coordinate interval $d>0$.

From straightforward algebra, it can be shown

$x_{1}=d\frac{1-v_{2}}{v_{2}-v_{1}},\qquad\qquad
t_{1}=v_{1}d\frac{1-v_{2}}{v_{2}-v_{1}},$ (116)

where $v_{i}\equiv\frac{t_{i}}{x_{i}}$ is the coordinate speed of the Rindler
observers matching the emitter and receiver. This yields a chain of observers,
with the endpoints coincident with the emitter and receiver observers, who
remain at constant proper-frame distance. Thus each would locally interpret
the infinitesimal redshift as a purely gravitational shift. And thus it is an
overall gravitational shift, in this interpretation.

The photon takes nonzero coordinate time $t$ to transit, which allows more
flexibility. For instance, there is no frame in which part of the Rindler rod
travels left, and part travels right. But due to the finite photon travel
time, a photon path which traversed from some position at $t<0$ to another
position at $t>0$ could include this property.

###  8.2 Arbitrary accelerations in special relativity

The case of arbitrary accelerations in general curved spacetimes is difficult,
so a natural precursor is arbitrary accelerations in Minkowski space. Because
it is just in $1+1$ dimensions, the situation is considerably simplified.

Noether and Herglotz both showed the motion of a rigid body in special
relativity is determined by a single particle. But one may wonder questions
like, does the local length-contraction depend only on the leading particle’s
instantaneous speed, or its full worldline history? From the particular case
of Rindler coordinates, the velocity of a single particle is not sufficient to
determine the behaviour of the rod, as inspection of the chart shows. Along a
radial line, $T$ and hence the velocity is constant, but all these particles
have different accelerations.

In the Rindler chart, the following properties hold, and one might wonder if
they hold for accelerated rigid motion more generally:

  * $\centerdot$

to stay rigid individual proper accelerations differ

  * $\centerdot$

to stay rigid individual speeds differ

  * $\centerdot$

speed depends on both speed and acceleration of adjoining particle

  * $\centerdot$

there is a limit on the extent, could possibly call a “rigidity horizon” or
similar. Can’t extend rigid structure indefinitely through space under the
given requirements.

Back to arbitrary accelerations (but still $1$-dimensional and collinear),
suppose $x^{\mu}$ are inertial coordinates corresponding to an observer. Let
$s$ be a material (“Lagrangian”) coordinate, with the additional requirement
that it be scaled to proper length. Label by $v\equiv\frac{dx}{dt}$ the
coordinate speed, and $a\equiv\frac{dv}{dt}$ for the coordinate (not proper)
acceleration. $x$, $v$, and $a$ are all functions of both $s$ and $t$. The
local or infinitesimal length-contraction is: (Rindler, 1977, p50–51)

$\boxed{\frac{\partial x}{\partial s}=\gamma^{-1}=\sqrt{1-v^{2}}}$ (117)

The higher the speed, the more material fits into a given coordinate interval,
as measured from the observer’s frame. Differentiate both sides with respect
to $t$. Then $\frac{d^{2}x}{dsdt}=\frac{\partial(\gamma^{-1})}{\partial t}$.
The right hand side is

$\frac{\partial}{\partial t}\gamma^{-1}=\frac{\partial}{\partial
t}\sqrt{1-v^{2}}=\frac{1}{2}(1-v^{2})^{-1/2}\cdot(-2)v\cdot\frac{\partial
v}{\partial t}=-v\gamma\frac{\partial v}{\partial t}=-v\gamma a,$ (118)

and the left hand side is
$\frac{d}{ds}\left(\frac{dx}{dt}\right)=\frac{dv}{ds}$. Hence

$\boxed{\frac{\partial v}{\partial s}=\frac{\partial\gamma^{-1}}{\partial
t}=-v\gamma a}$ (119)

This describes the propagation of the speed along the material, in a snapshot
of constant coordinate time. Differentiate again with respect to $t$, and
after a few lines one can show:

$\boxed{\frac{\partial a}{\partial s}=-\frac{\partial(v\gamma a)}{\partial
t}=-\gamma^{3}a^{2}-\gamma v\dot{a}.}$ (120)

This describes the propagation of acceleration along the material. The
$\dot{a}$ term is unappealing, but there is no point continuing this process
ad-infinitum. I derived the latter two independently, but they would certainly
have been derived before. Note they all resulted simply from the length-
contraction formula. As a check, the Rindler chart satisfies these equations.

From equation 117 and the definition of $v$ it follows that (Eriksen et al.,
1982, §2)

$\left(\frac{\partial x}{\partial s}\right)^{2}+\left(\frac{\partial
x}{\partial t}\right)^{2}=1.$ (121)

This equation is not explicitly covariant (tensorial), but it can be put into
a form which is: (Söderholm, 1982)

$\left(\frac{\partial s}{\partial x}\right)^{2}-\left(\frac{\partial
s}{\partial t}\right)^{2}=1.$ (122)

In fact the following features of the Rindler chart do carry through to
generalised $1$-dimensional accelerated rigid motion in special relativity.
There is a single frame in which the whole $1$-dimensional rigid body is at
rest (Eriksen et al., 1982, §4). If a given point has proper acceleration
$\frac{1}{X}$, then the maximum extent of the rigid body is $X$ in the
direction opposite to the acceleration. For such a limit point $X_{0}$, the
proper acceleration of other points is $(X-X_{0})^{-1}$ (Eriksen et al., 1982,
§5). The distribution of the acceleration across space is the same as for the
Rindler chart (Eriksen et al., 1982, §6). Contrast the approach in DeWitt
(2011, §2.4).

This analysis of rigid body dynamics in $1+1$ dimensions in flat spacetime is
linked to gravitational redshift interpretations, and is a preliminary step to
application in curved spacetime.

##  9 Gravitational redshifts — the Schwarzschild fishing line

Figure 10: A fishing line dangling over an unusually colourful black hole. The
line is moving radially, and is hence length-contracted as measured by
Schwarzschild observers. It is kept perfectly rigid by a large number of fixed
pulleys or winches (in pink), each rotating at a constant rate depending on
their position. This ensures the overall motion is Born-rigid.

This section describes the relativistic kinematics of a $1$-dimensional rigid
object in Schwarzschild spacetime, or put more simply, a “fishing line”
dangling outside a black hole and which is being reeled in or out. This rigid
body solution leads to gravitational redshift interpretations for radial
photons. This is an original result. It was claimed by Brotas (2006) who is
demonstrably incorrect.

The object could be visualised as an idealised fishing line, bar, rope,
string, etc. Technically, it is only the motion which is rigid and not the
object itself, as discussed in section 3.1. Hence, one could visualise a host
of miniature rocket boosters affixed at all points of the line, or a system of
pulleys which feed the line through like a conveyor belt. Historically, the
idea sprang out of discussion about “mining” black holes, for which one
proposal is to lower a box on a rope, fill it with energy, then raise it
again. This leads to inquiries about the tensilve strength and kinematics of
the rope.

One might assume the “proper distance”
$dR=\left(1-\frac{2M}{r}\right)^{-1/2}dr$ is the only important radial
distance, and hence that a rigid system must be described by a constant
$R$-spacing of particles in an object. However despite the ubiquity of this
length measure, it does not account for length-contraction effects due to
motion, as shown previously.

###  9.1 Initial calculations

The goal is to determine quantitative values such as the $4$-velocity of
particles on the line, and ultimately the proper metric. Firstly, I start with
insight from special relativity, to be later applied to local inertial frames
in Schwarzschild spacetime.

Let $K$ be a constant related to the speed of the line, defined as the rate of
proper length passing per unit coordinate time $t$, at any given point. This
definition is made by Brotas, who correctly asserts this leads to rigid
motion. See the pipe diagram for one justification, and the proper metric
calculation later. It seems to be the static nature of the geometry which
supports this feature. Brotas defines $0<K<\infty$, I use $-\infty<K<\infty$,
where as before positive $K$ refers to outward motion, and vice versa. This
unboundedness does not lead to unbounded velocities, as the following lemma
reassures. Let $L$ measure proper length along the rope, then

$\frac{dL}{dt}=K.$ (123)

Figure 11: Water flowing through oddly shaped pipes in Newtonian mechanics, an
analogy to the length-contracted fishing line in Schwarzschild geometry.
Assume the water flows into the pipe at a constant rate, and the flow is
streamlined, frictionless, and incompressible. Its speed through the thinner
sections is greater, and slower through the thicker sections. But the equation
of continuity says the mass flow rate is the same everywhere, i.e. the amount
of material or number of atoms passing any given $r$ remains constant. The
varying thickness of the pipe is analogous to the varying time and length
distortion of the fishing line in Schwarzschild spacetime. Stretching the pipe
(bottom diagram) in the lengthwise does not change the flow rate (arrows),
which is dependent only on the cross-section.

######  Lemma 9.1.

In Minkowski space, suppose a rigid rope lies in a straight line and moves
along its length, at constant speed $V$ relative to an inertial observer. Then
the amount of material (proper length) of rope passing the observer is given
by the rate

$\frac{dL}{d\tau}=V\gamma,$ (124)

where $L$ is the proper length of rope which has passed the observer, $\tau$
is the proper time of the observer, and $\gamma\equiv(1-V^{2})^{-1/2}$ is the
Lorentz factor.

###### Proof.

Recall “proper length” means the length of the rope as measured in its own
frame. The rope is length-contracted by a factor of $\gamma$, as measured by
the observer. In the observer’s frame, the rope will appear denser. Though the
speed is $V$, due to the increased density from length-contraction the proper
length will pass at the increased rate of $V\gamma$. ∎

######  Lemma 9.2.

Suppose $V\gamma=q$ for some number $q$ at a given point, where $\gamma(V)$ is
the Lorentz factor, and $q$ and $V$ may be positive or negative. Then

$\gamma=\sqrt{1+q^{2}},\qquad\qquad V=\frac{q}{\sqrt{1+q^{2}}}.$ (125)

###### Proof.

This follows from elementary algebra. Now
$V\gamma=\frac{V}{\sqrt{1-V^{2}}}=q$, so $\frac{V^{2}}{1-V^{2}}=q^{2}$. Now
$V=0\Longleftrightarrow q=0$, otherwise invert both sides so
$\frac{1-V^{2}}{V^{2}}=q^{-2}$, so
$\frac{1}{V^{2}}=1+q^{-2}=1+\frac{1}{q^{2}}=\frac{1+q^{2}}{q^{2}}$. Then
$V^{2}=\frac{q^{2}}{1+q^{2}}$, so $V=\frac{q}{\sqrt{1+q^{2}}}$, taking $V$ to
have the same sign as $q$. Then from the starting assumption $V\gamma=q$, it
follows $\gamma=\frac{q}{V}=\sqrt{1+q^{2}}$. ∎

Back to Schwarzschild geometry, the fishing line passes any stationary
observer at a proper length rate of $\frac{dL}{dt}=K$. The use of coordinate
time gives a convenient universal time with which to coordinate the motion.
Then this is related to the proper time $t_{\rm shell}$ of the local
Schwarzschild observer by

$\frac{dL}{dt_{\rm shell}}=\frac{dL}{dt}\frac{dt}{dt_{\rm
shell}}=K\left(1-\frac{2M}{r}\right)^{-1/2}.$ (126)

(The label $\tau$ will be is reserved for particles on the line.) Since
$t_{\rm shell}$ is a local inertial coordinate, the special-relativistic
result of lemma 9.2 is valid, taking $q=K\left(1-\frac{2M}{r}\right)^{-1/2}$.
It follows:

$\gamma=\sqrt{1+K^{2}\left(1-\frac{2M}{r}\right)^{-1}}=W(r)\left(1-\frac{2M}{r}\right)^{-1/2}\qquad\qquad
V=\frac{K}{W(r)},$ (127)

after simplifying. Here $W(r)$ is defined by Brotas and has the following
definition, and I also add a function $f(r)$ used shortly:

$W(r)\equiv\sqrt{1-\frac{2M}{r}+K^{2}},\qquad\qquad
f(r)\equiv\left(1-\frac{2M}{r}\right)^{-1}\sqrt{1-\frac{2M}{r}+K^{2}}=\left(1-\frac{2M}{r}\right)^{-1}W(r).$
(128)

One can show equation 127 implies $\gamma\geq 1$ and $\left\lvert
V\right\rvert<1$ as required.

