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Measurements in general relativity

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A physical theory is more than simply its mathematical structure. It also requires some mapping of those elements to real life measurements. The standard description of this process can be found for instance in [1][2], and can be roughly understood as, given a set of measurements performed on the real world, a physical theory is a mathematical model (in the sense of model theory) $F$, called the formalism, along with the mapping $\mathrm{CP}$, the correspondance principle.

\begin{equation} \mathrm{Meas.} \xrightarrow{\mathrm{CP}} \mathrm{F} \end{equation}

This mapping relates the physical measurements of a theory with the mathematical quantities in the formalism. The set of rules involved are called the rules of correspondance.

Although such a mapping is always difficult, or at least somewhat arbitrary, it is trickier than it appears in general relativity due to our lack of knowledge of the geometry of the overall spacetime. Many experiments in general relativity have some fairly broad assumptions to allow us to assume the spacetime metric, but is it actually possible to measure it with a minimal amount of assumptions?

This idea has been around for almost as long as general relativity has existed, but a more focused discussion of it goes back to a variety of work by Ruse[3], Synge[4], Reichenbach[5], Ehlers, Pirana, etc.

In this field, there are two main methods used to measure local spacetime quantities, the chronometric approach, where one trusts the onboard clock of observers to determine further quantities, and the structure approach, in which one looks for the relation between families of curves to find out the underlying geometry. Both of these have their benefits and drawback that we will look into.

1. Structures on a spacetime

Our first step is to determine exactly what we are trying to measure. The obvious candidates are of course the metric tensor and, if our connection isn't the Levi-Civita one, the connection associated to it. Different kinds of measurements may not necessarily give us enough details to actually go back to the metric tensor, but only more general equivalence classes of metrics. This is related to the notion of structures on a spacetime. The metric tensor itself is associated to the metric structure, where, given two spacetimes $(M, g)$ and $(M', g')$, we say that they share the same metric structure if there exists a diffeomorphism $f : M \to M'$ such that $f^*g = g'$. Other types of structures will have similar definitions

A set of further important structures for a spacetime are

• The conformal structure, where two spacetimes are equivalent if $f^*g = e^{2u}g'$ for some smooth function $u$
• The affine structure, where two spacetimes are equivalent if the geodesics are invariant, $f^* \nabla = \nabla'$
• The projective structure, where two spacetimes are equivalent if their unparametrized geodesics are invariant
• The causal structure, where two spacetimes are equivalent if the causal ordering is invariant.

Those structures are encoded in the notion of the Cartan geometry of the manifold, and on a local level, by the Klein geometry that the manifold is locally equivalent to.

1.1. Klein geometry

Every spacetime structure will correspond to some local geometric model called a Klein geometry. In the case of a spacetime, this will be some geometry based on the Lorentz vector space $\mathrm{Lor}^n$, which will be later on associated with the tangent space $T_pM$. The model geometry of our Lorentz space is $\mathrm{Lor}^n = (\mathbb{R}^{(n-1), 1}, \langle \cdot, \cdot\rangle_{\mathrm{Can}})$. If we equip it with the canonical basis $\{ e_\mu \}$ (with the associated dual basis $\{ \theta^\mu \}$ such that $\theta^{\mu}[e_\nu] = \delta^{\mu}_{\nu}$), the canonical quadratic form $\langle \cdot, \cdot\rangle_{\mathrm{Can}}$ is

\begin{equation} \langle e_\mu, e_\nu \rangle_{\mathrm{Can}} = \eta_{\mu\nu} \end{equation}

with $\eta$ the Minkowski metric $\eta = \mathrm{diag}(-1, 1, \ldots)$. A few other structures that we may consider on Minkowski space as a vector space are the canonical volume element $\varepsilon$,

\begin{equation} \varepsilon = \bigwedge_{\mu = 0}^{n-1} \theta^\mu \end{equation}

We will also consider more specifically the spatial volume form, $\varepsilon^{(n-1)}$

\begin{equation} \varepsilon^{(n-1)} = \bigwedge_{\mu = 1}^{n-1} \theta^\mu \end{equation}

We will consider some timelike form, such as the canonical one,

\begin{equation} T = \theta_0 \end{equation}

and the light cones, defined by

\begin{equation} \mathcal{C} = \left\{ v \in \mathrm{Lor}^n | \eta(v, v) = 0 \right\} \end{equation}

We will also consider the past and future light cones

\begin{equation} \mathcal{C}^{\pm} = \left\{ v \in \mathrm{Lor}^n | \eta(v, v) = 0, v^0 \in \mathbb{R}^{\pm} \right\} \end{equation}

Not all our Klein geometries will be based directly on $\mathrm{Lor}^n$, some will be based on derived spaces such as the projective Lorentz space, corresponding to the quotient of the deleted Lorentz space ($\mathrm{Lor}^n \setminus \{ 0 \}$) with the equivalence relation $v \sim v'$ such that $v' = k v$ for $k \in \mathbb{R}^*$.

\begin{equation} P\mathrm{Lor}^n = \mathrm{Lor}^n \setminus \{ 0 \} / \sim \end{equation}

Members of the projective Lorentz space are then equivalence classes

\begin{equation} [u] = \left\{ v \in \mathrm{Lor}^n \setminus \{ 0 \} | \exists k \in \mathbb{R}^*. u = kv \right\} \end{equation}

We of course have the injection of the deleted Lorentz space in the deleted Lorentz space via the canonical projection

\begin{eqnarray} P\mathrm{Lor}^n &\hookrightarrow& \mathrm{Lor}^n \setminus \{ 0 \} \\ &\mapsto& \end{eqnarray}

For the conformal structure of our spacetime, we will need to define the Möbius space of Minkowski space.

