out at rest at infinity: In mapping each observer's radial progression in terms of the static coordinate time ${t}$, we adjust initial conditions, specifically: the choice of initial speed at some fixed time ${t}$, in just the right way that the radial coordinate speed goes to zero for each observer in the same way as $r\to\infty.$

It is true that talking about ``infalling'' observers already reflects our expectation that our solution should describe the spacetime of a spherically symmetric mass.  As we know from the Newtonian limit, such a mass attracts test particles in its vicinity. It should be noted, though, that all our calculations would also be compatible with the limit of no mass being present. In that case, ``infalling'' would be a misnomer, as our family of observers would merely hover in empty space at unchanging positions in $r$.

We can imagine infinitesimal local coordinate systems associated with our observers --- think of the observer mapping out space and time by defining three orthogonal axes, and by measuring time with a co-moving clock. We assume all such little coordinate systems to be non-rotating --- otherwise, we would break spherical symmetry, since rotation would locally pick out a plane of rotation that is distinguishable from the other planes. The radial direction is a natural choice for the first space axis of those little free-falling systems. The other directions, we take to point to observers falling side by side with our coordinate-defining observer --- and to remain pointed at a specific such other observer, once the choice of direction is made.

We assume our infalling observers' clocks to be synchronised at some fixed radius value $r$. By spherical symmetry, those clocks should then be synchronised at {\em all} values of $r$. Anything else would indicate direction-dependent differences for the infalling observers and their clocks, after all. Hence, at any given static time ${t}$, all the infalling observers who are at radius value $r$ show the same proper time $T$ on the ideal clocks travelling along with them.

Once our definition is complete, our static, spherically symmetric spacetime is filled with infalling observers from that family: Whenever we consider an event $\cal E$, there will be an observer from that family passing by at that time, at that location.

Now, consider the coordinate speed of those infalling observers. If we position ourselves at some constant radius value $r$ and watch the falling observers fly by, then we can express both their proper time rate and their coordinate speed in the $r$ direction in terms of $r$ and ${t}$. We can combine the two pieces of information to obtain the rate of change in radial position $r$ with proper time $T$ for those infalling observers. But since the initial conditions for those observers are the same, and since our spacetime is, by assumption, static, the resulting function can only depend on $r$, and not explicitly on ${t}$. Let us rescale that function with the speed of light to make it dimensionless, give it an overall minus sign to make it positive for infalling particles, and call it $\beta(r)$,
\begin{equation}
\beta(r)\equiv -\frac{1}{c}\frac{\mathrm{d} r}{\mathrm{d} T}(r).
\label{betaDefinition}
\end{equation}

Recall from section \ref{SymmetriesCoordinates} that we also still have the freedom to decide on the physical meaning of $r$. We make the choice of making $\mathrm{d} r$ the physical length measured by one of our infalling observers at the relevant location in spacetime, at constant time $T$. Via our angular coordinates, that implies that length measurements orthogonal to the radial direction, $r\cdot\mathrm{d}\vartheta$ and $r\cdot\sin\vartheta\:\mathrm{d}\varphi$ inherit the same physical interpretation.

As a next step, we transform our metric (\ref{StaticForm}) from the static form into the form appropriate for our coordinate choice $r$ and $T$. We do so by writing the static time coordinate as a function ${t}(T,r)$ in terms of infalling observer time and radius value. In consequence,
\begin{equation}
\mathrm{d} {t} = \frac{\partial{t}}{\partial T}\cdot\mathrm{d} T+ \frac{\partial {t}}{\partial r}\cdot\mathrm{d} r,
\end{equation}
and our new metric now has the form
\begin{align}
 \mathrm{d} s^2 = {} & -c^2 F(r)\left(\frac{\partial t}{\partial T}\right)^2\mathrm{d} T^2 \nonumber \\[0.2em]
 & -2c^2F(r)\left(\frac{\partial t}{\partial T}\right)\left(\frac{\partial t}{\partial r}\right)\mathrm{d} T\:\mathrm{d} r \nonumber \\[0.2em]
 & +\left[G(r)-c^2F(r)\left(\frac{\partial t}{\partial r}\right)^2\right]\mathrm{d} r^2+r^2\:\mathrm{d}\Omega^2.
 \end{align}
At face value, this looks like we are moving the wrong way, away from simplification, since we now have more functions, and they depend on two variables instead of one.

But in fact, this new formulation paves the way for an even simpler form of the metric. Consider a specific event, which happens at given radius value $r$. In a small region around that event, we will introduce a new coordinate $\bar{r}$ to parametrize the radial direction. We want this coordinate to be co-moving with our infalling observers at $r$; each such observer then has a position $\bar{r}=const.$ that does not change over time.

Key to our next step is that we {\em know} the metric for the local length and time measurements made by any one of our free-falling observers. By Einstein's equivalence principle, the metric is that of special relativity. Locally, namely whenever tidal effects can be neglected, spacetime geometry for any non-rotating observer in free fall is indistinguishable from Minkowski spacetime as described by a local inertial system.

Since we have chosen both the time coordinate $T$ and the physical meaning of the radial coordinate $r$ so as to conform with the measurements of the local infalling observer, the transformation between $\bar{r}$ and $r$ is particularly simple: It has the form of a Galilei transformation
\begin{equation}
\mathrm{d}\bar{r}= \mathrm{d} r + \beta(r)c\:\mathrm{d} T.
\label{barRshift}
\end{equation}
In that way, as it should be by definition, radial coordinate differences at constant $T$ are the same in both systems, while for an observer at constant $\bar{r},$ with $\mathrm{d} \bar{r}=0$, the relation between $\mathrm{d} r$ and $\mathrm{d} T$ is consistent with the definition of the function $\beta(r)$ in (\ref{betaDefinition}).

Are you surprised that this is not a Lorentz transformation, as one might expect from special relativity? Don't be. We are not transforming from one local inertial coordinate system to another. The $T$ is already the time coordinate of the infalling observers, so both coordinate systems have the same definition of simultaneity, and time dilation plays no role in this particular transformation. Also, we have chosen $r$ intervals to correspond to length measurements of the infalling observers, so there is no Lorentz contraction, either. It is the consequence of these special choices that gives the relation (\ref{barRshift}) its simple form.

Last but not least, when we analyse specifically an infinitesimal neighbourhood of the point $r,\vartheta,\varphi$, let us make the choice that directly at our point of interest, we make $\bar{r}$ coincide with $r$. Since before, we had only fixed the differential $\mathrm{d} \bar{r}$, we do have the remaining freedom of choosing a constant offset for $\bar{r}$ that yields the desired result.