The $4$-velocity in Schwarzschild coordinates follows from equation 93,

$\boxed{u^{\mu}=\left(f(r),K,0,0\right)}$ (129)

after simplifying. In particular, $u^{r}=\frac{dr}{d\tau}=K$, so integrating
gives

$\boxed{r=K\tau+X,}$ (130)

where $X$ is the constant of integration and is clearly the value of $r$ when
$\tau=0$.

Similarly, $u^{t}=\frac{dt}{d\tau}=f(r)$. A relation between $t$, $\tau$, and
$r$ is needed. But $\frac{dr}{d\tau}=K$, so
$f(r)=\frac{dt}{d\tau}=\frac{dt}{dr}\frac{dr}{d\tau}=K\frac{dt}{dr}$, so
$\frac{dt}{dr}=\frac{1}{K}f(r)$. Integrate to get:

$\boxed{t=\frac{1}{K}\int_{X}^{r}f(r^{\prime})dr^{\prime}}$ (131)

where the lower bound follows from choosing $\tau=0$ to coincide with $t=0$,
when the $r$-coordinate is $X$. A primitive function for the integral is messy
but provided in Appendix B. It is not required for the calculations which can
be done from the above expression.

Note that since the upper limit in the integral for $t$ is $r=X+K\tau$, the
new variables $(\tau,X,\ldots)$ are expressed in terms of the Schwarzschild
coordinates $(t,r,\ldots)$, where $\theta$ and $\phi$ are shared by both
systems. The coordinate velocity

$\frac{dr}{dt}=\frac{K}{f(r)}$ (132)

follows directly from the $4$-velocity.

The proper length of rope contained in interval $dr$ is, from equation 127,

$\gamma
dR=\frac{W(r)}{\left(1-\frac{2M}{r}\right)^{1/2}}\left(1-\frac{2M}{r}\right)^{-1/2}dr={W(r)\left(1-\frac{2M}{r}\right)^{-1}}dr=f(r)dr.$
(133)

When $\left\lvert K\right\rvert$ is large, the interval tends to $\left\lvert
K\right\rvert dr$, so is unbounded. When $K=0$ the interval is $dr$ as
expected. The $4$-accelerationis
$\mathbf{a}\equiv\nabla_{\mathbf{u}}\mathbf{u}$, which evaluates to

$\mathbf{a}=\left(KM(r-2M)^{-2}f(r)^{-1},\frac{M}{r^{2}},0,0\right).$ (134)

The first term may also be expressed
$KMr^{-2}\left(1-\frac{2M}{r}\right)^{-2}f(r)^{-1}$ or
$\frac{KM}{r(r-2M)\sqrt{1-\frac{2M}{r}+K^{2}}}$. For $K=0$ this reduces to the
expression for Schwarzschild observers $a^{\mu}=(0,\frac{M}{r^{2}},0,0)$ as
expected (Hartle, 2003, p459). The magnitude describes the “felt”
acceleration:

$\sqrt{\mathbf{a}\cdot\mathbf{a}}=\frac{M}{r^{2}}\left(1-\frac{2M}{r}+K^{2}\right)^{-1/2}=\frac{M}{r^{2}W(r)}.$
(135)

The form is reassuringly similar to that of Schwarzschild observers, for whom
$\sqrt{\mathbf{a}\cdot\mathbf{a}}=\frac{M}{r^{2}}\left(1-\frac{2M}{r}\right)^{-1/2}$
(Hartle, 2003, p459), and as $K\rightarrow 0$ this limit is obtained. Note the
acceleration is always less for fishing line particles than Schwarzschild
observers, for $K\neq 0$.

###  9.2 Proper metric

The projection tensor is given by equation 30, with $V$ and $\gamma$ from
equation 127. Then,

$P_{\mu\nu}=\begin{pmatrix}K^{2}&-Kf(r)&0&0\\\ -Kf(r)&f(r)^{2}&0&0\\\
0&0&r^{2}&0\\\ 0&0&0&r^{2}\sin^{2}\theta\\\ \end{pmatrix},$ (136)

after simplifying.

Figure 12: Many fishing lines dangling over a black hole. The picture is
basically the same in the Schwarzschild frames, the fishing line frames, and
$\mathbb{R}^{3}$, as the only difference is a radial length-contraction.
Clearly this is not a rigid system in the angular coordinates - if the lines
were connected sideways, they would rip apart when reeled in from above. But
for a thin line which is not glued to others, only the radial direction
matters.

The proper metric (see section 3.4) requires the Schwarzschild coordinate
position $x^{\mu}$ of particles to be expressed in terms of material
coordinates $X^{\nu}$ including the proper time $\tau$. Now $\theta$ and
$\phi$ are fixed for any given particle, so they can double as material
coordinates: $X^{\theta}\equiv x^{\theta}$ and $X^{\phi}\equiv x^{\phi}$. Also
equation 130 is $r=K\tau+X$, so choose $X$ as a radial material coordinate:
$X^{r}\equiv X$. Equation 131 gives $t$ as a function of the $X^{\nu}$. Thus
the Schwarzschild coordinates $x^{\mu}$ are expressed in terms of the material
coordinates $X^{\nu}$, as required.

The derivatives $\frac{\partial x^{\mu}}{\partial X^{i}}$ are straightforward,
apart from $\frac{\partial t}{\partial X}$. To repeat,
$t=\frac{1}{K}\int_{X}^{r}f(r^{\prime})dr^{\prime}$. It is easy to miss that
the upper limit $r=K\tau+X$ is also dependent on $X$. Hence, by a double
application of the First Fundamental Theorem of Calculus which states
$\frac{d}{dx}\int_{c}^{x}h(s)ds=h(x)$,

$\frac{\partial t}{\partial X}=\frac{f(r)-f(X)}{K},$ (137)

since $\frac{d}{dX}(I)=\frac{d}{dr}(I)\frac{dr}{dX}$ by the chain rule,
writing $I$ for the integral expression. This is confirmed using the closed
form expression for $t$.

It follows the proper metric is, after simplifying,

$\gamma_{ij}=\begin{pmatrix}f(X)^{2}&0&0\\\ 0&r^{2}&0\\\
0&0&r^{2}\sin^{2}\theta\\\ \end{pmatrix}.$ (138)

This gives the proper-frame distance as perceived by particles in the fishing
line, and is interpreted as follows. In the radial direction the perceived
distance between particles separated by $dX$ in the material manifold, is
$f(X)dX$. This distance is measured in the orthogonal hypersurface to the
motion. This formula applies over all time, so the proper-frame distance
between these particles remains context, and thus they remain Born-rigid. $X$
is also the initial $r$-coordinate at $t=0$, so a Schwarzschild radial
distance of $dX$ is perceived as $f(X)dX$ by the particles.

The angular components $\gamma_{\theta\theta}$ and $\gamma_{\phi\phi}$ are
unchanged from those in the spacetime metric $g_{\mu\nu}$. This is an
illustration of the familiar principle that there are no length-contraction
(or expansion) effects orthogonal to the motion. However, $r$ is not constant
over time for a given particle, for $K\neq 0$. Hence the radial distances
change over time, and there is not rigidity in this direction. A Born-rigid
object extended in $3$-dimensions will rip apart under fishing line motion (if
$K\neq 0$). However the idealised $1$-dimensional fishing line remains Born-
rigid. An alternate approach is to accept the line width as nonzero but small,
and then allow for slight deviations from Born-rigidity, where the latter
remains an extremely good approximation.

Limiting cases provide helpful confirmation. As $K\rightarrow 0$,
$g_{11}\rightarrow\left(1-\frac{2M}{r}\right)^{-1/2}=dR$ as expected, which is
the proper distance for Schwarzschild observers. As $M\rightarrow 0$ and/or
$r\rightarrow\infty$, $\gamma_{11}\rightarrow\sqrt{1+K^{2}}$. This implies
Minkowski space, and indeed this fits lemma 9.2 with $q=K$. If in addition
$K\rightarrow 0$, $\sqrt{1+K^{2}}\rightarrow 1$ as expected.

Natário, J. (2014, §6) also discusses the fishing line.

###  9.3 Coordinate transform

There are many coordinate systems for the Schwarzschild geometry already in
common use. Each is chosen for a particular application, for instance
geometric reasons, or following light paths or free-falling particles. I
provide a new coordinate system $(\tau,X,\theta,\phi)$ which is adapted to the
fishing line kinematics. $X$ is constant for a particle on the line and
corresponds to the Schwarzschild $r$-coordinate when $\tau=0$. $\tau$ is the
proper time for particles on the line. $\theta$ and $\phi$ are the usual
angular spherical coordinates. They are synchronous comoving coordinates for a
Born-rigid fishing line.

After substitution and simplification, one ultimately obtains

$ds^{2}=-d\tau^{2}+\frac{2}{K}\left(f(X)\sqrt{1-\frac{2M}{r}+K^{2}}-1\right)d\tau
dX+\left(\left(1-\frac{2M}{r}\right)^{-1}-\frac{1}{K^{2}}\left(1-\frac{2M}{r}\right)\left(f(r)-f(X)\right)^{2}\right)dX^{2}+r^{2}d\Omega^{2}$
(139)

which is valid for $K\neq 0$. Here $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta
d\phi^{2}$, and $r$ is the Schwarzschild radial coordinate but interpreted as
a function of the coordinates, $r=X+K\tau$, which I continue to use for
brevity of expression. Note the cross-term $d\tau dX$, due to motion in the
Schwarzschild case.

This metric has vanishing Ricci tensors $R_{\mu\nu}$ as required for any
coordinatisation of the Schwarzschild geometry, as Brotas (2006) mentions.
This follows from the tensor transformation law
$R^{\prime}_{\alpha\beta}=\frac{\partial x^{\gamma}}{\partial
x^{\prime\alpha}}\frac{\partial x^{\delta}}{\partial
x^{\prime\beta}}R_{\gamma\delta}$, so since the $R_{\gamma\delta}$ all vanish
in Schwarzschild coordinates, they must vanish in all coordinate systems for
this geometry. Similarly, the curvature scalar $R=0$ everywhere for this
geometry.

###  9.4 Infinite length of line

Figure 13: The function $f(r)$ (blue line) compared with the usual “proper
distance” interval $\left(1-\frac{2M}{r}\right)^{-1/2}$, for $M=1$ and $K=1$.
Visually, it is quite plausible the integral of $f$ — the area under the curve
— is infinite near $r=2M$.

