我尝试修复运行头中最近的章节标记。但章节标记在彩色盒子环境。下面是我的最低工作命令。在此处输入图片描述
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\chapter{Chapter title on running head}
In the previous chapter, we looked at the Cauchy problem using a
special coordinate system. In this chapter, we will look at a more
geometrical approach to the problem. We will start by looking at the
way in which a hypersurface $\Sigma$ is embedded in a space-time, and
the geometric data that one needs to reconstruct the curvature of the
space-time from this. The key quantities are the \textbf{induced
metric} and the \textbf{extrinsic curvature} of the hypersurface
$\Sigma$.
We then go on to look at the way in which one can introduce
a time function $t$ and use this to slice up the space-time $M$ by a
family of spacelike hypersurfaces, or \textbf{foliation},
$\Sigma_t$. The geometry of the foliation can be described in terms of
the \textbf{lapse} function, and the extrinsic curvature can be given
an alternative description in terms of the rate of change of the
induced metric. The final ingredient is to introduce a timelike vector
field, or \textbf{fibration}, whose integral curves may be used to
identify points on neighbouring, $t={\rm constant}$,
hypersurfaces. The geometry of the fibration can then be defined in
terms of the \textbf{shift} vector.
The lapse and shift are freely
specifiable and encode the fourfold coordinate freedom in describing
the geometry. This machinery allows one to view general relativity
from a different perspective as a dynamical theory in which Einstein's
equations are encoded in a pair of first-order differential equations
which describe the way in which the dynamical variables, the induced
metric, and the extrinsic curvature, evolve with the time function $t$. A
knowledge of the induced metric and the lapse and shift may be used to
reconstruct the space-time metric and hence the geometry of the
space-time. A slightly different form of these equations, called the
\textbf{ADM formalism}, was derived by Arnowitt, Deser, and Misner (1959) from
their Hamiltonian formulation of general relativity and formed the
basis for what Wheeler (1963) called \textbf{geometrodynamics}. In
\S\ref{S13B.13} we look at an alternative approach called the
\textbf{$2+2$ formalism}, in which one decomposes space-time into two
families of spacelike 2-surfaces. This approach identifies the
gravitational degrees of freedom in a geometrically transparent way
and is also particularly suited to situations where the initial data is
given on a null surface.
\section{A heal head on running head}\label{S13B.2}
Let $\Sigma$ be a smooth spacelike hypersurface. Then the unit normal
$n^a$ to $\Sigma$ is timelike, may be taken to be future pointing, and
satisfies
In the previous chapter, we looked at the Cauchy problem using a
special coordinate system. In this chapter, we will look at a more
geometrical approach to the problem. We will start by looking at the
way in which a hypersurface $\Sigma$ is embedded in a space-time, and
the geometric data that one needs to reconstruct the curvature of the
space-time from this. The key quantities are the \textbf{induced
metric} and the \textbf{extrinsic curvature} of the hypersurface
$\Sigma$.
We then go on to look at the way in which one can introduce
a time function $t$ and use this to slice up the space-time $M$ by a
family of spacelike hypersurfaces, or \textbf{foliation},
$\Sigma_t$. The geometry of the foliation can be described in terms of
the \textbf{lapse} function, and the extrinsic curvature can be given
an alternative description in terms of the rate of change of the
induced metric. The final ingredient is to introduce a timelike vector
field, or \textbf{fibration}, whose integral curves may be used to
identify points on neighbouring, $t={\rm constant}$,
hypersurfaces. The geometry of the fibration can then be defined in
terms of the \textbf{shift} vector.
The lapse and shift are freely
specifiable and encode the fourfold coordinate freedom in describing
the geometry. This machinery allows one to view general relativity
from a different perspective as a dynamical theory in which Einstein's
equations are encoded in a pair of first-order differential equations
which describe the way in which the dynamical variables, the induced
metric, and the extrinsic curvature, evolve with the time function $t$. A
knowledge of the induced metric and the lapse and shift may be used to
reconstruct the space-time metric and hence the geometry of the
space-time. A slightly different form of these equations, called the
\textbf{ADM formalism}, was derived by Arnowitt, Deser, and Misner (1959) from
their Hamiltonian formulation of general relativity and formed the
basis for what Wheeler (1963) called \textbf{geometrodynamics}. In
\S\ref{S13B.13} we look at an alternative approach called the
\textbf{$2+2$ formalism}, in which one decomposes space-time into two
families of spacelike 2-surfaces. This approach identifies the
gravitational degrees of freedom in a geometrically transparent way
and is also particularly suited to situations where the initial data is
given on a null surface.
In the previous chapter, we looked at the Cauchy problem using a
special coordinate system. In this chapter, we will look at a more
geometrical approach to the problem. We will start by looking at the
way in which a hypersurface $\Sigma$ is embedded in a space-time, and
the geometric data that one needs to reconstruct the curvature of the
space-time from this. The key quantities are the \textbf{induced
metric} and the \textbf{extrinsic curvature} of the hypersurface
$\Sigma$.