By Einstein's equivalence principle, the metric in terms of the locally co-moving coordinates $T,\bar{r},\vartheta,\varphi$ is the spherical-coordinate version of the Minkowski metric,
\begin{equation}
\mathrm{d} s^2 = -c^2\mathrm{d} T^2 + \mathrm{d}\bar{r}^2 + \bar{r}^2\mathrm{d}\Omega.
\end{equation}
This version can, of course, be obtained by taking the more familiar Cartesian-coordinate version
\begin{equation}
\mathrm{d} s^2=-c^2\mathrm{d} T^2 + \mathrm{d} X^2 + \mathrm{d} Y^2 + \mathrm{d} Z^2,
\label{CartesianMinkowski}
\end{equation}
applying the definition of Cartesian coordinates $X,Y,Z$ in terms of spherical coordinates $\bar{r},\vartheta,\varphi$
\begin{equation}
x= \bar{r}\:\sin\vartheta\:\cos\varphi, \;\;
y= \bar{r}\:\sin\vartheta\:\sin\varphi, \;\;
z= \bar{r}\:\cos\vartheta,
\end{equation}
to express $\mathrm{d} X, \mathrm{d} Y, \mathrm{d} Z$ in terms of $\mathrm{d} \bar{r}, \mathrm{d}\vartheta, \mathrm{d}\varphi$, and substitute the result into (\ref{CartesianMinkowski}).

By noting that we have chosen $\bar{r}$ so that, at the specific spacetime event where we are evaluating the metric, $\bar{r}=r$, while, for small radial coordinate shifts around that location, we have the relation (\ref{barRshift}), we can now write down the same metric in the coordinates $T, r, \vartheta,\varphi$, namely as
\begin{equation}
\mathrm{d} s^2 = -c^2\left[
1-\beta(r)^2
\right] \mathrm{d} T^2+2c\beta(r)\mathrm{d} r\:\mathrm{d} T
+\mathrm{d} r^2+r^2\mathrm{d}\Omega^2.
\label{preMetric}
\end{equation}
Since we can repeat that local procedure at any event in our spacetime, this result is our general form of the metric, for all values of $r$. This, then is the promised simplification: By exploiting the symmetries of our solutions as well as the properties of infalling observers, we have reduced our metric to a simple form with no more than one unknown function of one variable, namely $\beta(r)$.

So far, what I have presented is no more than a long-form version of the initial steps of the derivation given by Visser in his heuristic derivation of the Schwarzschild metric.\cite{Visser2005} In the next section, we will deviate from Visser's derivation.

\section{$\beta(r)$ from tidal deformations}
\label{TidalSection}

In the previous section, we had exploited symmetries and Einstein's equivalence principle. In order to determine $\beta(r)$, we need to bring in additional information, namely the Einstein equations, which link the matter content with the geometry of spacetime. For our solution, we only aim to describe the spacetime metric outside whatever spherically-symmetric matter distribution resides in (or around) the center of our spherical symmetry. That amounts to applying the {\em vacuum Einstein equations}.

More specifically, we use a particularly simple and intuitive form of the vacuum Einstein equations, which can be found in a seminal article by Baez and Bunn:\cite{BaezBunn2005} Consider a locally flat free-fall system around a specific event $\cal E$, with a time coordinate $\tau$, local proper time, where the event we are studying corresponds to $\tau=0$. In that system, describe a small sphere of freely floating test particles, which we shall call a {\em test ball}. The particles need to be at rest relative to each other at $\tau=0$. Let the volume of the test ball be $V(\tau)$. Then the vacuum version of Einstein's equations states that
\begin{equation}
\left.\frac{\mathrm{d}^2 V}{\mathrm{d}\tau^2}\right|_{\tau=0} = 0.
\label{EinsteinVacuum}
\end{equation}
In words: If there is no matter or energy inside, the volume of such a test ball remains constant in the first order (those were our initial conditions) and the second order (by eq.~[\ref{EinsteinVacuum}]).

If you are familiar with Wheeler's brief summary of Einstein's equations, ``spacetime grips mass, telling it how to move'' and ``mass grips spacetime, telling it how to curve'',\cite{Wheeler1990} you will immediately recognise that this is a specific way for the structure of spacetime telling the test ball particles how to move. The calculation later in this section provides the second part: It will amount to using (\ref{EinsteinVacuum}) to determine the structure of spacetime, namely the still missing function $\beta(r)$, and that is the way for mass, in this case: for the absence of mass, to tell spacetime how to curve.

Note that equation (\ref{EinsteinVacuum}) also holds true in Newtonian gravity. So in a way, this version of Einstein's equation can be seen as a second-order extension of the usual Einstein equivalence principle: Ordinarily, the equivalence principle is a statement about physics in the absence of tidal forces. Equation (\ref{EinsteinVacuum}) adds to this that the lowest-order correction for tidal forces in a freely falling reference frame is that specified by Newtonian gravity. This makes sense, since by going into a free-fall frame, and restricting our attention to a small spacetime region, we have automatically created a weak-gravity situation. In such a situation, tidal corrections are approximately the same as those described by Newton. This argument can serve as a heuristic justification of (\ref{EinsteinVacuum}).

In 2017, Kassner made use of the Baez-Bunn form of Einstein's vacuum equation to derive the Schwarzschild solution, starting from what we have encountered as the static form of the metric (\ref{StaticForm}).\cite{Kassner2017} We follow the same general recipe, but using the infalling coordinates introduced in section \ref{Sec:InfallingObservers}, which makes our derivation even simpler.

Consider five test particles in a small region of space. Let the motion of each be the same as for the local representative from our coordinate-defining family of infalling observers. We take the central particle $C$ to be at radial coordinate value $r=R$ at the time of the snapshot shown in Fig.~\ref{TestParticlesOutside}. The other four are offset relative to the central particle: As described in the local inertial system that is co-moving with the central particle, one of the particles is shifted by $\Delta l$ upwards in the radial direction, another downward, while two of the particles are offset orthogonally by the same distance.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.5\linewidth]{01-free-fall-particles.pdf}
\caption{Five test particles in our spherically-symmetric spacetime}
\label{TestParticlesOutside}
\end{center}
\end{figure}
The $\Delta l$ is meant to be infinitesimally small, so while Fig.~\ref{TestParticlesOutside} is of course showing a rather large $\Delta l$ so as to display the geometry of the situation more clearly, we will in the following only keep terms linear in $\Delta l$.