The length-contracted fishing line has the curious property than an infinite
proper length will fit between the event horizon $r=2M$ and any point $r>2M$.
This follows from the integral of $f(r)dr$, which describes the proper length,
as given in Appendix B. One might guess a realistic material would break under
the extreme conditions near the event horizon. But as $r\rightarrow 2M$, the
$4$-accelerationmagnitude approaches $\frac{1}{4M\left\lvert K\right\rvert}$,
which curiously is finite. Perhaps a real line could even withstand the
tension, aided by time-dilation, as Brown (2013, §2) expresses “the weight of
the rope redshifts away.”

The infinite length is _not_ simply due to $f(r)$ diverging to infinity as
$r\rightarrow 2M$. Though this is true, this property does not imply the
integral also diverges — compare Taylor & Wheeler (2000, p2-49). Consider the
familiar setting of functions from $\mathbb{R}$ to $\mathbb{R}$, e.g.
$\frac{1}{x}$ and $\frac{1}{\sqrt{x}}$. Both are “infinite” at $x=0$ (more
rigorously, both are undefined at $x=0$ but the limit is $+\infty$ when
approaching from the positive side). And yet one integral diverges when the
other converges. for $\int_{0}^{1}\cdot dx$.

The point $r=2M$ is not included in the integral. But the integration is still
the same $\int_{2M}{\cdot}$, whether or not this endpoint is included.
(Technically, under Lebesgue integration, the “measure” of a single point
within an interval of $\mathbb{R}$ is $0$. The convention is $0$ times
$\infty$ is $0$, under Lebesgue integration. Thus one point makes no
difference. The divergence is due to the behaviour _near_ $r=2M$, not the
point itself.) Another way of expressing this is that the set of lengths
$\int_{r_{1}}^{3M}f(r^{\prime})dr^{\prime}$ for $2M<r_{1}<3M$ is unbounded.
The upper limit is an arbitrary choice, but close (in Schwarzschild
coordinates) to $2M$.

Figure 14: A rigid Schwarzschild fishing line or rope as measured in various
coordinate systems. In the top image, the rope is stationary in an inertial
frame in flat spacetime. The scale shows the rope’s regularly spaced one-unit
markings, which serve as a visual ruler. The bottom pair is Schwarzschild
spacetime with $M=1$, and here the scale represents the $r$-coordinate. In the
middle image the rope is stationary and almost touching the event horizon. A
proper length of $11$ units fits into a Schwarzschild $r$-coordinate interval
of just over $2\frac{1}{2}$ units, due to the different geometry. In the lower
image the rope is moving with $K=1$. Just $28$ segments are depicted of its
infinite proper length.

The various radial length measures compare as follows, for $K\neq 0$:

$dL>dR>dr,$ (140)

since

$\frac{dL}{dr}=f(r)>\left(1-\frac{2M}{r}\right)^{-1}\sqrt{1-\frac{2M}{r}}=\left(1-\frac{2M}{r}\right)^{-1/2}=\frac{dR}{dr}.$
(141)

See the diagram. If $K=0$ then $dL=dR>dr$.

###  9.5 Application

The kinematics of the rigid fishing line can be copied by a chain of observers
to interpret redshifts as gravitational, or at least adjust the usual
interpretation.

_Example 1_. Suppose a pair of observers are stationary on the same radial
line, at $r_{1}$ and $r_{2}$. Suppose a photon passes between them. Interpret
the redshift as purely gravitational.

This example is chosen for familiarity, and to show consistency with the usual
interpretation. A fishing line with $K=0$ has particle motion coinciding with
the observers. Because this is a rigid system, every particle/observer on the
line perceives constant proper-frame distance to its neighbours, and thus the
photon redshift observed must be purely gravitational in origin since it can’t
be a Doppler shift. So the overall redshift is purely gravitational.

_Example 2_. Suppose the $3$-velocity of an emitter at location $r_{1}$ is
$V_{1}$ with respect to the local Schwarzschild observer there, and speed
$V_{2}$ of a receiver at location $r_{2}$ relative to its local Schwarzschild
observer. Use the convention $V_{i}>0$ for outward motion, as before. Suppose
the emitter sends out a photon, which the emitter measures to have initial
energy $E_{1}$. Firstly, the usual convention is to quantify redshift around
what stationary Schwarzschild observers measure, a choice I affirm as the most
“natural”. Thus we convert the velocities to a stationary observer
perspective.

At the emission event, the local Schwarzschild observer would measure a
Doppler shift between themselves and the emitter. If $V_{1}>0$, the stationary
observer would measure increased energy, which is
$\sqrt{\frac{1+V_{1}}{1-V_{1}}}E_{1}$ from equation 11. Later, the photon
arrives at $r_{2}$, experiencing an overall gravitational redshift by an
energy factor $\sqrt{\frac{1-\frac{2M}{r_{1}}}{1-\frac{2M}{r_{2}}}}<1$. Then
another Doppler correction is needed in converting to the receiver’s frame, a
factor of $\sqrt{\frac{1-V_{2}}{1+V_{2}}}$ which is a decrease in energy if
$V_{2}>0$. Thus the breakdown is:

$\displaystyle\rm{Doppler}\qquad$
$\displaystyle\sqrt{\frac{1+V_{1}}{1-V_{1}}}\sqrt{\frac{1-V_{2}}{1+V_{2}}}$
(142) $\displaystyle\rm{Gravitational}\qquad$
$\displaystyle\sqrt{\frac{1-\frac{2M}{r_{1}}}{1-\frac{2M}{r_{2}}}}$ (143)

This standard interpretation is the most natural, based on the symmetries of
the spacetime. Nevertheless, one can manipulate the interpretation using the
equivalence principle.

Suppose for the same setup that a rigid fishing line moves concurrently with
the emitter. One can solve for
$K=V_{1}\gamma_{1}\left(1-\frac{2M}{r_{1}}\right)^{1/2}$. There is no initial
conversion Doppler shift in this case, as the emitter and other observer at
the event have the same motion. Then all along the photon trajectory,
frequency shifts are interpreted as purely gravitational by observers moving
with the line. At the receiver location $r_{2}$ the line has speed
$V^{\prime}=\frac{K}{W(r_{2})}$ relative to the local stationary observer by
equation 127. In general this is not the same speed as the receiver, so an
endpoint Doppler adjustment is needed, which can be deduced from a combined
shift “back” to the local Schwarzschild observer then “forward” to the
receiver. These factors are
$\sqrt{\frac{1+\frac{K}{W(r_{2})}}{1-\frac{K}{W(r_{2})}}}$ and
$\sqrt{\frac{1-V_{2}}{1+V_{2}}}$ respectively. The gravitational component
must be whatever energy shift is unaccounted for. Comparing to the above
calculations, overall:

$\displaystyle\rm{Doppler}\qquad$
$\displaystyle\sqrt{\frac{1+\frac{K}{W(r_{2})}}{1-\frac{K}{W(r_{2})}}}\sqrt{\frac{1-V_{2}}{1+V_{2}}}$
(144) $\displaystyle\rm{Gravitational}\qquad$
$\displaystyle\sqrt{\frac{1+V_{1}}{1-V_{1}}}\sqrt{\frac{1-\frac{2M}{r_{1}}}{1-\frac{2M}{r_{2}}}}\sqrt{\frac{1-\frac{K}{W(r_{2})}}{1+\frac{K}{W(r_{2})}}}$
(145)

Let’s put in some numbers. Suppose $r_{1}=3M$ and $r_{2}=4M$,
$V_{1}=\frac{1}{2}$ and $V_{2}=\frac{1}{5}$. Then the usual gravitational
redshift factor is
$\sqrt{\frac{1-\frac{2M}{r_{1}}}{1-\frac{2M}{r_{2}}}}=\sqrt{\frac{2}{3}}$. The
$V_{1}$ special-relativistic Doppler factor is $\sqrt{3}$, and
$\gamma(V_{1})=\frac{2\sqrt{3}}{3}$. The $V_{2}$ Doppler factor is
$\sqrt{\frac{2}{3}}$. Then the line matching the emitter has
$K=\frac{1}{2}\frac{2}{\sqrt{3}}\frac{1}{\sqrt{3}}=\frac{1}{3}$. Then
$W(r_{2})=\sqrt{\frac{11}{18}}$, so $\frac{K}{W(r_{2})}=\sqrt{\frac{2}{11}}$,
and after simplifying the
$\sqrt{\frac{1+\frac{K}{W(r_{2})}}{1-\frac{K}{W(r_{2})}}}=\frac{\sqrt{11}+\sqrt{2}}{3}\approx
1.58$.

The usual natural interpretation is of a Doppler energy shift by
$\sqrt{3}\sqrt{\frac{2}{3}}=\sqrt{2}\approx 1.41$, and a gravitational shift
by $\sqrt{\frac{2}{3}}\approx 0.82$ as above. The overall factor is an energy
shift of $\frac{2}{\sqrt{3}}\approx 1.15$, a slight increase or blueshift.

Using the fishing line interpretation, the Doppler component is
$\frac{\sqrt{11}+\sqrt{2}}{3}\sqrt{\frac{2}{3}}=\frac{\sqrt{66}+2\sqrt{3}}{9}\approx
1.29$, and the gravitational component $\frac{\sqrt{22}-2}{3}\approx 0.90$.
The overall shift is the same, but there is a significant difference in its
interpretation and makeup!

A possible extension to this analysis would be to use an arbitrary winding
rate $K$ which is not necessarily matched to either emitter nor receiver.

###  9.6 Alternate L coordinates

The previous coordinate system used a radial coordinate which is comoving for
particles on the line. Brotas makes a different choice of radial coordinates,
using the proper length along the line, which he labels $\overline{x}$ and I
label $L$. The purpose here is to repeat Brotas’ construction and to compare
the results.

Let $r_{a}$ be the Schwarzschild radial coordinate of a winch or fishing reel,
as in Brotas. Natural choices such as $r_{a}=2M$ or $r_{a}=\infty$ do not
work, as the proper length from these to any point within $2M<r<\infty$ would
be infinite. One must pick some arbitrary $2M<r_{a}<\infty$.

For a given particle on the line, $\frac{dL}{dt}=K$, so $L=Kt+Y$ where $Y$ is
the initial $L$-value at $t=0$. Given $X$ as the starting $r$-coordinate used
previously, then

$Y=\int_{r_{a}}^{X}f(r^{\prime})dr^{\prime}.$ (147)

As for relating $L$ to $r$, $\frac{dL}{dr}=f(r)$, so

$L=\int_{r_{a}}^{r}f(r^{\prime})dr^{\prime}.$ (148)

Then

$t=\frac{L-Y}{K}=\frac{1}{K}\int_{X}^{r}f(r^{\prime})dr^{\prime}.$ (149)

By comparison, Brotas has
$t=\frac{L}{K}-\frac{1}{K}\int_{r_{a}}^{r}f(r^{\prime})dr^{\prime}$ which
differs only in the swapping of $X$ and $r_{a}$. It appears Brotas’ setup
implies all particles must start at $r=r_{a}$ at $t=0$. This system has
limited flexibility, and certainly his coordinates would not cover all of
Schwarzschild spacetime. Yet in fairness these formulae are extremely similar,
and perhaps Brotas intends the winch position $r_{a}$ to be unique for every
fishing line, in which case the formulae would coincide. Brotas (2006,
equation 4) also has $r=2M+KL-K\tau$, where it remains to be further checked.
These formulae are given without proof nor even a hint of their derivation,
but merely the statement “[t]he derivation/justification of these formula[e]
is somewhat complicated” Brotas (2006).