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All these structures of Minkowski spacetime have their own associated Klein geometry, via groups leaving those structures invariant. In particular, the Klein geometry associated with the metric tensor is the one defined by the Poincaré group and its Lorentz subgroup,

\begin{equation} P = \mathbb{R}^{(n-1), 1} \rtimes \mathrm{SO}(n-1, 1) \end{equation}

Causal structure

Like the conformal structure, the causal structure even of Minkowski space requires some care to cover.

Generalized twistors, correspondence

1.2. Connections and Cartan geometries

As a brief reminder on connections, a connection on a fiber bundle

At a geometric level, those structures are defined by a Cartan geometry. On a bundle level, a Cartan geometry is composed of a principal bundle $P$ with gauge group $G$, a connection on this bundle $\nabla$, a subgroup $H$ with inclusion $\iota : H \hookrightarrow G$, and a local model space formed by the coset $G/H$.

The most common Cartan geometry used in general relativity is the Lorentz geometry, where we consider the model space $\mathbb{R}^{(n-1), 1}$, with the Poincaré gauge group $P = \mathbb{R}^{(n-1), 1} \rtimes \mathrm{SO}(n-1, 1)$ and its Lorentz subgroup $\mathrm{SO}(n-1, 1)$

Connection as a horizontal distribution on $TP$

A vector $X$ is said to be parallely transported along a curve $\gamma$ if, given our connection $\nabla$, we have

\begin{equation} \nabla_{\dot{\gamma}} X(\gamma) = 0 \end{equation}

The parallel transport operator on a curve $\gamma$ from points of a curve $\lambda_1$ to $\lambda_2$ is defined by a linear isomorphism between the tangent space at those points

\begin{equation} P_{\gamma}^{\lambda_1, \lambda_2} : T_{\gamma(\lambda_1)} M \to T_{\gamma(\lambda_2)} M \end{equation}

That maps vectors at $\gamma(\lambda_1)$ to their parallel transport at $\gamma(\lambda_2)$, so that every value between $\lambda_1$ and $\lambda_2$ defines a vector field on $\gamma$ that obeys the parallel transport condition. On an infinitesimal level, this corresponds to the notion that

\begin{equation} \nabla_{\dot{\gamma}(\lambda)} X = \frac{d}{ds} \left[ P_{\gamma}^{\lambda, s} X \right]_{s = 0} \end{equation}

It can be shown that this differential equation for the parallel transport can be solved via the iterated integral

\begin{equation} P_{\gamma}^{\lambda_1, \lambda_2} = \exp\left[ \int \gamma^* \omega \right] \end{equation}

The parallel transport operator obeys the following properties :

1. $$P_{\gamma}^{\lambda, \lambda} = \mathrm{Id}$$
2. $$P_{\gamma}^{\lambda_1, \lambda_2} \circ P_{\gamma}^{\lambda_2, \lambda_3} = P_{\gamma}^{\lambda_1, \lambda_3}$$

Holonomy

Geodesics

In the neighbourhood of a given point $p$, there exists a Cauchy normal neighbourhood $U_p$ in which, for any $q \in U_p$, there exists a unique geodesic connecting $p$ and $q$, noted $\vec{pq}$, such that $\vec{pq}(0) = p$ and $\vec{pq}(1) = q$.

Exponential map

World function

One quantity that will serve us quite often throughout this paper

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Structures

The structures on a spacetime can all be defined in terms of higher order $G$-structures, which can be described as a type of Cartan connection where the principal bundle we consider is the (higher order) frame bundle $\mathrm{Fr}^k M$, with structure group $G^k_n$, the jet group. For a few structures, the case $k = 1$ will be sufficient, where $G^k_n = \mathrm{GL}(n)$, but for the general case, as well as for some practical reasons, it is best to stick with the general case, although there won't be any need to go beyond the $k = 3$ case.

First, we need to introduce the jet equivalence of curves. Two curves $\gamma$ and $\gamma'$, defined on some interval $I, I'$ such that $0$ belongs to both, then those two curves have $k$-th order contact, noted $\gamma \sim_k \gamma'$, if for any smooth real function $\phi$, we have

\begin{equation} \forall i \leq k,\ \left( \frac{d^i}{d\lambda^i} \left[\phi \circ \gamma - \phi \circ \gamma'\right] \right)_{\lambda = 0} = 0 \end{equation}

This is true in particular for a smooth real function defined by the projection of a coordinate map onto a single component, $x^\mu = \mathrm{pr}_\mu \circ x$, meaning that in any coordinate system, the two curves share the same components for their derivatives at $0$.