Consider a generic particle, which moves as if it were part of our coordinate-defining family of infalling observers, and which at the time $T_0$ is at $r=r_0$. By a Taylor expansion, that particle's subsequent movement is given by
\begin{equation}
r(T) = r_0 + \frac{\mathrm{d} r}{\mathrm{d} T}(T_0) \cdot \Delta T +\frac12 \frac{\mathrm{d}^2 r}{\mathrm{d} T^2}(T_0) \cdot \Delta T^2
\label{TaylorREvo}
\end{equation}
where $\Delta T\equiv T-T_0$. We know from (\ref{betaDefinition}) that the derivative in the linear term can be expressed in terms of $\beta(r)$; by the same token,
\begin{equation}
\frac{\mathrm{d}^2 r}{\mathrm{d} T^2} = -c\frac{\mathrm{d}\beta}{\mathrm{d} T}=-c\beta' \frac{\mathrm{d} r}{\mathrm{d} T} = c^2\beta\cdot\beta',
\end{equation}
where the prime denotes differentiation of $\beta$ with respect to its argument. Since, in the following, the product of $\beta$ and its first derivative will occur quite often, let us introduce the abbreviation
\begin{equation}
B(r) \equiv \beta(r)\cdot\beta'(r).
\label{BigBDefinition}
\end{equation}
With these results, can rewrite the Taylor expansion (\ref{TaylorREvo}) as
\begin{equation}
r(T) = r_0 -c\beta(r_0)\cdot\Delta T + \frac12 c^2B(r_0)\cdot\Delta T^2.
\label{RadialOrbitTime}
\end{equation}
In order to find $r_C(T)$ for our central particle, we simply insert $r_0=R$ into that expression. If, on the other hand, we want to write down the time evolution for particles $U$ and $D$, let us denote it by $r_{U,D}(T)$, we need to evaluate the expression (\ref{RadialOrbitTime}) at the initial location $r_0=R\pm\Delta l$. Since $\Delta l$ is small, we can make a Taylor expansion of $\beta(r)$ and its derivative around $r=R$, and neglect everything beyond the terms linear in $\Delta l$. The result is
\begin{multline}
r_{U,D}(T)=R \pm\Delta l-c\left[
\beta(R)\pm\beta'(R)\Delta l
\right]\Delta T \\[0.2em]
+\frac{c^2}{2}\big[
B(R)\pm B'(R)\Delta l
\big]\Delta T^2
\end{multline}
In consequence, the distance between the upper and lower particle, $d_{\parallel}(T)\equiv r_U(T)-r_D(T),$ changes over time as
\begin{equation}
d_{\parallel}(T) =  2\Delta l\left[
1-c\beta'(R)\Delta T+\frac12c^2 B'(R)\Delta T^2
\right].
\label{dParallel}
\end{equation}
Next, let us look at how the distance between the particles $L$ and $R$ changes over time. The initial radial coordinate value for each of the particles is
\begin{equation}
r(T_0) = \sqrt{R^2+\Delta l^2}=R\left[1+\frac12\left(\frac{\Delta l}{R}\right)^2\right]\approx R,
\end{equation}
that is, equal to $R,$ as long as we neglect any terms that are higher than linear in $\Delta l$. In consequence, $r_{L,R}(t)$ is the same function as for our central particle, given by eq.~(\ref{RadialOrbitTime}) with $r_0=R$. The transversal (in Fig.~\ref{TestParticlesOutside}: horizontal) distance $d_{\perp}(T)$ between the particles $L$ and $R$ changes in proportion to the radius value,
\begin{align}
d_{\perp}(T) &= 2\Delta l\cdot\frac{r_{L}(T)}{R} \nonumber \\
                 &=2\Delta \left[1-\frac{c\beta(R)}{R}\Delta T+\frac{c^2}{2}\frac{B(R)}{R}\Delta T^2\right].
                \label{dPerp}
\end{align}
With these preparations, consider the vacuum Einstein equation (\ref{EinsteinVacuum}) for the volume of a test ball. Initially, our particles $C, U, D, L, R$ define a circle, which is deformed to an ellipse. By demanding rotational symmetry around the radial direction, we can construct the associated ellipsoid, which is initially a spherical surface. That ellipsoid has one axis in radial direction, whose length is $d_{\parallel}(T)$, and two axes that are transversal and each have the length  $d_{\perp}(t)$. But that ellipsoid is not quite yet the test ball we need. After all, the particles of the test ball need to be at rest initially, at time $T_0$, in the co-moving system defined by the central particle $C$. Our defining particles are not, as the terms linear in $\Delta T$ in both (\ref{dParallel}) and (\ref{dPerp}) show, where the coefficients of $\Delta T$ correspond to the particles' initial velocities.

In order to define our test ball, we need to consider particles at the same location, undergoing the same acceleration, but which are initially at rest relative to the central particle $C$.

We could go back to the drawing board, back to Fig.~\ref{TestParticlesOutside}, make a more general Ansatz that includes initial velocities which measure the divergence of the motion of our test ball particles from that of the infalling-observer particles, and repeat our calculation while including those additional velocity terms. But there is a short-cut. The only consequence of those additional velocity terms will be to change the terms linear in $\Delta T$ in equations (\ref{dParallel}) and (\ref{dPerp}). And we already know the end result: We will choose the additional terms so as to cancel the terms linear in $\Delta T$ in the current versions of (\ref{dParallel}) and (\ref{dPerp}). But by that reasoning, we can skip the explicit steps in between, and write down the final result right away. The time evolution of the radial-direction diameter of our test ball, let us call it $L_{\parallel}(T)$, must be the same as $d_{\parallel}(T)$, but without the term linear in $\Delta T$. Likewise, the time evolution $L_{\perp}(T)$ of the two transversal diameters must be equal to $d_{\perp}(T)$, but again without the term linear in $\Delta T$. The result is
\begin{align}
L_{\parallel}(T)  &=  2\Delta l \left[1+\frac12c^2B'(R)\Delta T^2\right] \\
L_{\perp}(T) &= 2\Delta l \left[1+\frac{c^2}{2}\frac{B(R)}{R}\Delta T^2\right].
\end{align}
Thus, our test ball volume is
\begin{align}
V(T) &= \frac{\pi}{6}L_{\parallel}(T) L_{\perp}^2(T) \\
       &= \left.\frac{4\pi}{3}\Delta l^3\left[1+{c^2}\left( \frac{B(r)}{r} + \frac{B'(r)}{2}\right)\Delta T^2\right]\right|_{r=R}
\end{align}
For the second time derivative of $V(T)$ to vanish at the time $T=T_0$, we must have
\begin{equation}
\frac{B(r)}{r} + \frac{B'(r)}{2}= 0
\label{VolumeConditionR}
\end{equation}
for all values of $r$. This is readily solved by the standard method of separation of variables: We can rewrite (\ref{VolumeConditionR}) as
\begin{equation}
\frac{\mathrm{d} B}{B} = -2\frac{\mathrm{d} r}{r},
\end{equation}
which is readily integrated to give
\begin{equation}
\ln(B) = -\ln(r^{2}) + const.  \;\; \Rightarrow \;\; \ln(Br^2) = C',
\end{equation}
with a constant $C'$, which upon taking the exponential gives us
\begin{equation}
Br^2= C,
\label{BSolution}
\end{equation}
with a constant $C$. Note that the constant $C$ can be negative --- there is no reason the constant $C'$ needs to be real; only our eventual function $B(r)$ needs to be that, and it is clear that (\ref{BSolution}) satisfies the differential equation
(\ref{VolumeConditionR}) for any constant $C$, positive, zero, or negative. By (\ref{BigBDefinition}), the solution (\ref{BSolution}) corresponds to the differential equation
\begin{equation}
\beta(r)\beta'(r) = \frac{C}{r^2}
\end{equation}
for our function $\beta$; with another separation of variables, we can re-write this as
\begin{equation}
\beta\cdot\mathrm{d}\beta=C\frac{\mathrm{d} r}{r^2}.
\end{equation}
Both sides are readily integrated up; we can solve the result for $\beta(r)$ and obtain
\begin{equation}
\beta(r) = \sqrt{
-\frac{2C}{r} +2D
},
\end{equation}
where $D$ is the second integration constant, and where we have chosen the proper sign, since we know that $\beta(r)>0$. That brings us to the last step: The requirement that, for large values of $r$, the description provided by our solution should correspond to the results from Newtonian gravity. First of all, we note that our initial condition for the infalling observers, which had those observers start out at zero speed at infinity, means that we must choose $D=0$. Then, as we would expect, $\beta(r)$ for large values of $r$ becomes very small, corresponding to small speeds. But at slow speeds, time and length intervals as measured by the infalling observer will become arbitrarily close to time and length intervals as measured by an observer at rest in our static coordinate system at constant $r$, using the static time coordinate ${t}$. As is usual, we identify these coordinates with those of an approximately Newtonian description. In that description, the radial velocity is
\begin{equation}
v(r) = \sqrt{\frac{2GM}{r}},
\end{equation}
which follows directly from energy conservation for the sum of each observer's kinetic and Newtonian-gravitational potential energy. This fixes the remaining integration constant as
\begin{equation}
C = -\frac{GM}{c^2},
\end{equation}
and the final form of our function $\beta(r)$ becomes
\begin{equation}
\beta(r) = \sqrt{\frac{2GM}{rc^2}}.
\end{equation}
Inserting this result in (\ref{preMetric}), we obtain the metric
\begin{equation}
\mathrm{d} s^2 = -c^2\left[
1-\frac{2GM}{rc^2}
\right]\mathrm{d} T^2+2\sqrt{\frac{2GM}{r}}\mathrm{d} r\:\mathrm{d} T+\mathrm{d} r^2+r^2\mathrm{d}\Omega^2.
\label{GPMetric}
\end{equation}
This is known as the Gullstrand-Painlev\'e version of the Schwarzschild metric.\cite{Martel2001,Visser2005,HamiltonLisle2008} A  last transformation step brings us back to the traditional Schwarzschild form. Recall our discussion in sec.~\ref{SymmetriesCoordinates}, leading up to the explicitly static form (\ref{StaticForm}) of the metric? The main difference between our current form and the static version is the mixed term containing $\mathrm{d} r\:\mathrm{d} T$ in (\ref{GPMetric}). Everything else already has the required shape. Inserting the Ansatz
\begin{equation}
\mathrm{d} T = \mathrm{d} t + \xi(r) \mathrm{d} r
\end{equation}
into the metric (\ref{GPMetric}), it is straightforward to see that the mixed term vanishes iff our transformation is
\begin{equation}
\mathrm{d} T = \mathrm{d} t +\frac{\sqrt{2GM/r}}{c^2\left(1-\frac{2GM}{rc^2}\right)}\mathrm{d} r.
\label{TtTrafo}
\end{equation}
Substitute this into (\ref{GPMetric}), and the result is the familiar form of the Schwarzschild metric in Schwarzschild's original coordinates $t,r,\vartheta,\varphi$,
\begin{equation}
\mathrm{d} s^2 = -c^2\left(1-\frac{2GM}{c^2 r}
\right)\mathrm{d} t^2 + \frac{\mathrm{d} r^2}{\left(1-\frac{2GM}{c^2 r}
\right)} + r^2\mathrm{d}\Omega^2.
\end{equation}