As Brotas states, the Ricci tensors $R_{\alpha\beta}$ must vanish for every
coordinate system for the Schwarzschild geometry. However his own coordinate
system fails this. Also the curvature scalar $R$ must be identically $0$ for
any Schwarzschild coordinate system, but this is not the case for Brotas’
metric.

$\frac{dL}{d\tau}=\frac{dL}{dt}\frac{dt}{d\tau}=Kf(r)$. So in the
$L$-coordinates, a given particle has $4$-velocity

$u^{\mu}=(1,Kf(r),0,0).$ (150)

Brotas claims the $r=2M$ pathology of the Schwarzschild coordinates is moved
to $r=\frac{2M}{1+K^{2}}$, but I do not find this result.

###  9.7 Conclusion

The description of a Born-rigid fishing line supports flexible interpretations
of redshifts in the Schwarzschild geometry.

##  10 Rigid rotation in Schwarzschild spacetime

I derive the dynamics of a light rigid object rotating on its axis, in
Schwarzschild geometry. The object is a spinning solid ball, with a black hole
inside it at the centre, with all of the ball’s matter outside the event
horizon. In general, the particles do not follow geodesics. This research is
original, and demonstrates that accelerated $3$-dimensional rigid motion does
exist in curved spacetimes. This may also be used to interpret redshifts as
gravitational, within the Schwarzschild geometry.

Orient Schwarzschild coordinates so that the rotation axis corresponds to the
$\theta=0$ direction. Then the “latitude” or $\theta$-coordinate will remain
fixed for a given particle, whereas the “longitude” or $\phi$-coordinate will
change. Suppose the rotation rate is

$\frac{d\phi}{dt}=\Omega$ (151)

for all particles, employing the universality of coordinate time in an attempt
to orchestrate the motion. Particles travel on circular paths, however these
are not geodesics unless (i) the particle is on the “equator” and (ii)
$\Omega$ matches the freefall rate, and in particular $r>3M$. Thus imagine the
structure held together by either (i) its large spatial extent, extending over
$180^{\circ}$, and balanced so that the whole structure moves in “freefall”
even though the individual particles do not follow geodesics, or (ii) a rocket
booster attached to every small clump of particles.

###  10.1 Proper metric

Assume the object is small enough or light enough not to affect the metric. A
given particle’s $4$-position is

$x^{\mu}=(t,r,\theta,\phi_{0}+\Omega t),$ (152)

where $\phi_{0}$ is the $\phi$-coordinate at $t=0$. One can compute the
$4$-velocity as

$u^{\mu}=u^{t}\left(1,0,0,\Omega\right),$ (153)

where the time component is

$u^{t}=\left(1-\frac{2M}{r}-r^{2}\Omega^{2}\sin^{2}\theta\right)^{-1/2}.$
(154)

The proper metric is, _in Schwarzschild coordinates_ :

$\gamma_{ij}=\begin{pmatrix}\left(1-\frac{2M}{r}\right)^{-1}&0&0\\\
0&r^{2}&0\\\ 0&0&\left(1-\frac{2M}{r}\right)(u^{t})^{2}r^{2}\sin^{2}\theta\\\
\end{pmatrix}$ (155)

Note that $r$, $\theta$, $\Omega$, and $u^{t}$ are all constant for a given
particle. Hence the proper metric is constant over time, and the system is
rigid in all $3+1$-dimensions. The $\gamma_{rr}$ and $\gamma_{\theta\theta}$
components are unchanged from the Schwarzschild metric. But the motion is in
the $\phi$-direction, and so the proper ruler distance is increased, relative
to $g_{\phi\phi}=r^{2}\sin^{2}\theta$.

There is a dependency on $\Omega$. This means we cannot change the rotation
rate and maintain Born-rigidity. The “proper geometry” is not Schwarzschild
for $\Omega\neq 0$, but nonetheless we can in principle assemble the object
while in rotation. These properties are identical for the spinning disc in
Minkowski space.

###  10.2 Limited extent

There is a limit to the spatial extent of the rotating object. We require

$u^{t}=1-\frac{2M}{r}-r^{2}\Omega^{2}\sin^{2}\theta>0.$ (156)

A “sufficient” (but not “necessary”) condition is
$1-\frac{2M}{r}-r^{2}\Omega^{2}>0$, since $0\leq\sin^{2}\theta\leq 1$. If the
structure doesn’t extend to the equator, there will be a looser bound. In the
extreme case, particles at the poles don’t move at all. Rearranging the
equation,

$\Omega<\frac{1}{r}\sqrt{1-\frac{2M}{r}}.$ (157)

Differentiating, the maximum of this function occurs at $r=3M$, which has
value $\frac{1}{3\sqrt{3}M}\approx 0.19M^{-1}$. Hence there is a global limit
$\Omega<\frac{1}{3\sqrt{3}M}$, but the body will be limited by its weakest
link.

Figure 15: The maximum (“supremum”) spin rate $\Omega$ for particles at the
equator, which is $\frac{1}{r}\sqrt{1-\frac{2M}{r}}$ for $M=1$. At large $r$,
a particle must move quickly through space to keep up — think classically
where we would have $v=r\Omega$. At small $r>2M$, time dilation is extreme and
so even a low coordinate velocity seems much faster to a particle there. Also
the acceleration must be increasingly angled outwards from the black hole, so
there is less leeway for revolution.

####  10.2.1 Application: Orbiting planets

An orbiting body such as a planet will measure an increased orbit length due
to length-contraction of its rulers, at least from the perspective of
Schwarzschild observers. Assume the orbit is circular, then
$\theta=\frac{\pi}{2}$ and the rotation rate is given by Kepler’s law (Hartle,
2003, p200)

$\Omega^{2}=\frac{M}{r^{3}}.$ (158)

(Recall freefall was not assumed prior to this point.) This also adds the
restriction $r>3M$. Then $u^{t}$ becomes $(1-\frac{3M}{r})^{-1/2}$ as in
(Hartle, 2003, p201). Then
$\gamma_{\phi\phi}=r^{2}\frac{1-\frac{2M}{r}}{1-\frac{3M}{r}}$. The ruler
length interval is $\sqrt{\gamma_{\phi\phi}}$, with total orbit length

$\sqrt{\frac{1-\frac{2M}{r}}{1-\frac{3M}{r}}}\cdot 2\pi r.$ (159)

For the Earth, $r\approx 1.5\times 10^{8}km$, and the Sun’s mass is $M\approx
1.5km$. The expansion factor is $\approx 1+10^{-8}$, an increase of $\approx
9km$.

For a millisecond pulsar, which is both extremely dense and rotating extremely
quickly, the length-contraction effects are likely extreme — relative to, say,
the Kerr metric. Does the literature account for the increased density due to
length-contraction?

###  10.3 Redshift

This rigid system gives the framework or coordinate system for interpreting a
certain class of redshifts as gravitational shifts. One must simply choose the
value of $\Omega$, then plot a null worldline through the system. The
particles encountered on this worldline all measure constant distances to
their neighbours, and thus interpret the redshift as purely gravitational. Due
to our setup, there are no restrictions on the path as being radial etc.,
however limits due to endpoint motions.

###  10.4 Application to Ehrenfest paradox — the spinning disc

The Ehrenfest paradox of the rigidly spinning $2$D disc in Minkowski space is
a special case with $M=0$ and $\theta=\frac{\pi}{2}$. Then the proper metric
reduces to

$\gamma_{ij}=\begin{pmatrix}1&0&0\\\ 0&r^{2}&0\\\ 0&0&(u^{t})^{2}r^{2}\\\
\end{pmatrix}$ (160)

where $u^{t}$ simplifies to

$u^{t}=\left(1-r^{2}\Omega^{2}\right)^{-1/2},$ (161)

so can reexpress $\gamma_{\phi\phi}=\frac{r^{2}}{1-r^{2}\Omega^{2}}$.

The $\theta$-component can be ignored, since there is no extent in this
direction. An alternative is to use cylindrical coordinates $(r,\phi,z)$ along
with $t$. This concurs with the result
$\operatorname{diag}(1,\frac{r^{2}}{1-\omega^{2}r^{2}},1)$ of DeWitt (2011,
§2.5), after rearranging the coordinate order.

Einstein realised the spinning disc would have non-Euclidean geometry, and for
him this was a heuristic motivation in developing general relativity.

##  11 Gravitational redshifts — the FLRW galactic cable

Figure 16: A rigid cable is tethered to the Milky Way galaxy. How far could it
reach?

An interesting question is how far a rigid cable could reach through space
before expanding space destroys it. This is interesting for two reasons.
Firstly, using rigidity to define a fixed distance allows us to interpret
cosmological redshifts as gravitational in origin. Secondly, when considering
whether space “expands” it is pertinent to ask whether it is possible to make
an infinitely long rigid rod. If not, it would seem that space really does
expand, because something has to be pulling the rod apart, and there is no way
to avoid that interpretation. This would be similar to the _ergosphere_ of a
spinning black hole, in which a stationary observer is impossible due to
“spinning” of space around the spinning black hole.

We investigate how far a rigid cable could reach through space before
expanding space destroys it. As discussed previously, rigidity allows
gravitational redshift interpretations. Also illustrative of expansion of
space, in fact this example adds evidence to that interpretation. This section
includes original results.

To begin with, I assume local proper velocities $v=-HD$ exactly counteract the
Hubble speed. While this is insufficient in general, it is a good starting
place.

######  Lemma 11.1.

The changing “proper distance” $\dot{D}$ from the origin equals the local
peculiar velocity of the distant object.

###### Proof.

Rigorously, we seek $\dot{D}=0$, where dots refer to differentiation by cosmic
time. $\dot{D}=\dot{R}\chi+R\dot{\chi}$ (Davis et al., 2003, §2), so
$\dot{\chi}=-\frac{\dot{R}}{R}\chi=-H\chi$. This is a coordinate speed, but
what speed would a local Hubble comover measure? From equation 108,
$\frac{d\chi}{dt}=\frac{v}{R(t)}$, so $v=-\dot{R}\chi=-HD$ is the speed
relative to the Hubble LIF. ∎

###  11.1 Cable length for constant Hubble parameter

Suppose $H(t)$ is constant over time. This implies either $H(t)=0$, a static
universe; or a universe with cosmological constant and no other matter-energy,
which grows exponentially and is known as _de Sitter space_. The former is
trivial, so we assume the latter. This could be a model of inflation, or of
the distant future of our universe.

Suppose the cable has one end anchored with the local Hubble flow, say to the
Milky Way galaxy. Suppose it lies in a straight line (geodesic), is Born-
rigid, and can withstand any tension so long as its particles remain
subluminal. (To be precise, to achieve Born-rigidity we assume adjustments to
its displacement are not causally connected, but caused by a series of pre-
programmed rocket boosters all along the cable. See section 3.1.)

Now, the change in proper distance $v\equiv\dot{D}$ is given by the Hubble
flow $v=HD=\dot{R}\chi$. Hence for the cable to remain rigid its particles
must move against the flow as $v=-HD$ to maintain equilibrium. Note, this will
_only_ result in a rigid system if $H(t)$ is constant.