The jet spaces we will be most interested in will be the jet spaces of curves and maps, those of the mapping spaces $\mathrm{Map}(\mathbb{R}, M)$ and $\mathrm{Map}(\mathbb{R}^n, M)$. of $C^\infty(M,M)$, $J^k(M,M)$, or $J^k(M)$ for short, as well as the jet space $J^k(\mathbb{R}^n, M)$. $J^k(M)$ will be involved in the

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Our principal bundle is therefore the $k$-th order frame bundle $\mathrm{Fr}^k M$, with structure group $G^k_n$. A higher order $G$-structure in this context is then a reduction of the structure group to some subgroup of $G^k_n$

If we have two structures defined by the subgroups $H$ and $H'$, such that $H'$ is a subgroup of $H$, the two structures are compatible

[Condition on the sections of the associated homogeneous bundles $G/H$ and $G/H'$]

Causal hierarchy and splitting structure

While less fundamental and somewhat arbitrary compared to the other structures we've seen, there are a few structures associated to observers and therefore measurements that are important. These can be constructed with varying degrees of complexity depending on the spacetime and observers involved.

The fundamental basis for a splitting structure is a congruence of observers foliating the spacetime, modeled by a foliation of the spacetime by timelike curves. As any spacetime admits a timelike line element field[Steenrod], there always exists a foliation by timelike curves, although any additional structure on the foliation is not guaranteed. Even in the one-dimensional case, we are not guaranteed the existence of a typical leaf, as the spacetime may be partially foliated by closed timelike curves.

This foliation can be defined equivalently by a foliation $\mathcal{F}$ of the manifold by unparametrized curves, or by the one-dimensional timelike distribution that they span. Given a local parametrization $f : (a,b) \to \mathcal{F}_p$ of a leaf, the distribution is simply

\begin{equation} \Delta_t = \mathrm{Span}(d f) \end{equation}

Our congruence of observers induces a splitting of the tangent bundle, where we are free to choose a linearly independent spacelike $(n-1)$-distribution $\Delta_\sigma$ such that

\begin{equation} TM = \Delta_t \oplus \Delta_\sigma \end{equation}

This will give us locally the existence of a frame for a given observer, but in our current context, this distribution is not guaranteed to be integrable or even orientable.

To get a more practical structure to work with, causality conditions must be imposed first. The most common one being time orientability, which implies that any such foliation can be generated by a timelike vector field $\mathfrak{t}$ of which the foliation is the set of integral curves. There is a variety of such vector fields, associated to the possible parametrizations of each curves, but we will usually choose the proper time parametrization such that $g(\mathfrak{t}, \mathfrak{t}) = -1$, which can be gotten by any other timelike vector field via $X \to X / \sqrt{|g(X,X)|}$. We will define in this context the observer space $\mathcal{O}$ of future-oriented timelike vector of unit norm, which will define equivalently the observer foliations on $M$.

[Volume functions $t^\pm$]

If the spacetime is chronological, the foliation by observer is composed of leaves $L \cong \mathbb{R}$

If the spacetime is stably causal, it admits a time function $t$, such that for any two points $p, q$ on a leaf $L$, we have that

If the spacetime is globally hyperbolic, there exists a Cauchy hypersurface $\Sigma$, a submanifold of codimension $1$, such that every curve of the foliation intersects it orthogonally exactly once.

The typical case of a synchronization structure occurs for globally hyperbolic spacetimes, where one could define a temporal function $t : M \to \mathbb{R}$ such that $t$ is smooth and strictly increasing over every future-directed causal curve, with $dt$ a past-directed $1$-form. This also defines a foliation on our spacetime via $t^{-1}(p) = \Sigma_t$.

Any spacetime can be foliated by timelike curve. Such a choice of foliation (also called an observer field) gives it the structure of a principal bundle, with principal group $\mathbb{R}$

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With this discussion of observers, there are two possible, somewhat complementary, structures that we can define. We can either foliate the spacetime by timelike curves or by spacelike hypersurfaces.

As we've seen, we can always foliate the spacetime into timelike curves, but while we do not need to generally stick to the globally hyperbolic case, we will usually at least ask that our spacetime has the structure of an $\mathbb{R}$-principal bundle. In this case, the bundle structure is defined by $\pi : M \to \Sigma$, with $\Sigma = M / \mathcal{F}$ the leaf space of our foliation, and the typical fiber being the timelike curve of an observer.

\begin{equation} \pi : M \to \Sigma \end{equation}

The left action on the bundle $\ell : G \times L \to L$ corresponds to a time translation along the observer's line, and the

The fibers will be spacelike hypersurfaces $\Sigma_t = \pi_t^{-1} (t)$. As a fiber bundle, there is a local trivialization

This type of structure is called a splitting structure. They occur in particular for the study of foliations. Given a $k$-foliated $n$-manifold, we have a reduction of the structure group

\begin{equation} \mathrm{GL}(k) \times \mathrm{GL}(n - k) \hookrightarrow \mathrm{GL}(n) \end{equation}

As we are working on a bundle made from the spacetime itself, the structures of the vertical and horizontal space are directly in the tangent bundle, which simply split along the tangent of our time function.

\begin{equation} V_tT \cong \mathrm{R}^{n-1} \end{equation}

As our horizontal space is one-dimensional, there is no freedom in the choice of a horizontal distribution

Abstract measurements of our structures

On a mathematical level, there are several things which are equivalent to the measurements of our various structures, which will later on be useful for the physical measurement of them.

Given a connection form $\omega$ on a principal bundle $P$, we have a one. to one correspondance with the parallel transport operator.

Coordinate fixing of the observer field

Given an appropriate observer field, the coordinates on the manifold may be fixed.