\section{Conclusion}
Using coordinates adapted to the symmetries, we were able to write down the spherically symmetric, static spacetime metric. On this basis, and using the family of infalling observers that is characteristic for the Gullstrand-Painlev\'e solution, we wrote down the metric in the form (\ref{preMetric}), with a single unknown function $\beta(r)$. From the simplified form (\ref{EinsteinVacuum}) of the vacuum Einstein equations, as applied to a test ball in free fall alongside one of our family of observers, we were able to determine $\beta(r)$, up to two integration constants. By using the Einstein equation, we escape the restrictions imposed on simplified derivations by Gruber et al.\cite{Gruber1988}

From the initial condition for our infalling observers, as well as from the Newtonian limit at large distances from our center of symmetry, we were able to fix the values of the two intergration constants. Our derivation does not require knowledge of advanced mathematical concepts beyond the ability to properly interpret a given metric line element $\mathrm{d} s^2$. Even our analysis of tidal effects proceeds via a simple second-order Taylor expansion, leading to differential equations for $\beta(r)$ that are readily solved using two applications of the method of separation of variables.

What is new about the derivation presented here is the combination of the Baez-Bunn equations with the infalling coordinates typical for the Gullstrand-Painlev\'e form of the metric --- this combination is what, in the end, makes our derivation particularly simple. In turn, this simplicity is what should make the derivation particularly useful in the context of teaching general relativity in an undergraduate setting.

The derivation proceeds close to the physics, and gives ample opportunity to discuss interesting properties of Einstein's theory of gravity. Students who are presented with this derivation, either as a demonstration or as a (guided) exercise, will come to understand the way that symmetries determine the form of a metric, the deductions that can be made from Einstein's equivalence principle, and last but not least that we need to go beyond the equivalence principle, and consider tidal forces, to completely define our solution.

\section*{Acknowledgements}

I would like to thank Thomas M\"uller for helpful comments on an earlier version of this text.

Answer: Using the vacuum Einstein equation and the Baez-Bunn form.


Input: What is Professor Tulis's forthcoming book?