Now denote the proper-length along the rope by $s$, where $s=0$ at the anchor.
This is a Lagrangian- or material coordinate, which would elsewhere be denoted
$X$, and is fixed for any given particle of the rope. In the local Hubble
frames, the rope is length-contracted, and so more of it will fit into a given
“proper distance” $D$. From equation 111,

$\displaystyle\frac{dD}{ds}$
$\displaystyle=\gamma^{-1}=\sqrt{1-v^{2}}=\sqrt{1-H^{2}D^{2}}$ (162)
$\displaystyle\int ds$ $\displaystyle=\int\frac{dD}{\sqrt{1-H^{2}D^{2}}}$
(163) $\displaystyle s$ $\displaystyle=\frac{1}{H}\sin^{-1}(HD),$ (164)

using the initial condition $s=0$ when $D=0$. Locally, particle proper motion
cannot exceed light speed, so we have the following:

$\left\lvert v\right\rvert<1\qquad\qquad HD<1\qquad\qquad
D<H^{-1}\qquad\qquad\chi<\dot{R}^{-1}.$ (165)

At this maximum (technically “supremum”), we have

$s_{\rm max}=H^{-1}\sin^{-1}(1)=\frac{\pi}{2}H^{-1}.$ (166)

$H^{-1}$ is the Hubble radius. This means the longest possible cable, if one
end is anchored with the Hubble flow, is $H^{-1}$ in “proper distance”, but
longer than this by a factor of $\frac{\pi}{2}\approx 1.6$ in terms of its
proper length — that is, the amount of cable.

_Example:_ In our universe, roughly $70\%$ of the matter-energy content is
dark energy, so the above calculations (which are only valid for $100\%$ dark
energy) seem a reasonable approximation. At the present, $H=H_{0}$ which in
geometrical units is $\frac{H_{0}}{c}$; writing this in SI units gives
$\approx 10^{-26}m^{-1}$. Thus the Hubble radius is $H_{0}^{-1}\approx
10^{26}m\approx 14Glyr$. A cable stretching to both horizons would have
“proper distance” $D=2H_{0}^{-1}$, and proper length $\pi H_{0}^{-1}\approx
4\times 10^{26}m\approx 40Glyr$. Towards the endpoints, even this idealised
cable can barely resist the expansion. This test scenario gives a sense that
space really “is” expanding. In Newtonian cosmology by contrast, such a cable
could have any extent.

This is not just a failure of our metric. (Incidentally, one extension to this
problem would be the effect of the cable on the metric — which here assumed
inconsequential). What is supplying the force? In flat spacetime e.g. Rindler
case, acceleration is the force ripping the system apart (if were extended too
far). Here, endpoint has no acceleration, however due to expansion of space,
i.e. curvature of spacetime — the change in how the LIFs are “joined”
together.

We can rearrange to get:

$D=\frac{1}{H}\sin(sH)\qquad\qquad\qquad\chi=\dot{R}^{-1}\sin(sH).$ (167)

Note that for $sH\ll 1$, $D\approx s$ as expected since the counteracting
Hubble velocities are small and thus length-contraction is minimal. Also in
the limit $H\rightarrow 0$ we have $D\rightarrow s$ (using L’Hôpital’s rule),
as expected. Even though $\gamma$ increases without bound as $D\rightarrow
H^{-1}$, the total length remains finite, in contrast to the “fishing line”
case.

###  11.2 Application

The galactic cable can be used to interpret redshifts as a certain composite
of Doppler and gravitational effects, by setting up observers or a coordinate
system with the same kinematics. Firstly, perform a Doppler shift from a
comoving galaxy frame to the frame corresponding to the endpoint of a rigid
cable at the same event. Then, the infinitesimal redshifts all along the cable
are interpreted as gravitational. There is no Doppler shift adjustment at
home, the coordinate origin, because the cable is stationary with respect to
the origin.

Suppose we have a dark energy dominated universe, and a comoving galaxy at
$\chi$ emits a photon of wavelength $\lambda_{0}$ towards the coordinate
origin. If $\chi<\dot{R}^{-1}$ then the galaxy is within the Hubble radius
(equation 165), and we could set up a rigid cable stretching from the origin
to this point, or else a matching coordinate system or chain of observers. The
endpoint at $\chi$ has local speed $\dot{R}\chi$ relative to the galaxy.

The local cable frame will observe a frequency $(1+z_{\rm
Doppler})\lambda_{0}$ where the factor is

$1+z_{\rm Doppler}=\sqrt{\frac{1+\dot{R}\chi}{1-\dot{R}\chi}}$ (168)

by equation 11. Then, for the entirety of the photon’s journey, the cable
particles interpret the infinitesimal redshifts as gravitational, because they
maintain constant proper-frame distance from each-other. There is no Doppler
shift adjustment at the home galaxy (coordinate origin), because the cable is
comoving with the flow there. Thus, this observer or coordinate system
interprets the overall redshift as made up of a mixed $1+z_{\rm Doppler}$
component, and a gravitational component, which is nonzero in general (Davis
et al., 2003, §3).

This shows the redshift interpretation is flexible. For an emitter galaxy
beyond the Hubble sphere, light could never reach the home galaxy in a de
Sitter universe.

For faster than exponential expansion, the event horizon is smaller than the
Hubble sphere. So if this equation still held, you could make a rigid rod
whose end we would never see! However this is a more general case and must be
solved from the system of differential equations which follows.

###  11.3 General case

If $H$ is decreasing, the particles will require less peculiar velocity to
counteract the expansion, so their length-contraction effects will diminish,
and so the cable will expand outwards (as measured by the “proper distance” of
Hubble flow comovers). The competing requirements of moving against the flow,
and partial acquiescence to compensate for changing $H$, result in a feedback
mechanism. This would be sensitive to the initial setup, although quite
possibly might form a stable equilibrium also.

Using the same variables as previously, but with $D$ and $v$ now each
functions of $s$ and $t$:

$\boxed{\frac{\partial D}{\partial s}=\gamma^{-1}=\sqrt{1-v^{2}},}$ (169)

which is nothing more than local length-contraction. We also have

$\boxed{\frac{\partial D}{\partial t}=HD+v}$ (170)

This follows from $\dot{D}=\dot{R}\chi+R\dot{\chi}=HD+v$ since
$\dot{\chi}=\frac{d\chi}{dt}=\frac{v}{R(t)}$ which follows from the
$4$-velocity in equation 108. We also have initial conditions including
$D(t,0)=0$ and $v(t,0)=0$. In the varying $H$ case, $v\neq-HD$ in general,
although we would expect it to remain close to this value if $H$ changes only
gently.

We could eliminate $v$ to give

$\boxed{\left(\frac{\partial D}{\partial s}\right)^{2}+\left(\frac{\partial
D}{\partial t}-HD\right)^{2}=1.}$ (171)

This complements equation 121 for Minkowski space which does not have the
“$-HD$” term.

I have not analysed these equations further. We likely need to specify further
initial conditions, such as the initial velocity of the cable as it was
unreeled. For instance, it could be assumed to unroll from Earth at a constant
rate $K$, analogous to the variable used for the Schwarzschild fishing line.
Assuming there is flexibility in the system, we could go further and consider
acceleration of an endpoint, analogous to the Rindler chart. This would
achieve great flexibility in the interpretation of cosmological redshifts.

###  11.4 Further exploration

One could compute the forces on the cable, or even consider elasticity and the
propagation of waves along it. Could one set up a “galaxy hook”, by which I
dub a cable which would “hook” to the Hubble flow and tow a spacecraft? This
is meant as an analogy to “skyhooks”, which are a speculative concept for
escaping Earth’s gravitational pull by a long rotating cable.

##  12 Conclusions

There are lines of evidence to interpret the expansion of the universe as both
“stretching of space” and motion through space. Parallels with the Milne model
and Newtonian gravity might encourage the kinematic view, as does the natural
interpretation of cosmological redshifts as purely Doppler. On the other hand,
the superluminal recession velocity of distant galaxies, and the limited
extent of a galactic cable suggest the expansion cannot be kinematic. Overall,
the stretching of space is indeed a helpful metaphor, though it has benefited
from critiques of misinterpretations. It does not describe any novel physical
law, but is simply a manifestation of curved spacetime. It might be said the
expansion is locally kinematic but not globablly so, at least there is no
global inertial frame.

The nature of redshifts is highly flexible in interpretation, however in the
Schwarzschild and FLRW cases symmetries of the matter and spacetime single out
the standard interpretation as most natural. Distances measured by an observer
are not given simply by the spacetime metric in general, but depend also on
the observer’s motion. Length-contraction does occur in general relativity,
which has major conceptual consequences including the need to reevaluate
certain astronomical calculations. Rigid acceleration of extended
$3$-dimensional bodies is possible in curved spacetimes, in specific cases.

##  13 Appendix A: Rigidity distance

I propose a new distance measure “rigidity distance” or “ruler distance” as
follows. Given the worldline of a particle as the starting event, extend a
Born-rigid ruler out to the specified end event, if possible. Then the
rigidity distance is the proper length of the ruler. The required inputs are:

  * $\centerdot$

motion of start particle (instantaneous position, velocity, and acceleration)

  * $\centerdot$

a time slicing $t=const$ (default assumed in FLRW, Schwarzschild)

  * $\centerdot$

position of end particle, at least at same time $t$.

This assumes the ruler follows a spacetime geodesic. It is highly dependent on
the start particle worldline. Distance is relative to the frame it is measured
in, but there may be an orthogonal or other natural frame as discussed
previously in section 6.4, or such a frame may be specified directly. There
are limits to the extent of a rigid body, depending on the acceleration of the
start particle and the spacetime geometry. In some cases such an extension is
impossible, for instance from inside a black hole event horizon to outside.

The motion of the far end of the ruler will not match the motion of the
destination location or particle, in general. Thus the measure is asymmetric
in the sense that if is between two particles, it depends on the motion of the
chosen start particle, but not the motion of the other. For nice properties,
this measure has both existence and uniqueness.

One application would be to the “tethered galaxy” scenario, where in this case
the tether is taken more literally.

##  14 Appendix B: Schwarzschild distance integrals

The Schwarzschild radial “proper distance”
$dR=\left(1-\frac{2M}{r}\right)^{-1/2}dr$ may be integrated via the
substitution $r=z^{2}$ (Taylor & Wheeler, 2000, p2-28) or computer algebra to
obtain a primitive function

$\sqrt{r(r-2M)}+2M\ln\left(\sqrt{r}+\sqrt{r-2M}\right).$ (172)

Then the proper distance is given by evaluating between some $r_{1}$ and
$r_{2}$.

The proper length interval of fishing line is
$f(r)dr=\left(1-\frac{2M}{r}\right)^{-1}\sqrt{1-\frac{2M}{r}+K^{2}}dr$, as
measured by the local Schwarzschild observer. The following closed-form
primitive function $F(r)$ satisfying $\frac{dF(r)}{dr}=f(r)$ follows from
computer algebra plus subsequent manual simplifying:

$F(r)=rW(r)-2KM\ln\left(1+2Kf(r)+2K^{2}\left(1-\frac{2M}{r}\right)^{-1}\right)+\frac{M(1+2K^{2})}{\sqrt{1+K^{2}}}\ln\left(r\left(\frac{M}{r}+W(r)^{2}+\sqrt{1+K^{2}}W(r)\right)\right).$
(173)

As above, the length is given by evaluation between some $r_{1}$ and $r_{2}$.