We then go on to look at the way in which one can introduce
a time function $t$ and use this to slice up the space-time $M$ by a
family of spacelike hypersurfaces, or \textbf{foliation},
$\Sigma_t$. The geometry of the foliation can be described in terms of
the \textbf{lapse} function, and the extrinsic curvature can be given
an alternative description in terms of the rate of change of the
induced metric. The final ingredient is to introduce a timelike vector
field, or \textbf{fibration}, whose integral curves may be used to
identify points on neighbouring, $t={\rm constant}$,
hypersurfaces. The geometry of the fibration can then be defined in
terms of the \textbf{shift} vector.
\section{Second A heal head on running head}\label{S13B.22}
The lapse and shift are freely
specifiable and encode the fourfold coordinate freedom in describing
the geometry. This machinery allows one to view general relativity
from a different perspective as a dynamical theory in which Einstein's
equations are encoded in a pair of first-order differential equations
which describe the way in which the dynamical variables, the induced
metric, and the extrinsic curvature, evolve with the time function $t$. A
knowledge of the induced metric and the lapse and shift may be used to
reconstruct the space-time metric and hence the geometry of the
space-time. A slightly different form of these equations, called the
\textbf{ADM formalism}, was derived by Arnowitt, Deser, and Misner (1959) from
their Hamiltonian formulation of general relativity and formed the
basis for what Wheeler (1963) called \textbf{geometrodynamics}. In
\S\ref{S13B.13} we look at an alternative approach called the
\textbf{$2+2$ formalism}, in which one decomposes space-time into two
families of spacelike 2-surfaces. This approach identifies the
gravitational degrees of freedom in a geometrically transparent way
and is also particularly suited to situations where the initial data is
given on a null surface.
In the previous chapter, we looked at the Cauchy problem using a
special coordinate system. In this chapter, we will look at a more
geometrical approach to the problem. We will start by looking at the
way in which a hypersurface $\Sigma$ is embedded in a space-time, and
the geometric data that one needs to reconstruct the curvature of the
space-time from this. The key quantities are the \textbf{induced
metric} and the \textbf{extrinsic curvature} of the hypersurface
$\Sigma$.
We then go on to look at the way in which one can introduce
a time function $t$ and use this to slice up the space-time $M$ by a
family of spacelike hypersurfaces, or \textbf{foliation},
$\Sigma_t$. The geometry of the foliation can be described in terms of
the \textbf{lapse} function, and the extrinsic curvature can be given
an alternative description in terms of the rate of change of the
induced metric. The final ingredient is to introduce a timelike vector
field, or \textbf{fibration}, whose integral curves may be used to
identify points on neighbouring, $t={\rm constant}$,
hypersurfaces. The geometry of the fibration can then be defined in
terms of the \textbf{shift} vector.
The lapse and shift are freely
specifiable and encode the fourfold coordinate freedom in describing
the geometry. This machinery allows one to view general relativity
from a different perspective as a dynamical theory in which Einstein's
equations are encoded in a pair of first-order differential equations
which describe the way in which the dynamical variables, the induced
metric, and the extrinsic curvature, evolve with the time function $t$. A
knowledge of the induced metric and the lapse and shift may be used to
reconstruct the space-time metric and hence the geometry of the
space-time. A slightly different form of these equations, called the
\textbf{ADM formalism}, was derived by Arnowitt, Deser, and Misner (1959) from
their Hamiltonian formulation of general relativity and formed the
basis for what Wheeler (1963) called \textbf{geometrodynamics}. In
\S\ref{S13B.13} we look at an alternative approach called the
\textbf{$2+2$ formalism}, in which one decomposes space-time into two
families of spacelike 2-surfaces. This approach identifies the
gravitational degrees of freedom in a geometrically transparent way
and is also particularly suited to situations where the initial data is
given on a null surface.
In the previous chapter, we looked at the Cauchy problem using a
special coordinate system. In this chapter, we will look at a more
geometrical approach to the problem. We will start by looking at the
way in which a hypersurface $\Sigma$ is embedded in a space-time, and
the geometric data that one needs to reconstruct the curvature of the
space-time from this. The key quantities are the \textbf{induced
metric} and the \textbf{extrinsic curvature} of the hypersurface
$\Sigma$.
We then go on to look at the way in which one can introduce
a time function $t$ and use this to slice up the space-time $M$ by a
family of spacelike hypersurfaces, or \textbf{foliation},
$\Sigma_t$. The geometry of the foliation can be described in terms of
the \textbf{lapse} function, and the extrinsic curvature can be given
an alternative description in terms of the rate of change of the
induced metric. The final ingredient is to introduce a timelike vector
field, or \textbf{fibration}, whose integral curves may be used to
identify points on neighbouring, $t={\rm constant}$,
hypersurfaces. The geometry of the fibration can then be defined in
terms of the \textbf{shift} vector.