Context:
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next profile Jeffrey Tulis Associate Professor — Ph.D.,
E-mail: tulis@austin.utexas.edu
Office: MEZ 3.152
Political Theory and American Politics
Professor Tulis's interests bridge the fields of political theory and American politics, including more specifically, American political development, constitutional theory, political philosophy and the American presidency. His publications include The Presidency in the Constitutional Order (LSU, 1981; Transaction, 2010), The Rhetorical Presidency (Princeton, 1987), The Constitutional Presidency (Johns Hopkins 2009), The Limits of Constitutional Democracy (Princeton, 2010) and recent journal articles and chapters on constitutional interpretation, the logic of political change, and the meaning of political success. Four collections of essays on The Rhetorical Presidency with responses by Tulis have been published, including a special double issue of Critical Review: An Interdisciplinary Journal of Politics and Society, (2007), where his book is described as "one of the two or three most important and perceptive works written by a political scientist in the twentieth century."
He has served as President of the Politics and History Section of the American Political Science Association. He received the President's Associates Teaching Excellence Award at the University of Texas. He has held research fellowships from NEH, ACLS, Olin Foundation, Harvard Law School, and the Mellon Preceptorship at Princeton University, where he taught before moving to Texas. He has held visiting positions at Notre Dame and Harvard. He has served as associate chair of the Department of Government from 1989-2001 and was acting chair during 1992-93. and for part of each year between 1989 and 2001. During the academic year 2008-09, he was a Laurance S. Rockefeller Visiting Fellow at the University Center for Human Values at Princeton. During Spring 2016, he was a Dahrendorf Visiting Fellow at the London School of Economics and Political Science.
His forthcoming books include: Legacies of Losing in American Politics, with Nicole Mellow (University of Chicago Press, Fall 2017), and an expanded edition of The Rhetorical Presidency in the Princeton Classics series (Princeton, Fall 2017). For two decades he served as co-editor of the Johns Hopkins Series in Constitutional Thought, and he currently co-edits (with Sanford Levinson) Constitutional Thinking, a Series at the University Press of Kansas.
GOV 370L • Pres In Constitutional Ord 38840 • Spring 2017 Meets MW 2:30PM-4:00PM CAL 221 show description
GOV 370 Seminar: The Presidency in the Constitutional Order
Spring 2017 Unique # 38840
MW 2:30 to 4pm GDC 2.402
Jeffrey K. Tulis
In this Seminar we will discuss a series of constitutional problems including: the problem of executive energy in the American Constitution; presidential selection and the problem of political legitimacy; separation of powers; delegation of powers, the constitutional status of war and foreign affairs, administration and bureaucracy and the meaning of leadership in the constitutional order.
Seminar will meet twice a week and regular attendance and thorough preparation for discussion is expected. Unexcused absence from more than three classes will result in failure of the participation component of the course. There will also be pop quizzes on the reading that will count as part of your participation grade. In addition to class participation, course requirements include four short analytic essays, and one in-class test. The course grade will be calculated as follows:
Seminar participation: 20%
In-class test: 20%
Three analytic essays 60% (20% each)
Class participation is especially important. Preparation for seminar and for your in-class test will be enhanced by careful note taking on the readings. If students appear to be unprepared, pop quizzes will be given and the grades on them will affect the participation component of your course grade.
Texts: (tentative)
Joseph M. Bessette and Jeffrey K. Tulis, The Constitutional Presidency
Michael Nelson, The Presidency in the Political System (tenth edition)
Richard Ellis and Michael Nelson, Debating the Presidency (third edition)
The Federalist (any edition, or online) GOV 310L • American Government-Honors 38335 • Fall 2016 Meets TTH 3:30PM-5:00PM BEN 1.106 show description
GOV 310 (Honors) (38335) Fall 2016
TTH 3:30-5:00pm, BEN 1.106
This honors seminar offers an introduction to American politics that emphasizes the confluence of ideas, mores, institutions, and interests, in the constitutional system. This course covers more theory, and the readings are more demanding, than other versions of GOV 310. One of the main objectives of the course is to deepen your understanding of the practical aspects of contemporary public affairs by developing your ability to understand the theoretical foundations of American politics. Although we cover the nuts and bolts of politics there is much more theory in this version of GOV 310. If you have registered for this section mainly because 310 is a legislative requirement that you need to fulfill, this is not the right version for you. There is a substantial workload in this class.
Regular attendance, thorough and timely preparation, and active participation are all necessary to do well.
Four essays (approximately 1000 words each). Three of these will be assigned analytic essay topics. The last will be a book review of a title chosen by the student from a long list of provided possibilities. (15% each essay, 60% of total course grade)
Two in-class tests. These will count 15% each, 30% of total course grade.
Class participation. (10% of course grade). Both informed participation and occasional leadership of the seminar will be graded.
No make-up exams or late papers, except for documented medical or other emergencies.
Mark Landy and Sidney M. Milkis, American Government: Enduring Principles, Critical Choices, Third Edition
Mary Nichols and David Nichols, Readings in American Government, Ninth Edition
Thomas Mann and Norman Ornstein, Its Even Worse Than It Looks: How the American Constitutional System Collided With the New Politics of Extremism
Bruce Ackerman,Before the Next Attack: Preserving Civil Liberties in an Age of Terrorism GOV 381L • Constitutional Conflict 38660 • Fall 2016 Meets W 3:30PM-6:30PM BAT 5.102 show description
GOV 381L Fall 2016
Constitutional Conflict
W 3:30-6:30pm, BAT 5.102
Many of the most important debates regarding the nature and character of contemporary American politics are essentially arguments regarding the structure of separation of powers. In this seminar we will consider such questions as whether the American system is prone to deadlock of stalemate in the construction of national policy; whether conflict is a hindrance to institutional responsibility or an essential attribute of responsibility; whether there are “political questions” especially suitable to resolution between President and Congress; how one can distinguish salutary from pathological conflict, and whether it is truly possible to harness the ambition of office holders to the duties of their office.
More specifically, we will review literature and arguments regarding constitutional reform; divided government; separation of powers theory; and case studies of Supreme Court appointments; the budget process; and war powers and foreign affairs. In these contexts we will also discuss current controversies surrounding war authorization, intelligence and secrecy, sequestration, government shut downs and budget resolutions, and debt ceiling politics.
The course is designed to accommodate two different student needs: it will provide a good overview of important literature relevant to the comprehensive examination in American politics and it will provide opportunities for research. This subject area is a treasure trove of “hot” topics, publication possibilities, subjects for MA theses and Ph.D. dissertations. I will tailor the written requirements to the objectives of individual students.
1. All students will prepare a short analytic essay early in the semester, and an annotated bibliography at mid-semester. These assignments will count (30%) of the grade.
2. Students interested primarily in exam preparation will complete an examination near the end of the semester based on study questions assigned in advance. OR
Students interested in research will write a 20-25 page paper. (60%)
3. A basic requirement of the course is that students prepare for each seminar by carefully reading the material assigned for that week. Class discussion is an essential component of the course. (10%)