##  15 Appendix C: Counteracting coordinates

For an expanding FLRW universe, one could define a new coordinate system which
attempts to “remove” the expansion in which comoving coordinates remain at
constant “proper distance” $D$. The particles could be conceived to be tied by
rigid ropes, or affixed with rocket boosters, to maintain constant $D$, as
they are not freefalling in general. These coordinates are inspired in part by
an online physics forum post, for which I have lost the source. But the
derivation is done independently.

Actually this coordinate system fails to achieve its purpose, except in the
specific case of a Hubble constant $H$ fixed over time. The model fails to
consider length-contraction (as measured by Hubble comovers) due to motion
against the Hubble flow. It is included primarily as a negative result, to
argue against its validity should it be proposed by others.

The “proper distance” from $\chi$ to the origin is $R\chi$. Define a new
radial coordinate by $X=R(t)\chi$, and keep the other coordinates the same.
Taking the total derivative,

$dX=\dot{R}\chi dt+Rd\chi.$ (174)

Rearranging, $Rd\chi=dX-\dot{R}\chi dt$, and so

$R^{2}d\chi^{2}=dX^{2}+\dot{R}^{2}\chi^{2}dt^{2}-2\dot{R}\chi dXdt.$ (175)

Substituting the Hubble parameter $H\equiv\dot{R}/R$,

$R^{2}d\chi^{2}=dX^{2}+H^{2}X^{2}dt^{2}-2HXdXdt.$ (176)

Then $\chi$ is eliminated from the right-hand side. Now substitute this
expression into the FLRW metric:

$\displaystyle ds^{2}$
$\displaystyle=-dt^{2}+dX^{2}+H^{2}X^{2}dt^{2}-2HXdtdX+R^{2}S_{k}^{2}d\Omega^{2}$
(177)
$\displaystyle=-(1-H^{2}X^{2})dt^{2}-2HXdtdX+dX^{2}+R^{2}S_{k}^{2}d\Omega^{2}.$
(178)

Note the coordinate degeneracy when $g_{tt}=-(1-H^{2}X^{2})=0$, which occurs
when $X=\frac{1}{H}$, that is at the Hubble radius. This is to be expected,
because the Hubble radius is where the Hubble recession velocity is $c$, so a
particle attempting to stay at constant proper distance from the origin would
require a proper velocity of $c$ inwards. So this coordinate singularity
corresponds to a true physical limitation of the setup.

The energy-momentum tensor $T$ will transform accordingly.

Also the particles experience increasing time dilation with distance from the
origin, as measured by Hubble flow observers who proper time is $t$. For a
particle stationary in the new coordinates, $dX=d\theta=d\phi=0$, so its
$4$-velocity$\mathbf{u}$ satisfies

$-1=g(u,u)=-(1-H^{2}X^{2})(u^{t})^{2},$ (179)

so

$u^{t}=(1-H^{2}X^{2})^{-1/2}.$ (180)

This relates the coordinate and proper times via
$u^{t}\equiv\frac{dt}{d\tau}$. So the harder they attempt to resist the
expansion, the more time-dilated they are relative to comoving objects

Naïvely it may seem this achieves the goal of setting up observers at constant
distance. However this metric distance does not consider length-contraction
effects, in contrast to the “galaxy cable” of section 11.

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I gratefully acknowledge use of the following free images: the rocket by
Andreas Preuss, ruler and chain by 3lian, spiral by hmime, and the surface
with normal vectors by Oleg Alexandrov. Hyperlinks are present here and
elsewhere in the electronic version of this document.


11institutetext: a EPFL, Section of Financial Engineering
b University of Geneva, GSEM, Information Science Institute
c Cyber-Defence Campus, armasuisse Science and Technology
d EPFL, Mathematics Section

# TechRank: A Network-Centrality Approach for Informed Cybersecurity-
Investment

Anita Mezzettia; Dimitri Percia Davidb,c; Santiago Anton Morenod; Thomas
Maillartb; Michael Tsesmelisc ; Alain Mermoudc

###### Abstract

The cybersecurity technological landscape is a complex ecosystem in which
entities – such as companies and technologies – influence each other in a non-
trivial manner. Measuring the influence between entities is a tenet for
informed technological investments in critical infrastructure. To study the
mutual influence of companies and technologies from the cybersecurity field,
we consider a bi-partite graph that links both sets of entities. Each node in
this graph is weighted by applying a recursive algorithm based on the method
of reflection. This endeavor helps to measure the impact of an entity on the
cybersecurity market. Our results help researchers measure more precisely the
magnitude of influence of each entity, and allows decision-makers to devise
more informed investment strategies, according to their portfolio preferences.
Finally, a research agenda is suggested, with the aim of allowing tailor-made
investments by arbitrarily calibrating specific features of both types of
entities.

Acknowledgement: This research is supported by armasuisse Science and
Technology.

Corresponding author<EMAIL_ADDRESS>
###### Keywords:

critical infrastructure technology monitoring bipartite networks centrality
measure optimal investment.

##  1 Introduction

The operational continuity of critical infrastructure (CI) is central to a
functioning modern society. However, CIs are managed and monitored by
interdependent information systems, exposing CIs to cascading failures [5]. In
this high-risk environment, the security of information systems is crucial
[9].111The hack of the Colonial Pipeline in May, 2021 is one of the most
significant cyberattacks on a national CI in history. This case illustrates
the high-risk environment CIs operate in and hence the importance of making
informed technological investments that improve the (cyber) security of this
type of infrastructure. Source: https://www.bbc.com/news/technology-57063636
However, developing and implementing an effective defence strategy to protect
information-systems is a challenging task. The cybersecurity market is
continuously exposed to complex and fast-paced technological developments.
Research and development in the field is a costly and risky endeavour that
leads to sunk costs and poor development prospects. Hence, uncertainties
related to cybersecurity technologies – and companies that develop them – are
ubiquitous [1]. Statistics point to a start-up failure rate of 90%. In 42% of
cases, such failures are due to a misreading of market-demand, while in 29% of
cases, startups fail because they run out of funding [11]. The decision-making
process associated with cybersecurity investment and procurement has to work
in this uncertain environment, which makes it hard to achieve high returns on
investment.

Our objective is to help CI decision-makers to make more informed investments
when scouting and acquiring in the cybersecurity industry. For that purpose,
we first model and map the ecosystem of entities (i.e., technologies,
companies) from Crunchbase. This mapping reflects the cybersecurity market
through a bi-partite network. Then, we evaluate their relative influence in
the whole ecosystem by adapting a recursive algorithm that returns a network-
centrality measure. This should help decision-makers and investors
quantitatively assess the influence of each entity in the cybersecurity
ecosystem, thus reducing potential investment uncertainties and optimizing the
procurement process.

The remainder of this paper will be structured as follows: Section 2 presents
the related work; Section 3 presents the data and methods; Section 4 shows the
preliminary results; Section 5 sets the agenda for future work and discusses
the limitations on current research; Finally, Section 6 concludes the whole.

##  2 Related Work

Network-centrality measures – i.e., measures that assess the
importance/influence of nodes in networks – have been widely investigated (see
[2, 7] for extensive literature reviews in the field of social networks and
complex systems). Some research has focused on extensions and re-adaptations
of the well-know Google PageRank [8] algorithm, which also includes a bi-
partite network to improve its indexing capability. For instance, Hidalgo et
al. [4] developed the reflection method: an algorithm that characterizes the
structure of bi-partite graphs by iteratively calculating the mean value of
previous-step properties of a node’s neighbors. Similarly, previous work by
Klein at al. [6] assessed the relationship between editors and articles on
Wikipedia by extending the PageRank centrality measure. They developed a
recursive algorithm to measure how editor expertise influences the quality of
articles, and how contributions to articles influence editors’ expertise.
Their work shed light on cooperation and coordination dynamics, and the study
of these systems has helped better solve common interaction problems that
emerge within such social structures.

To the best of our knowledge, despite a prolific literature related to
economic valuation of cybersecurity investments (see [10] for an extensive
literature review), little work has tried to assess the importance/influence
of entities in a network, especially in the field of cybersecurity and CIs.
The idea of measuring the global rank of a node based on local information is
therefore still an open research gap. This work tries to use an approach
similar to Klein et al. to investigate the following research question: How
does an entity’s influence, predicated on its relations with other entities,
allow for more informed investment strategies?

##  3 Data and Methodology

We use a Crunchbase222https://www.crunchbase.com/; data downloaded on April
28th, 2021. (CB) dataset to build our bi-partite network, which is composed of
two node types : companies and technologies. CB encompasses information about
company activity and financing by leveraging big data and open-source
information in a semi-automated fashion. Data is sourced from investors and
the community of contributors. As emphasized by Dalle et al. [3], CB is widely
used by researchers because of the quality of its data and the usability of
the platform. For the sake of brevity, we refrain from a detailed dataset
description which can be found on the CB enterprise API.
333https://app.swaggerhub.com/apis-docs/Crunchbase/crunchbase-enterprise˙api/

Figure 1: Bi-partite network for selected cybersecurity companies (red nodes)
and technologies (blue nodes) they are working on. The nodes’ size represents
the number of neighbors.

Figure 1 gives an idea of the structure of the bi-partite network that
describes which technologies each company is working on.

We adapt the recursive algorithm developed by Klein at al. [6], based on the
method proposed by Hidalgo et al. [4]. This new method models the complex
structure of cooperation and competition occuring in the cybersecurity space.
The algorithm produces a ranking of technologies and companies from most to
least influential. This ranking thus condenses the positive influence of
experienced companies on technologies as well as the positive impact of
newborn companies on novel fields. In the same way, important technologies
will positively influence companies that are linked to them, and this process
iteratively increases the rank of both the companies and the technologies.
Moreover, in their paper on Wikipedia editorial work, Klein et al. [6] claim
that too many editors working on an article can sink the value of the
entities. Our network analysis also investigates this phenomenon of negative
influence in the context of cybersecurity technologies. This would mean for
instance that if too many companies work on the same technology, the business
gap narrows, and companies lose market share. This indisputably reduces a
company’s value.

Starting from Klein at al. [6], we build an adjacency matrix
$M_{c,t}\in\mathbb{R}^{N_{c},N_{t}}$ that takes value 1 if a company $c$ works
on a technology $t$ and 0 otherwise. $N_{c}$ represents the total number of
cybersecurity companies we are considering and $N_{t}$ the total number of
technologies. The aim of the algorithm is to assign a weight to every node,
which sums up its relevance within the graph: a contribution value for each
company and a quality value for each technology. The starting point consists
in measuring the expertise of a company ($w_{c}^{0}$) by summing up the number
of technologies it works on: $w_{c}^{0}=\sum_{t=1}^{N_{t}}M_{c,t}=k_{c}$. The
same holds for technologies: $w_{t}^{0}=\sum_{c=1}^{N_{c}}M_{c,t}=k_{t}$.

First, we recall that this algorithm is a Markov process: the step $w^{n}$
depends only on information available at the previous step $w^{n-1}$. At each
step, the method incorporates information about the expertise of companies and
the relevance of technologies, leveraging the bi-partite network structure.
The whole process can be seen as a random walker that jumps with a transition
probability that is zero in case $M_{c,t}=0$. We need to define two matrices
that explain how we move from one step to another one: they represent the
probability of jumping from technology $t$ to company $c$ and depends on the
initial conditions.