The obvious fixed coordinate is that, given an observer field $\mathcal{O}$ with a time function $h$, we can

2. Spacetime experiments

There are a variety of experiments we can perform in spacetime (all experiments are somewhat experiments on spacetime in some way), but we will focus on the ones where they may be idealized by clocks, massive particles and light rays. Other types of mapping of experiments to theoretical objects, such as fields, extended objects or Hilbert space vectors, may be more accurate to the idea we have of it, but as a first approximation, mapping them to causal curves is much more tractable.

2.1. Clocks in general relativity

A clock in general relativity consists in a future-directed timelike curve $\gamma$, along with a scalar function $h$ on that curve, such that $h$ is a strictly increasing function of the parameter. In other words, our function $h$ maps spacetime points of the curve to $I \subset \mathbb{R}$ via $h(\gamma(\tau))$. Being strictly increasing, and timelike curves being themselves homeomorphisms (if they are causal), there is therefore a corresponding inverse function $h^{-1}$.

\begin{equation} h^{-1}(h(\gamma(\tau))) = \gamma(\tau) \end{equation}

If you take the reparametrization defined by $f = \gamma^{-1} \circ h^{-1}$, the new curve $\gamma' = \gamma \circ f$ is such that

\begin{eqnarray} h(\gamma'(\tau')) &=& h(\gamma \circ \gamma^{-1} \circ h^{-1}(\tau'))\\ &=& \tau' \end{eqnarray}

Therefore, for a causal curve and clocks defined on the proper range of values, a clock always corresponds to a parametrization of the curve.

A clock is a standard clock if it corresponds to a proper time parametrization, so that, if $\tau$ is such a parametrization,

\begin{equation} h(\gamma(\tau)) = \alpha \tau + t_0 \end{equation}

with only some freedom of the scale $\alpha$ and origin $t_0$. Otherwise, a non-standard clock will be related to standard time via some reparametrization. As we have the relation that for affine parametrizations,

\begin{eqnarray} \sigma_{\gamma}(\tau_1, \tau_2) &=& \int_{\tau_1}^{\tau_2} d\tau\\ &=& \tau_2 - \tau_1 \end{eqnarray}

Standard clocks give us a measure of the spacetime interval of our observers, up to some rescaling. Non-standard clocks cannot define it without knowing the metric itself, though. In addition, if our observer is geodesic, we have

\begin{eqnarray} \sigma(\tau_1, \tau_2) &=& \frac{1}{2} \sigma^2_{\gamma}(\tau_1, \tau_2) &=& \frac{1}{2} (\tau_2 - \tau_1)^2 \end{eqnarray}

for the world function $\sigma$.

On a more practical level, actual clock measurements will usually be discrete, such as counting the number of oscillations for the radiation of a given atom.

On a more basic structural level, a clock can also simply serve as a away of ordering points on the curve, if we define the order relation via $\gamma(a) < \gamma(b)$ if $a < b$.

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Asserting the timing of events in experimental general relativity is the first thing we need to do, for all the other measurable quantities will usually stem from the foliation of our spacetime into spacelike hypersurfaces, so that we can use proper Riemannian geometry to compute the distances and angles.

While there are certainly quantities related to time for theoretical entities in general relativity, these are fairly arbitrary, although this is also true of classical theories, where the point of origin, direction and scale of time measurements are somewhat arbitrary. There's no way to provide measurements of the time coordinate from the manifold's atlas, nor from any time function we may define, as we can change any such quantities via diffeomorphisms or a different slicing of our manifold. We can't even guarantee that such a foliation of our spacetime into spacelike hypersurfaces exist, so that our slicing even makes sense.

As we (probably) cannot get global informations from our spacetime manifold, all definition will therefore have to be local. Therefore, we'll require all our definitions to be associated with an observer $\gamma$, represented by a timelike curve. An obvious number we can associate with the observer is the parameter $\tau$, such that we can define our time for the observer as

\begin{equation} \forall p \in \text{Im}(\gamma),\ t(p) = \tau \text{ such that } \gamma(\tau) = p \end{equation}

but of course this parameter is fairly arbitrary, as we can simply redefine a parameter for our curve. Therefore, we have to tie our definition of time to a measurable process.

If we look at modern physical theories, there doesn't seem to be any process which is exactly periodic, but this does not matter. Our notion of time is defined independently of any such considerations. We'll therefore define a notion of clock here :

In other words, every such occurences of the event is defined to be $T$ of our measuring unit from the previous one. If we define some original event $e_0$ such that $t(e_0) = 0$, then we end up with the measurement of time

\begin{equation} t(e_{k}) = k T \end{equation}

or more generally $t_0 + kT$ if we define $t(e_0) = t_0$.

Atomic clocks

A particularly useful standard clock is the one given for the second.

Therefore, if we have an atomic clock along with our observer, we'll define a metric (as in metric system here, not metric tensor) parametrization of the curve if, for some initial time parameter $\tau_0$, and given the original proper time parametrization $\tau$ and the interval between two maximal amplitudes as $T$, our new parametrization is

\begin{equation} \tau_m = \frac{\tau - \tau_0}{T} \end{equation}

Unfortunately, all we have so far is a way to measure only the observer's own time, which is of limited use. The next step is to find a way to assign a time to non-local events.

Light clocks

Another common way of measuring time in general relativity is the notion of light clocks.