Tentative Texts:
Jones, Separate But Equal Branches
Silverstein, Imbalance of Powers
Wilson & Schram, Separation of Powers and Good Government
Burgess, Contest for Constitutional Authority
Farrier, Passing the Buck: Congress, the Budget and Deficits
Weissman, A Culture of Deference
Zeisberg, War Powers: The Politics of Constitutional Authority
Fisher, Congressional Abdication on War and Spending
Lowi, The End of Liberalism GOV 379S • Regime Persp Amer Poltc-Honors 38105 • Spring 2016 Meets TH 3:30PM-6:30PM GAR 1.134 (also listed as CTI 335, LAH 350) show description
GOV 379S Regime Perspectives on American Politics
This is a seminar on American politics and culture. Two purposes govern the selection of texts for the course and guide our discussion of them. All of our texts attempt to look at American politics as a whole. Most books and courses on America look at only a part, such as the Presidency, or elections, or popular culture. Here we attempt to think about how the parts of America fit together. Even when these texts speak about a part, for example an institution such as the presidency or the Congress, they present the topic from a vantage point on the whole polity. To see the polity as a whole also means that we will have to revisit and rethink aspects of our political life that we take for granted – that we don’t examine because those parts have become so natural or familiar to us. Seeing the polity whole enables us to render the familiar unfamiliar, to make what we take for granted strange and new.
To see the polity as a whole requires that we get some distance from our subject, much as to see the planet earth as a whole requires one to look at it from outer space. Just as it is difficult to get visual perspective on a place living within it, it is difficult to understand the promise or pathologies of a regime from within. To get critical distance from our politics, we will closely study three sets of texts that look at American politics from a distance. The first part of the course will recover the perspective of the founding debate between Federalists and Anti-federalists. This fundamental debate reveals what is a stake in the basic architecture of the American regime. The second part of the course is a close study of Tocqueville’s Democracy in America. Regarded by many as the best book ever written on democracy and the best book written on America, Tocqueville sees our polity whole because he looks at it from the vantage point of Europe, in general, and France, in particular. In the third part of the seminar we think about American politics from the perspective of thoughtful commentators who feel only nominally included in the polity. Half in and half out, these extraordinary black American writers reveal fissures and fault lines in the American regime. We end the class with a discussion of America’s place in the world today – examining a speech by a writer who articulately raises challenges to our self-understanding that are inarticulately expressed today in rage and ranting from enemies of the United States.
Three take home analytic essays, chosen from a list of topics I provide, each weighted 25% of the course grade. Late essays will not be accepted, except with a doctor’s excuse or a Dean’s excuse for family emergency.
OR as an option: you may write the two short essays (both together weighted 25%) and do a longer 15 page paper on a topic of your choice in consultation with me (weighted 50% of your course grade). Government honors students who are thinking of doing an honors thesis next year may prefer this option to begin to develop research and writing skills for longer work. Students who prefer this option will need to designate their preferred third short essay and have discussed with me a topic for their long paper by March 30. Texts:
Selected Anti-Federalist writings
Tocqueville, Democracy in America
Essays, speeches and articles by Frederick Douglass, W.E.B. Dubois, Booker T. Washington, James Baldwin and Ralph Ellison GOV 382M • Democratic Theory 38120 • Spring 2016 Meets M 3:30PM-6:30PM BAT 1.104 show description
GOV 382M (38120)
Democratic Theory Spring 2016
This is a graduate seminar on contemporary topics in democratic theory. Topics to be covered include: democratic epistemology; deliberative democracy; the meaning of the people; oracular democracy; agonistic democracy; and possibly new theories of republicanism, representation and partisanship.
Texts (tentative)
Helene Landemore, Democratic Reason
Jeffrey Edward Green, The Eyes of the People
Amy Gutmann and Dennis Thompson, Why Deliberative Democracy?
Alan Keenan, Democracy in Question
Jason Frank, Constituent Moments
Jason Frank, Publius and Political Imagination
Nadia Urbanati, Democracy Disfigured
Russell Muirhead, Partisanship in a Polarized Age
Bryan Garsten, manuscript
Active seminar participation; an annotated bibliography or review essay; a research/analytic paper. GOV 310L • American Government-Honors 37615 • Fall 2015 Meets TTH 2:00PM-3:30PM BEN 1.106 show description
TTH 2-3:30/BEN 1.106
Bruce Ackerman,Before the Next Attack: Preserving Civil Liberties in an Age of Terrorism GOV 370L • Presidency In Constitutl Order 37845 • Fall 2015 Meets TTH 5:00PM-6:30PM PAR 310 show description
GOV 370L (37845)
TTH 5-6:30 PAR 310
The Presidency in the Constitutional Order
A study of the place of the presidency in the American political order that stresses tension between power and accountability inherent in the office and the system. Topics include: separation of powers, presidential selection, impeachment, relations with Congress and bureaucracy, emergency powers, presidential character, and leadership.
This is a very demanding writing flag class. If you are enrolling in this class just in order to satisfy the writing flag, you are in the wrong class. Interest in political theory and willingness to work very hard are necessary for success in this class.
Joseph M. Bessette, The Constitutional Presidency
Andrew Rudalevige, The New Imperial Presidency
Bruce Ackerman, The Rise and Decline of the American Republic
Michael Nelson, ed., The Presidency in the Political System
Michael Nelson, ed., The Evolving Presidency
Louis Fisher, Constitutional Conflicts Between Congress and the President
Active and prepared class participation
Regular quizzes on the reading
Four analytic essays (approximately 1200 words).
One term paper, (approximately 5000 words). GOV 379S • Regime Persp On Amer Politics 38100 • Spring 2015 Meets T 3:30PM-6:30PM MEZ 1.104 (also listed as LAH 350) show description
Essays, speeches and articles by Frederick Douglass, W.E.B. Dubois, Booker T. Washington, James Baldwin and Ralph Ellison GOV 382M • Tocqueville 38135 • Spring 2015 Meets M 3:30PM-6:30PM BAT 5.102 show description
This graduate seminar will be devoted to close readings of two principal writings of Tocqueville: Democracy in America and The Ancien Regime and the Revolution. We will also assess some of the best secondary studies of Tocqueville, including work by Sheldon Wolin, Harvey Mansfield, Delba Winthrop, Jon Elster, Francois Furet, and a book by Pierre Manent.
Course requirements will include two very short analytic essays and one seminar paper (20-25 pages). GOV 310L • American Government-Honors 38722 • Fall 2014 Meets TTH 2:00PM-3:30PM GAR 2.112 show description
Joseph M. Bessette and John J. Pitney, American Government and Politics: Deliberation, Democracy and Citizenship
Mary Nichols and David Nichols, Readings in American Government
Bruce Ackerman,Before the Next Attack: Preserving Civil Liberties in an Age of Terrorism GOV 370L • Presidency In Constitutl Order 38977 • Fall 2014 Meets TTH 9:30AM-11:00AM CBA 4.332 show description
A study of the place of the presidency in the American political order that stresses
tension between power and accountability inherent in the office and the system.
Topics include: separation of powers, presidential selection, impeachment,
relations with Congress and bureaucracy, emergency powers, presidential
character, and leadership.
This is a very demanding writing flag class. If you are enrolling in this class just in order
to satisfy the writing flag, you are in the wrong class. Interest in political theory and willingness
to work very hard are necessary for success in this class.
One term paper, (approximately 5000 words). GOV 379S • Regime Persp On Amer Politics 39395 • Spring 2014 Meets T 3:30PM-6:30PM MEZ 1.104 (also listed as CTI 335, LAH 350) show description