$\begin{cases}G_{c,t}(\beta)=\frac{M_{c,t}k_{c}^{-\beta}}{\sum_{c^{\prime}=1}^{N_{c}}M_{c^{\prime},t}k_{c}^{\prime-\beta}}\\\
G_{c,t}(\alpha)=\frac{M_{c,t}k_{t}^{-\alpha}}{\sum_{t^{\prime}=1}^{N_{t}}M_{c,t^{\prime}}k_{t}^{\prime-\alpha}}\end{cases}$
(1)

Klein at al. [6] also introduce two parameters, $\alpha$ and $\beta$, that
measure how coordination generates value. Thanks to $G_{e,a}$, we get the
recursive step:

$\begin{cases}w_{c}^{n+1}=\sum_{t=1}^{N_{t}}G_{c,t}(\beta)w_{c}^{n}\\\
w_{t}^{n+1}=\sum_{c=1}^{N_{c}}G_{c,t}(\alpha)w_{t}^{n}\end{cases}$ (2)

Similarly to PageRank, the recursion ends when the rank stabilizes.

The CB platform assigns a rank to the top companies – according to their
algorithm – in each industry. The CB rank takes into account the entity’s
strength of relationships, funding events, news articles, acquisitions,
etc.444https://about.crunchbase.com/blog/influential-companies/. We compare
our results in cybersecurity with this rank. We investigate the strength and
direction of the association between the two scores using Spearman’s rank
correlation coefficients.

Figure 2: Convergence evolution of the rank algorithm.

##  4 Preliminary Results

Our first results are based on the selection of all companies whose
description contains at least two words related to the field of cybersecurity.
We get a total of 2,443 companies and 478 technologies. Figure 2 shows that
the recursive algorithm introduced in section 3 converges for both companies
and technologies after a sufficient number of iterations (421 and 538
respectively).

As mentioned in section 3, we compare our ranking with the CB rank, hereafter
designated as the baseline. Thus, to make the ranks comparable, we convert our
algorithm’s output into a ranking. The resulting Spearman’s correlation
(0.014) shows that the two ranks are not correlated: even if the goals of the
TechRank and the CB rank are similar, this outcome reflects their substantial
differences. First, we do not know the exact mechanism by which CB (which is
not open source) ranks the entities, which offers little information to
investors trying to make investments according to technology or market
preferences. Moreover, the CB score focuses more on the level of activity of
the company, rather than its influence on the market. Among all the factors
that influence the CB ranking, nothing is said about the influence that
technologies have on the company’s value and market importance. Moreover, CB
simply ranks companies, while our algorithm assigns a weight, which gives not
just an ordering of the companies, but also gives a quantitative measure of
the distance between one entity and the next one. These discrepancies lead to
different results and we believe that our less-is-more approach, which
directly depends on the capabilities of the companies, is a good way for
investors to analyze an entity’s development prospects. This, together with
the next steps explained in section 5, could lead to a personalisable and
transparent approach to company rankings.

Another benefit of our methodology is the technology ranking, which allows
investors to make portfolio decisions not just based on company insights, but
also on technology insights. The TechRank methodology thus enables them to
create the portfolio that better reflects their preferences. Finally, a last
finding of the research points to the fact that new fields have a good impact
on companies working on them and that too much competition on a technology has
a negative impact on its neighbors, thus confirming our secondary hypothesis.

##  5 Further Steps

Our research agenda will focus on two steps: first, including the influence of
exogenous factors – such as impact of incubators and social aspects– and
second, creating the optimal portfolio strategy. The first step incorporates
the impact of previous investments, captured by another bi-partite structure
composed of investors and companies: each company is linked to its investors
and edges are weighted by the investment amount. Tracking the investments is a
relevant factor for a follow the money strategy and describes the investor
trust in the company. Once we have defined the entities to invest in, for the
last step, we use modern portfolio theory to maximise returns while minimizing
variance. The final outcome will allow investors to personalize the algorithm
according to their preferences thanks to a transparent a customisable
platform.

##  6 Conclusion

The aforementioned methodology constitutes the first step towards a new data-
driven investment strategy, which gives investors a succint and easily
understable ranking of companies and technologies based on their influence in
the market.

Thanks to the interdisciplinary core of this solution, investors can undertake
a transparent decision making process when dealing with highly complex
scenarios, such as in the cybersecurity market. In finance, the efficacy of
both “classical” technical and fundamental analysis is disputed by the
efficient-market hypothesis. Our TechRank is the kickoff for a complementary
(or even alternative) modern technological portfolio analysis for CI
operators. We believe also that the algorithm is extendable to every large
organization dealing with a high level of uncertainty. Therefore, we expect
that the TechRank algorithm will be applied and further developed in other
fields.

## References

  * [1] Anderson, R.: Why information security is hard - an economic perspective. In: Seventeenth Annual Computer Security Applications Conference (2001). https://doi.org/10.1109/ACSAC.2001.991552
  * [2] Battiston, F., Cencetti, G., Iacopini, I., Latora, V., Lucas, M., Patania, A., Young, J.G., Petri, G.: Networks beyond pairwise interactions: structure and dynamics. Physics Reports (2020)
  * [3] Dalle, J.M., den Besten, M., Menon, C.: Using Crunchbase for economic and managerial research. OECD Science, Technology and Industry Working Papers 2017/08, OECD Publishing (Nov 2017). https://doi.org/10.1787/6c418d60-en, https://ideas.repec.org/p/oec/stiaaa/2017-08-en.html
  * [4] Hidalgo, C.A., Hausmann, R.: The building blocks of economic complexity. Proceedings of the National Academy of Sciences 106(26), 10570–10575 (2009). https://doi.org/10.1073/pnas.0900943106
  * [5] Hines, P., Talukdar, S., et al.: Controlling cascading failures with cooperative autonomous agents 3(1), 192 (2007)
  * [6] Klein, M., Maillart, T., Chuang, J.: The virtuous circle of wikipedia: Recursive measures of collaboration structures p. 1106–1115 (2015), https://doi.org/10.1145/2675133.2675286
  * [7] Landherr, A., Friedl, B., Heidemann, J.: A critical review of centrality measures in social networks. Business & Information Systems Engineering 2(6), 371–385 (2010)
  * [8] Page, L., Brin, S., Motwani, R., Winograd, T.: The pagerank citation ranking: Bringing order to the web. Tech. rep. (1999), http://ilpubs.stanford.edu:8090/422/
  * [9] Percia David, D.: Three Articles on the Economics of Information-Systems Defense Capability. Ph.D. thesis, Université de Lausanne (2020)
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  * [11] Steward, J.: The ultimate list of startup statistics for 2021 https://findstack.com/startup-statistics/


# Decoupling Maximal Inequalities

Aryeh Kontorovich
<EMAIL_ADDRESS>
###### Abstract

A maximal inequality seeks to estimate $\mathop{\mathbb{E}}\max_{i}X_{i}$ in
terms of properties of the $X_{i}$. When the latter are independent, the union
bound (in its various guises) can yield tight upper bounds. If, however, the
$X_{i}$ are strongly dependent, the estimates provided by the union bound will
be rather loose. In this note, we show that for non-negative random variables,
pairwise independence suffices for the maximal inequality to behave comparably
to its independent version. The condition of pairwise independence may be
relaxed to a kind of negative dependence, and even the latter admits
violations — provided these are properly quantified.

##  1 Motivation

Maximal inequalities are at the heart of empirical process theory (van Handel,
2014). The case of Gaussian processes is well-understood via the celebrated
generic chaining technique (Talagrand, 2016). There, a key role in the lower
bounds is played Slepian’s inequality, which allows one to approximate a
Gaussian process by an appropriate uncorrelated one. The absence of a generic
analog of Slepian’s inequality — say, for the kind of Binomal process
considered in Cohen and Kontorovich (2022) — can be a major obstruction in
obtaining tight lower bounds. Indeed, as Proposition 3 below shows, for
nonnegative $X_{i}$, any upper bound on
$\mathop{\mathbb{E}}\max_{i}\tilde{X}_{i}$, where $\tilde{X}_{i}$ is the “the
independent version” of $X_{i}$, automatically yields an upper bound on
$\mathop{\mathbb{E}}\max_{i}X_{i}$. The reverse direction, of course, fails
without additional structural assumptions. We discover that pairwise
independence suffices for the reverse direction, and that this condition can
be relaxed further.

##  2 The Bernoulli case

Let $X_{1},X_{2},\ldots,X_{n}$ and
$\tilde{X}_{1},\tilde{X}_{2},\ldots,\tilde{X}_{n}$ be two collections of
Bernoulli random variables, where the $\tilde{X}_{i}$s are mutually
independent (and independent of the $X_{i}$s), with
$X_{i},\tilde{X}_{i}\sim\mathrm{Bernoulli}(p_{i})$. Letting
$Z=\sum_{i=1}^{n}X_{i}$ and $\tilde{Z}=\sum_{i=1}^{n}\tilde{X}_{i}$, we have

$\displaystyle\mathop{\mathbb{E}}\max_{i\in[n]}X_{i}=\mathbb{P}(Z>0),\qquad\mathop{\mathbb{E}}\max_{i\in[n]}\tilde{X}_{i}=\mathbb{P}(\tilde{Z}>0).$

### Decoupling from above.

An elegant result of Pinelis (2022) (answering our question) shows that
$\mathbb{P}(Z>0)\lesssim\mathbb{P}(\tilde{Z}>0)$; his proof provided for
completeness:

######  Proposition 1 (Pinelis).

For $c=\mathrm{e}/(\mathrm{e}-1)$ and $X_{i},\tilde{X}_{i},Z,\tilde{Z},p_{i}$
as above, we have

$\displaystyle\mathbb{P}(Z>0)\leq c\mathbb{P}(\tilde{Z}>0).$

###### Proof.

Put $M=\mathbb{P}(Z>0)$, $\tilde{M}=\mathbb{P}(\tilde{Z}>0)$ and
$S=\sum_{i=1}^{n}p_{i}$, and observe that

$\displaystyle M\leq\min\left\\{S,1\right\\}\leq c(1-\mathrm{e}^{-S}).$

On the other hand,

$\displaystyle\tilde{M}=1-\prod_{i=1}^{n}(1-p_{i})\geq 1-\mathrm{e}^{-S},$

whence $M\leq c\tilde{M}$. ∎

Further, we note that Pinelis’s constant $c=\mathrm{e}/(\mathrm{e}-1)$ is
optimal. Indeed, consider the case where $p_{i}=1/n$, $i\in[n]$, and
$\mathbb{P}(Z=1)=1$. This makes $\mathbb{P}(\tilde{Z}>0)=1-(1-1/n)^{n}\to
1-1/\mathrm{e}$ as $n\to\infty$.

Despite its elegance, Proposition 1 will likely have limited applications,
since in practice, the techniques for upper-bounding
$\mathop{\mathbb{E}}\max_{i}X_{i}$ rely on the union bound and are insensitive
to the dependence structure of $X_{i}$ — in which case the technique employed
in upper-bounding $\mathop{\mathbb{E}}\max_{i}X_{i}$ automatically upper-
bounds $\mathop{\mathbb{E}}\max_{i}\tilde{X}_{i}$ as well.