2.2. Test particles

We have two kinds of test particles we can use : massive particles of mass $m$, represented by timelike curves, and massless particles, represented by null curves. Additionally, as we deal with particles, our curves may carry some additional informations, such as a polarization vector.

We assume that our test particles have, up to some uncertainty, a known boundary condition : an observer measures their passage at a time $t$, at an angle $\theta^a$, and, depending on their mass $m$, a momentum $p$ or frequency $\omega$. from the point of view of the observer, this is their causal curve intersecting at coordinates $(t, 0)$, with the momentum

\begin{eqnarray} p^\mu = (E, x^a(\theta^b) p) \end{eqnarray}

with $x(\theta)$ the function mapping angles to the Cartesian basis, so that, in 3 dimensions,

\begin{eqnarray} x^1(\theta, \varphi) &=& \sin(\theta) \cos(\varphi)\\ x^2(\theta, \varphi) &=& \sin(\theta) \sin(\varphi)\\ x^3(\theta, \varphi) &=& \cos(\theta) \end{eqnarray}

and $E = \sqrt{m^2 - p^2}$. Similarly, the frequency is

\begin{eqnarray} p^\mu = (E, x^a(\theta^b) p) \end{eqnarray}

If they also carry some kind of polarization

2.3. Observers

Our observers are composed of timelike curves $\gamma$, along with some or all of the following features :

• A clock $h$
• A local frame $e_a(\tau)$
• A spin vector $S(\tau)$

Our observer is defined, as far as our spacetime goes, by its initial conditions $\gamma(0)$, $\dot{\gamma}(0)$, $e_a(0)$, its parametrization $h$ of its onboard clock, its acceleration $a = \nabla_{\dot{\gamma}}\dot{\gamma}$, and its rotation $\Omega$ such that $D_\tau e_a = -\Omega(e_a)$.

If our observer has a standard clock, zero acceleration and a Fermi-Walker frame, we call it a standard observer.

Local spacetime experiments

Before we consider a more general case, we first need to work out what happens locally, in other words in some appropriate neighbourhoods which have the most common properties with Minkowski space.

3. Measurements in special relativity

As the model space from which all the local structures are gotten on a spacetime, it may be a good idea to first check that we can do so efficiently in special relativity.

Minkowski space as a manifold is quite simply $\mathbb{R}^{(n-1), 1}$ with the identity map, and it possesses a flat connection

Clocks and synchronization in special relativity

The simplest case we can work with here is of course special relativity, in Minkowski space. We'll consider the two dimensional case for simplicity, although this generalizes quite readily. In this case, let's consider two observers, $\gamma_1$ and $\gamma_2$. There's a few cases we should be mindful for.

Out of all of these, the simplest case is to consider two timelike geodesic curves. In Minkowski space, this is simply

\begin{equation} x_i(\tau) = (\gamma_i \tau + t_0, \beta_i \gamma_i \tau + x_0) \end{equation}

which indeeds obeys the geodesic equation, and is timelike if

\begin{equation} \gamma_i = \frac{1}{\sqrt{1 - \beta_i^2}} \end{equation}

our usual Lorentz factors. Also of use is the parametrization of a null geodesic,

\begin{equation} x_i(\tau) = (\tau + t_0, \pm \tau + x_0) \end{equation}

If we perform the synchronization by exchange of light signals, $\gamma_1$ emitting $\ell_1$ at $\tau_1$, $\gamma_2$ receiving it at $\tau_2'$ before emitting $\ell_2$ immediately which $\gamma_1$ receives at $\tau_1''$, then the Reichenbach synchronization is, for $\varepsilon \in (0,1)$,

\begin{equation} \tau_1' = \tau_1 + \varepsilon (\tau_1'' - \tau_1) \end{equation}

If we consider an observer $\gamma$ starting at $\gamma(0) = (0,0)$ emitting light signals in both directions at every instant $t$, $\ell^\pm_t$, we have that \begin{equation} \ell^\pm_t(\lambda) = (\tau + t_0, \pm \tau + x_0) \end{equation} any event $p = (t, x)$ will intersect $\ell^\pm_t$ as

\begin{equation} \tau_1' = \tau_1 + \varepsilon (\tau_1'' - \tau_1) \end{equation}

The more common synchronization is of course Einstein's synchronization, which is simply $\varepsilon = 1/2$

\begin{equation} \tau_1' = \tau_1 + \frac{1}{2} (\tau_1'' - \tau_1) \end{equation}

Local measurements in general relativity

The type of neighbourhood that will be the closest to approximate a measurement in Minkowski space is the radar neighbourhood. In any spacetime, there exists for any point $p$ a geodesically convex neighbourhood $U_p$ such that for any two points $q, r \in U_p$, there exists a unique geodesic $\vec{qr}$ entirely within $U_p$.

The photon bounce in a radar neighbourhood

One of the most fundamental measurement in general relativity is the photon bounce, or radar process, by which an observer is sending a light beam to an object which is then reflected.

Our observer is a timelike curve $\gamma$ equipped with a clock $h$, such that, at time $t_1$, a null geodesic $\ell_1$ departs from $\gamma(\tau_1) = p_1$, with tangent vector $k^1$, intersects a point $q$, and another null geodesic $\ell_2$ connects $q$ to another point $p_2 = \gamma(\tau_2)$, at time $t_2$. We therefore have $\ell_1 = \vec{p_1q}$ and $\ell_2 = \vec{q p_2}$.