Essays, speeches and articles by Frederick Douglass, W.E.B. Dubois, Booker T. Washington, James Baldwin and Ralph Ellison GOV 381L • Constitutional Conflict 39415 • Spring 2014 Meets M 3:30PM-6:30PM BAT 1.104 show description
Lowi, The End of Liberalism GOV 330K • The American President 39140 • Fall 2013 Meets MW 3:00PM-4:30PM MEZ B0.306 show description
This course offers an over view of the place of the presidency in the American political order. Topics covered include: constitutional design of the office; nominations and elections; legislative leadership; leadership of the bureaucracy; staffing and organizing the White House; the presidency and the judiciary; war and emergencies. We will spend extra time this fall on the presidential campaign and election of 2012.
Two in-class examinations (50% of the final grade)
One short (1000 word) take-home essay (30% of the final grade)
Class participation and quizzes (20% of the final grade)
Richard J. Ellis, The Development of the American Presidency (Routledge, 2012)
Richard J. Ellis and Michael Nelson, eds, Debating the American Presidency, (2nd edition, CQ Press, 2009)
Packet of selected primary texts (to be linked or posted on Blackboard). GOV 330K • The American President 39145 • Fall 2013 Meets MW 5:00PM-6:30PM MEZ B0.306 show description
Packet of selected primary texts (to be linked or posted on Blackboard). GOV 381L • American Founding 39040 • Spring 2013 Meets T 6:30PM-9:30PM BAT 1.104 show description
NOTE WELL: Course meets Tuesdays, 6:30 to 9:30pm
Batts Hall 1.104
This is a seminar on American political thought and constitutional design. It is designed for students of American politics and political theory. The principal themes include: 1) the nature of founding and its constitutive significance; 2) the relation of structure and power in American politics; 3) the meaning and significance of the Federalist/Anti-Federalist debate; 4) the philosophic background of the American founding; and 5) the relevance of the founding to debate to prospects for, and pathologies of, American politics today.
We will conduct a close reading of the Madison’s Notes, of The Federalist, and selected Anti-Federalist writings. We will also study a larger and growing body of secondary literature on the constitutional convention, ratification and early American political thought.
James Madison, Notes of the Debates: In the Federal Convention of 1787
The Federalist (Rossiter, ed.)
The Anti-Federalist (Storing, ed.)
David Brian Robertson, The Constitution and America’s Destiny (2005)
Pauline Maier, Ratification (2012)
Gordon Wood, The Idea of America (2011)
Jack Rakove, Original Meanings: Politics & Ideas in the Making of the Constitution
Herbert Storing, What the Anti-Federalists Were For (1981)
Numerous essays and articles (to be posted on line or gathered in packet)
Grading: Active seminar participation, including three short papers and presentations (40%) and one article-length seminar paper (60%) T C 357 • Amer Founding/Probs Const Des 43095 • Spring 2013 Meets M 3:30PM-6:30PM CRD 007B show description
The American Founding and Problems of Constitutional Design
Jeffrey Tulis, Associate Professor, Department of Government
Sanford Levinson, Professor, School of Law
This Plan II seminar will be built around a close reading of the debates that informed the drafting and ratification of the U.S. Constitution. We aim to recover the perspective of these founding thinkers -- their way of thinking -- as much as their concrete ideas, in order to raise fundamental questions about the American political order today. Are some of the most important pathologies of American politics today rooted in design features of our original political architecture? Are the original answers to basic founding questions (such as "how democratic is our Constitution?) still adequate for contemporary circumstances? What features of the Constitution should we preserve and what features should we amend, if possible? Would it be good for the polity as a whole to reconsider these questions in a new constitutional convention today, or would such an event be a political nightmare? Our reading will include notes from the founding conventions, writings by Federalists and Anti-Federalists, and present-day critiques of the American political order. Our aim will be to generate a dialogue between the thought of the founders and some of the best present day critics and supporters of the Constitution.
James Madison, Notes of the Debates in the Federal Convention
The Federalist, ed. Clinton Rossiter
The Anti-Federalist, ed. Herbert Storing
Pauline Maier, Ratification: The People Debate the Constitution, 1787-1788
Sanford Levinson, Framed: America’s 51 Constitutions and the Crisis of Governance
Bruce Ackerman, The Decline and Fall of the American Republic
Robert Goldwin, ed. How Democratic is the Constitution?
a course packet of selected articles, essays, and additional primary materials.
Class participation, including at least one presentation of a short discussion paper 25%
One take-home analytic essay 25%
One term paper 50%
About the Professors:
Professor Tulis's interests bridge the fields of political theory and American politics, including more specifically, American political development, constitutional theory, political philosophy and the American presidency. He received the President's Associates Teaching Excellence Award at the University of Texas. He has held research fellowships from NEH, ACLS, Olin Foundation, Harvard Law School, and the Mellon Preceptorship at Princeton University, where he taught before moving to Texas. He has held visiting positions at Notre Dame and Harvard. During the academic year 2008-09, he was a Laurance S. Rockefeller Visiting Fellow at the University Center for Human Values at Princeton.
Proefessor Levinson holds the W. St. John Garwood and W. St. John Garwood, Jr. Centennial Chair in Law, he joined the University of Texas Law School in 1980. Previously a member of the Department of Politics at Princeton University, he is also a Professor in the Department of Government at the University of Texas. The author of over 350 articles and book reviews in professional and popular journals--and a regular contributor to the popular blog Balkinization. He received the Lifetime Achievement Award from the Law and Courts Section of the American Political Science Association in 2010. He has been a visiting faculty member of the Boston University, Georgetown, Harvard, New York University, and Yale law schools in the United States and has taught abroad in programs of law in London; Paris; Jerusalem; Auckland, New Zealand; and Melbourne, Australia.
GOV 330K • The American President 38675 • Fall 2012 Meets MW 3:00PM-4:30PM MEZ B0.306 show description
Packet of selected primary texts (to be linked or posted on Blackboard). GOV 330K • The American President 38675 • Fall 2011 Meets MW 3:30PM-5:00PM WAG 420 show description
see syllabus GOV 330K • The American President 38680 • Fall 2011 Meets MW 5:30PM-7:00PM UTC 1.146 show description
see syllabus GOV 379S • Regime Persp On Amer Polit-Hon 39110 • Spring 2011 Meets W 3:30PM-6:30PM BAT 5.102 (also listed as CTI 326, LAH 350) show description
To see the polity as a whole requires that we get some distance from our subject, much as to see the planet earth as a whole requires one to look at it from outer space. Just as it is difficult to get visual perspective on a place living within it, it is difficult to understand the promise or pathologies of a regime from within it. To get critical distance from our politics, we will closely study three sets of texts that look at American politics from a distance. The first part of the course will recover the perspective of the founding debate between Federalists and Anti-federalists. This fundamental debate reveals what is a stake in the basic architecture of the American regime. The second part of the course is a close study of Tocqueville’s Democracy in America. Regarded by many as the best book ever written on democracy and the best book written on America, Tocqueville sees our polity whole because he looks at it from the vantage point of Europe, in general, and France, in particular. In the third part of the seminar we think about American politics from the perspective of thoughtful commentators who feel only nominally included in the polity. Half in and half out, these extraordinary black American writers reveal fissures and fault lines in the American regime. We end the class with a discussion of America’s place in the world today – examining a speech by a writer who articulately raises challenges to our self-understanding that are inarticulately expressed today in rage and ranting from enemies of the United States.