### Decoupling from below.

A more interesting and useful direction would be to obtain an estimate of the
form $\mathbb{P}(Z>0)\gtrsim\mathbb{P}(\tilde{Z}>0)$. Clearly, no such
dimension-free estimate can hold without further assumptions on the $X_{i}$.
Indeed, for a small $\varepsilon>0$, let
$\mathbb{P}(X_{1}=X_{2}=\ldots=X_{n}=1)=\varepsilon$ and
$\mathbb{P}(X_{1}=X_{2}=\ldots=X_{n}=0)=1-\varepsilon$. In this case,
$\mathbb{P}(Z>0)=\varepsilon$. On the other hand,
$\mathbb{P}(\tilde{Z}>0)=1-(1-\varepsilon)^{n}=n\varepsilon+O(\varepsilon^{2})$,
and so $\mathbb{P}(\tilde{Z}>0)/\mathbb{P}(Z>0)\to n$ as $\varepsilon\to 0$.
Nor can the ratio exceed $n$, since

$\displaystyle\mathop{\mathbb{E}}\max_{i\in[n]}\tilde{X}_{i}\leq\sum_{i=1}^{n}\mathop{\mathbb{E}}\tilde{X}_{i}\leq
n\max_{i\in[n]}\mathop{\mathbb{E}}\tilde{X}_{i}=n\max_{i\in[n]}\mathop{\mathbb{E}}X_{i}\leq
n\mathop{\mathbb{E}}\max_{i\in[n]}X_{i}.$

Let us recall the notion of pairwise independence. For the Bernoulli case, it
means that for each $i\neq j\in[n]$, we have
$\mathop{\mathbb{E}}[X_{i}X_{j}]=\mathop{\mathbb{E}}[X_{i}]\mathop{\mathbb{E}}[X_{j}]$.
The main result of this note is that pairwise independence suffices for
$\mathbb{P}(Z>0)\gtrsim\mathbb{P}(\tilde{Z}>0)$.

######  Proposition 2.

Let $X_{i},\tilde{X}_{i},Z,\tilde{Z},p_{i}$ be as above, and assume
additionally that the $X_{i}$ are pairwise independent. Then

$\displaystyle\mathbb{P}(Z>0)\geq\frac{1}{2}\mathbb{P}(\tilde{Z}>0).$

###### Proof.

By the Paley-Zygmund inequality,111 We thank Ron Peled for the suggestion of
applying Paley-Zygmund to $Z$.

$\displaystyle\mathbb{P}(Z>0)\geq\frac{(\mathop{\mathbb{E}}Z)^{2}}{\mathop{\mathbb{E}}[Z^{2}]}.$

Now $\mathop{\mathbb{E}}Z=\sum_{i=1}^{n}p_{i}$ and, by pairwise independence,

$\displaystyle\mathop{\mathbb{E}}[Z^{2}]=\sum_{i=1}^{n}p_{i}+2\sum_{1\leq
i<j\leq
n}p_{i}p_{j}=\sum_{i=1}^{n}p_{i}+\left(\sum_{i=1}^{n}p_{i}\right)^{2}-\sum_{i=1}^{n}p_{i}^{2}\leq\sum_{i=1}^{n}p_{i}+\left(\sum_{i=1}^{n}p_{i}\right)^{2}.$
(1)

Hence,

$\displaystyle\frac{(\mathop{\mathbb{E}}Z)^{2}}{\mathop{\mathbb{E}}[Z^{2}]}\geq\frac{\left(\sum_{i=1}^{n}p_{i}\right)^{2}}{\sum_{i=1}^{n}p_{i}+\left(\sum_{i=1}^{n}p_{i}\right)^{2}}.$

On the other hand, $\mathbb{P}(\tilde{Z}>0)$ is readily computed:

$\displaystyle\mathbb{P}(\tilde{Z}>0)=1-\prod_{i=1}^{n}(1-p_{i}).$

Therefore, to prove the claim, it suffices to show that

$\displaystyle
F(p_{1},\ldots,p_{n}):=2\left(\sum_{i=1}^{n}p_{i}\right)^{2}-\left(\sum_{i=1}^{n}p_{i}+\left(\sum_{i=1}^{n}p_{i}\right)^{2}\right)\left(1-\prod_{i=1}^{n}(1-p_{i})\right)\geq
0.$

To this end,222 This elegant proof that $F\geq 0$ is due to D. Berend, who
also corrected a mistake in an earlier, clunkier proof of ours. we factorize
$F=SG$, where $G=S+P+SP-1$, $S=\sum_{i}p_{i}$ and $P=\prod_{i}(1-p_{i})$.
Thus, $F\geq 0\iff G\geq 0$ and in particular, it suffices to verify the
latter. Now if $S\geq 1$ then obviously $G\geq 0$ and we are done. Otherwise,
since $P\geq 1-S$ trivially holds, we have $G\geq S(1-S)$. In this case,
$S<1\implies G\geq 0$. ∎

We conjecture that the constant $\frac{1}{2}$ in Proposition 2 is not optimal.
For a fixed $n$, define the joint pairwise independent distribution on
$(X_{1},\ldots,X_{n})$ — conjecturally, an extremal one for minimizing
$\mathbb{P}(Z=0)/\mathbb{P}(\tilde{Z}>0)$ — as follows: $p_{i}=1/(n-1)$,
$i\in[n]$, $\mathbb{P}(Z=0)=\frac{1}{2}-\frac{1}{2(n-1)}$, and
$\mathbb{P}(Z=2)=1-\mathbb{P}(Z=0)$. This makes
$\mathbb{P}(\tilde{Z}>0)=1-(1-1/(n-1))^{n}\to 1-1/\mathrm{e}$ as $n\to\infty$.
If our conjecture is correct, the optimal constant for the lower bound is
$c^{\prime}=\frac{\mathrm{e}}{2(\mathrm{e}-1)}$, or exactly twice Pinelis’s
constant.333 We thank Daniel Berend, Alexander Goldenshluger, and Yuval Peres
for raising the question of the constants. AG (and also Omer Ben-Porat)
pointed out a possible connection to prophet inequalities — and in particular,
the Bernoulli selection lemma in Correa et al. (2017)

### Relaxing pairwise independence.

An inspection of the proof shows that we do not actually need
$\mathop{\mathbb{E}}[X_{i}X_{j}]=p_{i}p_{j}$, but rather only
$\mathop{\mathbb{E}}[X_{i}X_{j}]\leq p_{i}p_{j}$. This condition is called
negative (pairwise) covariance (Dubhashi and Ranjan, 1998).

##  3 Positive real case

In this section, we assume that $X_{1},\ldots,X_{n}$ are nonnegative
integrable random variables and the $\tilde{X}_{1},\ldots,\tilde{X}_{n}$ are
their independent copies: each $\tilde{X}_{i}$ is distributed identically to
$X_{i}$ and the $\tilde{X}_{i}$ are mutually independent.

As a warmup, let us see how Proposition 1 yields
$\mathop{\mathbb{E}}\max_{i\in[n]}X_{i}\lesssim\mathop{\mathbb{E}}\max_{i\in[n]}\tilde{X}_{i}$:

######  Proposition 3.

Let $X_{1},\ldots,X_{n}$ be nonnegative and integrable with independent copies
$\tilde{X}_{i}$ as above. For $c=\mathrm{e}/(\mathrm{e}-1)$, we have

$\displaystyle\mathop{\mathbb{E}}\max_{i\in[n]}X_{i}\leq
c\mathop{\mathbb{E}}\max_{i\in[n]}\tilde{X}_{i}.$

###### Proof.

For $t>0$ and $i\in[n]$, put $Y_{i}(t)=\boldsymbol{1}[X_{i}>t]$,
$\tilde{Y}_{i}(t)=\boldsymbol{1}[\tilde{X}_{i}>t]$ and
$Z(t)=\sum_{i=1}^{n}Y_{i}(t)$, $\tilde{Z}(t)=\sum_{i=1}^{n}Y_{i}(t)$. Then

$\displaystyle\mathop{\mathbb{E}}\max_{i\in[n]}X_{i}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathbb{P}\left(\max_{i\in[n]}X_{i}>t\right)\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathbb{P}\left(Z(t)>0\right)\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle
c\int_{0}^{\infty}\mathbb{P}\left(\tilde{Z}(t)>0\right)\mathrm{d}t$
$\displaystyle=$ $\displaystyle
c\int_{0}^{\infty}\mathbb{P}\left(\max_{i\in[n]}\tilde{X}_{i}>t\right)\mathrm{d}t$
$\displaystyle=$ $\displaystyle
c\mathop{\mathbb{E}}\max_{i\in[n]}\tilde{X}_{i}.$

∎

For pairwise independent $X_{i}$, we have a reverse inequality:

######  Proposition 4.

Let $X_{1},\ldots,X_{n}$ be nonnegative and integrable with independent copies
$\tilde{X}_{i}$ as above. If additionally the $X_{i}$ are pairwise
independent, then

$\displaystyle\mathop{\mathbb{E}}\max_{i\in[n]}X_{i}\geq\frac{1}{2}\mathop{\mathbb{E}}\max_{i\in[n]}\tilde{X}_{i}.$

###### Proof.

The proof is entirely analogous to that of Proposition 3, except that
Proposition 2 is invoked in the inequality step. ∎

### Relaxing pairwise independence.

As before, the full strength of pairwise independence of the $X_{i}$ is not
needed. The condition
$\mathbb{P}(X_{i}>t,X_{j}>t)\leq\mathbb{P}(X_{i}>t)\mathbb{P}(X_{j}>t)$ for
all $i\neq j\in[n]$ and $t>0$ would suffice; it is weaker than pairwise
negative upper orthant dependence (Joag-Dev and Proschan, 1983).444 Thanks to
Murat Kocaoglu for this reference.

##  4 Back to Bernoulli: beyond negative covariance

What if the Bernoulli $X_{i}$ do not satisfy the negative covariance condition
$\mathop{\mathbb{E}}[X_{i}X_{j}]\leq p_{i}p_{j}$? Proposition 2 is not
directly inapplicable, but not all is lost. For $i\neq j\in[n]$, define
$\eta_{ij}$ by

$\displaystyle\eta_{ij}=\left(\mathop{\mathbb{E}}[X_{i}X_{j}]-p_{i}p_{j}\right)_{+}$

and put $\eta_{ii}:=0$. Thus, $\mathop{\mathbb{E}}[X_{i}X_{j}]\leq
p_{i}p_{j}+\eta_{ij}$, and, repeating the calculation in Eq. (1),

$\displaystyle\mathop{\mathbb{E}}[Z^{2}]\leq\sum_{i=1}^{n}p_{i}+\left(\sum_{i=1}^{n}p_{i}\right)^{2}+\sum_{i,j\in[n]}\eta_{ij}.$

Let us put $S=\sum_{i=1}^{n}p_{i}$, $A=S^{2}$, $B=S+S^{2}$,
$C=\frac{1}{2}\mathbb{P}(\tilde{Z}>0)$, and $H=\sum_{i,j\in[n]}\eta_{ij}$.
Now, for $A,B,C,H\geq 0$, we have

$\displaystyle\frac{A}{B}\geq C$ $\displaystyle\implies$
$\displaystyle\frac{A}{B+H}\geq c\left(1-\frac{H}{B+H}\right).$

and so we obtain a generalization of Proposition 2:

######  Proposition 5.

Let $X_{i},\tilde{X}_{i},Z,\tilde{Z},p_{i},B,H$ be as above. Then

$\displaystyle\mathbb{P}(Z>0)\geq\frac{1}{2}\left(1-\frac{H}{B+H}\right)\mathbb{P}(\tilde{Z}>0).$

When $H\lesssim B$, Proposition 5 yields useful estimates.

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