From a purely physical standpoint, what we can get out of this experiment (up to some idealizations) is, for both the emission and reception of the photon, the time $t_i$ at which the event occured, the angles $\theta_i^a$ with respect to the local tetrads at $p_i$, as well as possibly additional informations such as the frequency $\omega_i$ and polarization $\varepsilon_i$. With those informations, we need to find the mapping to the specific spacetime point $q$.

The calculation for this was performed by Synge[]. First, let's work out a bit on the mapping of our measurements. Our original clock measurements are $t_1$ and $t_2$. We have some reparametrization of our curve $\gamma' = \gamma \circ h$ so that $\gamma'$ is parametrized by proper time $\tau = h(t)$, with $\tau_i = h(t_i)$. Our angles $\theta_i$ give us the unit null vectors

First, we need to rework our parametrization. Let's consider the points $p_\tau = \gamma(\tau)$, and the unique geodesics connecting them to $q$, $\gamma_{\tau}$, with tangent vector $u_\tau$. From the fact that we're in a normal neighbourhood, $\gamma_\tau$ is a spacelike geodesic for $\tau \in (\tau_1, \tau_2)$, and equal to $\ell_1, \ell_2$ at the boundaries. $\gamma_\tau$ is a past-pointing null curve at $\tau_2$, and a future-pointing null curve at $\tau_1$, therefore, the function $g(\dot{\gamma}(\tau), u_\tau)$ is of different signs at the boundary, and reaches $0$ for some value of $\tau$ [UNIQUE?]. We will note this point $r$, connecting to $q$ via the curve $\sigma$ (of length $\sigma$), of tangent $\mu$, and shifting our parameter to $s$, so that $\gamma(0) = r$.

Those curves form an admissible family of curves $\Gamma$,

\begin{eqnarray} \Gamma : [\tau_1, \tau_2] \times [0,1] &\to& M\\ (\tau, \lambda) &\mapsto& \Gamma(\tau, \lambda) = \gamma_\tau(\lambda) \end{eqnarray}

By the symmetry lemma, we get that

\begin{eqnarray} D_\tau D_\lambda \Gamma = D_\lambda D_\tau \Gamma \end{eqnarray}

We define $T = \partial_\lambda \Gamma$, $S = \partial_\tau \Gamma$, so that $D_\tau T = D_\lambda S$.

Now let's consider the world function between $p_s$ and $q$, $\sigma(\gamma(s), q)$. As a scalar function of $s$, we can Taylor expand it to

\begin{eqnarray} \sigma(s) &=& \sigma(0) + s \dot{\sigma}(0) + \frac{s^2}{2} \ddot{\sigma}(0) + \frac{s^3}{6} \ddddot{\sigma}(0) + \frac{s^4}{24} \ddddot{\sigma}(0) + \mathcal{O}(s^5) \end{eqnarray}

All we need to do now is to work out these quantities with respect to our measurements. We have $\sigma(s_1)$ depending on $t_1$ and $\theta_1^a$. $\sigma(0)$ is for our curve $\sigma$

\begin{eqnarray} \sigma(0) &=& \frac{1}{2} \sigma^2 \end{eqnarray}

For the derivatives, let's consider an extension $\sigma(s, \lambda)$, defined by

\begin{eqnarray} \sigma(s, \lambda) &=& \frac{1}{2} (\lambda_2 - \lambda_1) \int_{\lambda_1}^{\lambda_2} g(T,T) d\lambda \end{eqnarray}

Fairly obviously $\sigma(s, \lambda) = \Omega(s)$.

\begin{eqnarray} \partial_s \sigma(s, \lambda) &=& \frac{1}{2} (\lambda_2 - \lambda_1) \int_{\lambda_1}^{\lambda_2} \partial_s g(T,T) d\lambda\\ &=& \frac{1}{2} (\lambda_2 - \lambda_1) \int_{\lambda_1}^{\lambda_2} g(D_s T,T) d\lambda\\ &=& \frac{1}{2} (\lambda_2 - \lambda_1) \int_{\lambda_1}^{\lambda_2} g(D_\lambda S,T) d\lambda\\ &=& \frac{1}{2} (\lambda_2 - \lambda_1) \int_{\lambda_1}^{\lambda_2} \left[ \partial_\lambda (g(S,T)) - g(S, D_\lambda T) \right] d\lambda\\ &=& -\frac{1}{2} (\lambda_2 - \lambda_1) \left[ g(S,T) \right]^{\lambda_2}_{\lambda_1}\\ \end{eqnarray}

Now at $\lambda_2$, $S = 0$, so that

\begin{eqnarray} \partial_s \sigma(s, \lambda) &=& -\frac{1}{2} (\lambda_2 - \lambda_1) g(\dot{\gamma}(s),T(s, \lambda_1)) \\ \end{eqnarray}

At $s = 0$, we just get

\begin{eqnarray} \partial_s \sigma(s, \lambda)|_{s = 0} &=& -\frac{1}{2} (\lambda_2 - \lambda_1) g(\dot{\gamma}(0), \mu) \\ \end{eqnarray}

...

Clock synchronization

Another common experiment is the idea of clock synchronization. For two observers $\gamma_1$, $\gamma_2$ carrying clocks $h_1$ and $h_2$, we have that our clocks intersect at two points, $p_1$ and $p_2$.