Four take home writing assignments. Analytic essays, each 1000-1500 words. (Grades weighted: 10%, 25%, 25%, and 25%) Late essays will not be accepted, except with a doctor’s excuse or a Dean’s excuse for family emergency. Regular preparation and class participation: 15%.
OR as an option: By prior arrangement with me by the due date of the second analytic essay, students may substitute one longer research paper (15 – 20 pages) for two of the last three analytic papers This paper will be on a topic of the students choosing , if I approve, and the due date will be the same as the last assigned analytic essay. This project would count 50% of the students course grade.
Selected writings by Frederick Douglass, W.E.B. Dubois, Ralph Ellison, James Baldwin
Solzhenitsyn, “A World Split Apart”
Tocqueville, Democracy in America GOV 382M • Tocqueville 39150 • Spring 2011 Meets T 6:30PM-9:30PM BAT 5.102 show description
See syllabus GOV 370L • President, Congress, And Court 38695 • Fall 2010 Meets TTH 8:00AM-9:30AM UTC 3.112 show description
Course Description: A Study of the political relationship of the President, Congress and Court in the American constitutional order. Has this relationship changed over the course of American history? Is American national politics prone to stalemate or deadlock between the branches regarding major issues of public policy? Do we have a new “imperial presidency?” Should the Court arbitrate disputes between the President and Congress over custody of their respective powers? Has Congress abdicated its constitutional responsibilities? We will examine questions like these in light of practical problems such as executive privilege and secrecy, the war on terror, budget politics and controversies regarding appointments to the Supreme Court. Grading:Three in class essay tests, for which study questions will be distributed in advance. The exam questions will be chosen from the list of study questions. (25% each) One short take home essay (10% each). Class participation and attendance (15%). Tentative Texts: The FederalistFisher, Congressional Abdication on War and SpendingRudalevige, The New Imperial PresidencyBessette and Tulis, The Constitutional PresidencySkowronek, Presidency in Political TimeGoldsmith, The Terror PresidencyA course packet of articles and essays GOV 370L • President, Congress, And Court 38700 • Fall 2010 Meets TTH 5:00PM-6:30PM UTC 3.122 show description
Course Description: A Study of the political relationship of the President, Congress and Court in the American constitutional order. Has this relationship changed over the course of American history? Is American national politics prone to stalemate or deadlock between the branches regarding major issues of public policy? Do we have a new “imperial presidency?” Should the Court arbitrate disputes between the President and Congress over custody of their respective powers? Has Congress abdicated its constitutional responsibilities? We will examine questions like these in light of practical problems such as executive privilege and secrecy, the war on terror, budget politics and controversies regarding appointments to the Supreme Court. Grading:Three in class essay tests, for which study questions will be distributed in advance. The exam questions will be chosen from the list of study questions. (25% each) One short take home essay (10% each). Class participation and attendance (15%). Tentative Texts: The FederalistFisher, Congressional Abdication on War and SpendingRudalevige, The New Imperial PresidencyBessette and Tulis, The Constitutional PresidencySkowronek, Presidency in Political TimeGoldsmith, The Terror PresidencyA course packet of articles and essays GOV 312L • Iss & Policies In Amer Gov-Hon 38698 • Spring 2010 Meets MW 3:30PM-5:00PM UTC 3.104 show description
Government 312L satisfies the second half of the mandated six hours of government that every UT student must take. Course covers analysis of varying topics concerned with American political institutions and policies, including the United States Constitution, and assumes basic knowledge of government from GOV 310L, which is a prerequiste. May be taken for credit only once. GOV 370L • President, Congress, And Court 38966 • Spring 2010 Meets MW 5:00PM-6:30PM MEZ B0.306 show description
Prerequisite: Six semester hours of lower-division coursework in government.
GOV 370L • President, Congress, And Court 39295 • Fall 2009 Meets TTH 2:00PM-3:30PM UTC 3.112 show description
GOV 370L • President, Congress, And Court 39435 • Spring 2008 Meets MW 3:00PM-4:30PM PAR 203 show description
GOV 312L • Iss & Policies In Am Gov-Hon-W 38615-38620 • Spring 2007 Meets MW 11:00AM-12:00PM MEZ B0.306 show description
Government 312L satisfies the second half of the mandated six hours of government that every UT student must take. Course covers analysis of varying topics concerned with American political institutions and policies, including the United States Constitution, and assumes basic knowledge of government from GOV 310L, which is a prerequiste. May be taken for credit only once. GOV 312L • Iss & Policies In Am Gov-Hon-W 37600-37605 • Spring 2006 Meets MW 11:00AM-12:00PM MEZ B0.306 show description
Government 312L satisfies the second half of the mandated six hours of government that every UT student must take. Course covers analysis of varying topics concerned with American political institutions and policies, including the United States Constitution, and assumes basic knowledge of government from GOV 310L, which is a prerequiste. May be taken for credit only once. GOV 312L • Iss & Policies In Am Gov-Hon-W 34900-34905 • Spring 2004 Meets MW 11:00AM-12:00PM BUR 134 show description
Government 312L satisfies the second half of the mandated six hours of government that every UT student must take. Course covers analysis of varying topics concerned with American political institutions and policies, including the United States Constitution, and assumes basic knowledge of government from GOV 310L, which is a prerequiste. May be taken for credit only once. GOV 312L • Iss & Policies In Am Gov-Hon-W 34495-34500 • Spring 2003 Meets MW 11:00AM-12:00PM UTC 1.130 show description
Government 312L satisfies the second half of the mandated six hours of government that every UT student must take. Course covers analysis of varying topics concerned with American political institutions and policies, including the United States Constitution, and assumes basic knowledge of government from GOV 310L, which is a prerequiste. May be taken for credit only once. Publications
Tulis, JK (2011), "Plausible Futures," in Dunn, Charles W. (ed.) The Presidency in the Twenty-First Century, University Press of Kentucky.Tulis, J.K. and Macedo, S. (2010) The Limits of Constitutional Democracy, Princeton University Press.Tulis, J.K. and Macedo, S. (2010) "Constitutional Boundaries," in The Limits of Constitutional Democracy, Princeton University Press.Tulis, JK (2010), "The Possibility of Constitutional Statesmanship," in Tulis, JK and Macedo, S (eds.) The Limits of Constitutional Democracy, Princeton University Press.Tulis, J. (2009) The Constitutional Presidency. Johns Hopkins University Press.Tulis, J. (2009) Impeachment in the Constitutional Order. In J. Tulis & J.M. Bessette (Eds.), The Constitutional Presidency. Johns Hopkins University Press.Tulis, J. & Bessette, J.M. (2009) On the Constitution, Politics, and the Presidency. In J. Tulis & J.M. Bessette (Eds.), The Constitutional Presidency. Johns Hopkins University Press.Tulis, J (and Bessette, J.M) (2010) The Presidency in the Constitutional Order: Historical Perspectives, Reissued Classics Series, Transaction Publishers,Tulis, J and Bessette, J.M. (2010, "Introduction to the Transaction Edition," The Presidency in the Constitutional Order: Historical Perspectives, Transaction Publishers.
Tulis, JK, (2009) "The Two Constitutional Presidencies," in Nelson, Michael (ed.) The Presidency in the Political System, Congressional Quarterly Press.Tulis, J. & Mellow, N. (2007) Andrew Johnson and the Politics of Failure. In S. Skowronek & M. Glassman (Eds.), Formative Acts: Reckoning with Agency in American Politics. Philadelphia: University of Pennsylvania Press.Tulis, J. (2007, September) The Rhetorical Presidency in Retrospect. Critical Review: An Interdisciplinary Journal of Politics and Society, 19(2&3). Curriculum Vitae

Answer: Legacies of Losing in American Politics and an expanded edition of The Rhetorical Presidency in the Princeton Classics series.