The lapse and shift are freely
specifiable and encode the fourfold coordinate freedom in describing
the geometry. This machinery allows one to view general relativity
from a different perspective as a dynamical theory in which Einstein's
equations are encoded in a pair of first-order differential equations
which describe the way in which the dynamical variables, the induced
metric, and the extrinsic curvature, evolve with the time function $t$. A
knowledge of the induced metric and the lapse and shift may be used to
reconstruct the space-time metric and hence the geometry of the
space-time. A slightly different form of these equations, called the
\textbf{ADM formalism}, was derived by Arnowitt, Deser, and Misner (1959) from
their Hamiltonian formulation of general relativity and formed the
basis for what Wheeler (1963) called \textbf{geometrodynamics}. In
\S\ref{S13B.13} we look at an alternative approach called the
\textbf{$2+2$ formalism}, in which one decomposes space-time into two
families of spacelike 2-surfaces. This approach identifies the
gravitational degrees of freedom in a geometrically transparent way
and is also particularly suited to situations where the initial data is
given on a null surface.
In the previous chapter, we looked at the Cauchy problem using a
special coordinate system. In this chapter, we will look at a more
geometrical approach to the problem. We will start by looking at the
way in which a hypersurface $\Sigma$ is embedded in a space-time, and
the geometric data that one needs to reconstruct the curvature of the
space-time from this. The key quantities are the \textbf{induced
metric} and the \textbf{extrinsic curvature} of the hypersurface
$\Sigma$.
We then go on to look at the way in which one can introduce
a time function $t$ and use this to slice up the space-time $M$ by a
family of spacelike hypersurfaces, or \textbf{foliation},
$\Sigma_t$. The geometry of the foliation can be described in terms of
the \textbf{lapse} function, and the extrinsic curvature can be given
an alternative description in terms of the rate of change of the
induced metric. The final ingredient is to introduce a timelike vector
field, or \textbf{fibration}, whose integral curves may be used to
identify points on neighbouring, $t={\rm constant}$,
hypersurfaces. The geometry of the fibration can then be defined in
terms of the \textbf{shift} vector.
The lapse and shift are freely
specifiable and encode the fourfold coordinate freedom in describing
the geometry. This machinery allows one to view general relativity
from a different perspective as a dynamical theory in which Einstein's
equations are encoded in a pair of first-order differential equations
which describe the way in which the dynamical variables, the induced
metric, and the extrinsic curvature, evolve with the time function $t$. A
knowledge of the induced metric and the lapse and shift may be used to
reconstruct the space-time metric and hence the geometry of the
space-time. A slightly different form of these equations, called the
\textbf{ADM formalism}, was derived by Arnowitt, Deser, and Misner (1959) from
their Hamiltonian formulation of general relativity and formed the
basis for what Wheeler (1963) called \textbf{geometrodynamics}. In
\S\ref{S13B.13} we look at an alternative approach called the
\textbf{$2+2$ formalism}, in which one decomposes space-time into two
families of spacelike 2-surfaces. This approach identifies the
gravitational degrees of freedom in a geometrically transparent way
and is also particularly suited to situations where the initial data is
given on a null surface.
\begin{spanbox}[breakable,enhanced,boxrule=0pt]
We then go on to look at the way in which one can introduce
a time function $t$ and use this to slice up the space-time $M$ by a
family of spacelike hypersurfaces, or \textbf{foliation},
$\Sigma_t$.
\section{Section title within tcolor box wants to running head}
\markboth{XXXX}{YYYYYYYY}
The geometry of the foliation can be described in terms of
the \textbf{lapse} function, and the extrinsic curvature can be given
an alternative description in terms of the rate of change of the
induced metric. The final ingredient is to introduce a timelike vector
field, or \textbf{fibration}, whose integral curves may be used to
identify points on neighbouring, $t={\rm constant}$,
hypersurfaces. The geometry of the fibration can then be defined in
terms of the \textbf{shift} vector.
\end{spanbox}
The lapse and shift are freely
specifiable and encode the fourfold coordinate freedom in describing
the geometry. This machinery allows one to view general relativity
from a different perspective as a dynamical theory in which Einstein's
equations are encoded in a pair of first-order differential equations
which describe the way in which the dynamical variables, the induced
metric, and the extrinsic curvature, evolve with the time function $t$. A
knowledge of the induced metric and the lapse and shift may be used to
reconstruct the space-time metric and hence the geometry of the
space-time. A slightly different form of these equations, called the
\textbf{ADM formalism}, was derived by Arnowitt, Deser, and Misner (1959) from
their Hamiltonian formulation of general relativity and formed the
basis for what Wheeler (1963) called \textbf{geometrodynamics}. In
\S\ref{S13B.13} we look at an alternative approach called the
\textbf{$2+2$ formalism}, in which one decomposes space-time into two
families of spacelike 2-surfaces. This approach identifies the
gravitational degrees of freedom in a geometrically transparent way
and is also particularly suited to situations where the initial data is
given on a null surface.
\end{document}