The particle bounce

In the case where the particle sent possesses. a non-zero mass,

The double bounce

For a more general case of what we want, it's instructive to see what happens if we perform two photon bounces one after the other. Our curve $\gamma$ has two such process, a photon bounce from $p_1$ to $p_2$, and one from $\bar{p}_1$ to $\bar{p}_2$.

The continuum limit

With all of the experiments we got from various measurements involving causal curves, our goal is to hopefully reconstruct our entire spacetime. As general relativity is a theory of spacetime, our basic measurements to consider will be distances, angles and durations. There are more quantities we could measure of course, but these will be the basic notions we'll need for informations on our metric, and other measurements will likely be derived from such quantities as well.

As the goal is the experimental measurement of a spacetime, things will be done in two steps here : First, given a known spacetime, try to find if this gives rise to unique measurements, and then, given a set of such measurements, try to find out which spacetime we are dealing with here.

While the attempt will try its best to remain broad, for our sanity, a few assumptions will remain here :

1. Neither atomic clocks nor rigid rods are impervious to the spacetime metric nor the elements, but we will assume that those indeed define the measurements we want. At most, we'll allow for some known bounded uncertainty $\Delta t$ and $\Delta d$

Some general considerations on causality and topology

One of the biggest potential obstacle in general relativity to reconstructing our spacetime is the possibility that our spacetime may not be causal, globally hyperbolic, topologically trivial or any other such issue that could cause large complications to our model. While overall we will ignore those issues and mostly suppose both global hyperbolicity and a somewhat topologically simple spacetime, let's consider for a moment what those could cause on a measurement level.

One of the most basic restriction on the structure of spacetimes is the imposition of a time orientation.

Measuring the metric

With the various possible experiments that we've seen previously, there are quite a lot of experiments we could have, but we'll consider the following scheme here. These are fairly idealized ideas, but we'll see later on what will happen when we decide for a more realistic set of measurements.

Through every point of spacetime, an observer, idealized by a timelike curve, passes through, defining an observer field $O$ of unparametrized curves foliating $M$. Each such observer is identified by some indexes (for instance if our spacetime has the Cauchy surface $\mathbb{R}^3$, then the index could be $(x,y,z)$). Each observer has on board a clock and accelerometer, and it can fire off either light or particle beams of arbitrary energy, mass and polarization. Each such beam is encoded as to contain the relevant informations : when it was fired, according to the local clock, at what angle it was fired, and the index of the observer firing it. Those beams can be received by other observers, but they will also continue on beyond that point (for a more realistic perspective, we can say that the observer intercepting it will simply relay it by firing it again, perhaps adding its own informations to it), and it will fire back to the emitter of that signal with its own informations.

In other words, the emitter will emit a signal of the form

\begin{equation} \text{Emitted at $t = T$ by $(x,y,z)$ at angle $\theta, \phi$} \end{equation}

And may receive something of the form

\begin{equation} \text{Signal originally Emitted at $t = T$ by $(x,y,z)$ at angle $\theta, \phi$, received at $t_R = T_R$ by observer $(x_R, y_R, z_R)$} \end{equation}

There are many reasons why this idealization cannot work : we obviously cannot have one such machine per point of space, idealizing such a complex machine as a single point is quite a lot, sending so many signals in infinitely many directions will both raise issues with the quantity of energy that you would need to store for such a thing as well as break the idealization of those observers in the test field limit, the quantity of data to transmit and store would be infinite at the precision we're requesting (both the clock value and the index which is part of $\mathbb{R}^3$, for a start), and of course all the quantities we're using have some inherent uncertainty from real measuring apparatus or simply the process of emission. Once we have the theory for the exact measurement using this idealization, we'll try our best, with minimal assumptions, to find out what a more realistic case looks like.

The more well-known case for all of this is of course Minkowski space, so a lot of our first examples will be (for simplicity) two-dimensional Minkowski space, in other words $(\mathbb{R}^2, \eta)$ with the canonical coordinates $(t,x)$ such that

\begin{equation} \eta = -dt \otimes dt + dx \otimes dx \end{equation}

As all our measurements are done with causal means, and that we probably cannot assume an experiment spanning the whole duration of our universe, what we essentially get is a measurement of spacetime equivalent to a causal diamond. If our main observer starts at $p_1$ and ends at $p_2$, the spacetime probed will be fundamentally $I^+(p_1) \cap I^-(p_2)$. In other words, we won't be able to probe anything outside the horizon of our observer, as is to be expected. We can hopefully perform some spacetime extension from our data that could be unique enough to have a reasonable idea of what it may be.

In the Minkowski case, we have the benefit of having a flat connection, so that the holonomy of any vector will simply be $\{ 1 \}$, and the Pythagorean theorem reduces to the simple case of

\begin{equation} = \end{equation}

Synchronization in general relativity

Things get much more complicated in general relativity due to the fact that we do not know the intermediate geometry. There are many things which can make clock synchronization painful here.

First, the local time at which we receive the signal back may happen before we've sent it, for instance by taking the Deutsch-Politzer spacetime :

A signal sent may never come back at all, for a variety of reasons. If we attempt to send a signal from $\gamma$ to $q$, $p_1 \in \gamma$ the time of signal, perhaps $q$ does not belong to $I^+(p_1)$, or $\gamma \cap I^+(q) = \varnothing$, or there may simply be no null geodesics connecting the two (consider Minkowski space with a point removed as an example).

A global synchronization may simply not exist. Even under the broadest definition of a synchronization (perhaps just assuming some foliation by spacelike hypersurfaces), some spacetimes, such as the Gödel spacetime, will not admit such a thing.

There may be some discontinuities in the light clock signals if the topology isn't trivial. Consider a wormhole spacetime with an outside observer. With our plan to send a signal at every point of our observer's curve, what we are basically getting if we retract our entire spacetime along its foliation is a family of loops, which will locally be contractible to the identity. As the fundamental group of a wormhole spacelike hypersurface isn't trivial, we will finally get a curve that cannot be contracted to the identity, and two light signals $\ell_{t}$, $\ell_{t + dt} can then arrive at arbitrarily far apart times.

To simplify things somewhat, first, let's assume that $(M, g)$ is globally hyperbolic. While not necessarily uncontroversial, this will help out immensely to reduce the number of pathological cases to consider, and it will be at least a valid assumption in some neighbourhood (although how far from it is of course matter to some debate).

We now get a rather useful set of theorems :

1. There exists a set of foliations of $M$ into Cauchy surfaces (achronal spacelike hypersurfaces) $\Sigma_t$, $M \cong \mathbb{R} \times \Sigma$, such that $\Sigma_t \cong \Sigma_s$ for every parameter $s,t$, such that every observer $\gamma$ crosses every Cauchy surface exactly once.
2. Every foliation into Cauchy surface induces a parametrization on every observer $\gamma$ by $\forall \tau \in \mathbb{R},\ \gamma(\tau) \in \mathbb{R}^3_\tau $
3. Globally hyperbolic spacetimes admit continuous time functions
4. For two causally related points $p \leq q$, there is a non-spacelike geodesics of maximal length
5. There is a foliation of $M$ by timelike geodesics
6. The spacetime is isometric to the product spacetime $\mathbb{R} \times \Sigma$ with the metric $$ds^2 = -\beta(t, x) dt^2 + g_{ab}(x,t) dx^a dx^b$$with $g_{ab}$ a Riemannian metric on $\Sigma_t$. $t$ defines a time function on this spacetime and $dt$ is a tangent $1$-form to it.

The set of null geodesics

A very important subset of our spacetime that we will require is the set of null geodesics. For this, we define by $N^\pm(p)$ the set of all points $q$ such that there exists a null geodesic connecting $p$ and $q$. As most of our measurements will be done using null geodesics, it will be a useful set to have.

Unfortunately, the set of null geodesics isn't quite as trivial as we would hope. In other words, $N^\pm(p)$ is not actually equivalent (in general) to the horismos, $H^\pm(p) = J^\pm(p) \setminus I^\pm(p)$. A very simple example for this is to consider the Minkowski cylinder $\mathbb{R} \times S$, where null geodesics are helixes spiraling around but for every $2\pi$ turns, any two such points on the helix can be obviously joined by a timelike curve.

We therefore have a point in between those two points on the light clock with which we can synchronize that event. Unfortunately, $p_1$ and $p_2$ are not necessarily on the light cone itself. Two counterexamples of $p \neq \partial J^+(q)$ despite a null geodesic between $p$ and $q$ are simply the Minkowski cylinder $\mathbb{R} \times S$, where a null geodesic will obviously lie in $I^+(q)$ after a single turn, and for a topologically trivial example, ultra-compact object spacetimes, in which the presence of a lightring makes it that any observer stationary on this lightring will be

For now, let's proceed backward : let's assume that we have a foliation $\mathfrak{t} : M \to \mathbb{R}$, such that $\mathfrak{t}^{-1}(t) = \Sigma_t$. Our goal here will be to find out if there exists a synchronization convention such that, using some manner of synchronization method, we can reproduce this foliation without using previous knowledge of our metric.

By definition of a time function in a globally hyperbolic spacetime, our time function $\mathfrak{t}$ is a surjective function that grows monotonously along every future-oriented timelike curve. At minima, we'd like such a function to at least agree with our main observer $\gamma_0$ : for every point $p \in \gamma_0$, $\mathfrak{t}(p)$ has the same value as $\gamma_0$'s onboard clock.

Let's now work backward for a bit : consider an existing time foliation of our spacetime by a time function $\mathfrak{t}$, with a preferred observer $\gamma_0$. For every point $q \in M$, we define a light trip by

\begin{equation} \ell_q = \ell_{p_1 \to q} \cup \ell_{q \to p_2} \end{equation}

Unfortunately, such trips may not be unique. What we will ask here, for now (this definition will not work in the most general of cases) is that

The simultaneity bundle

The situation that we have here may bring back memories of bundles here. We have a spacetime that is locally (and in this case, globally) diffeomorphic to a product, $M \cong \mathbb{R} \times \Sigma$, and for any point $p \in M$, we could map it to either $\mathbb{R}$ or $\Sigma$ : either $\pi_{t}(p) = t(p)$, where we have the time function $t$ playing the role of the bundle projection onto $\mathbb{R}$, or $\pi_{x}(p)$ is the position of our point on the spacelike hypersurface. A way to achieve this is to pick both a time function $t$ and a foliation by timelike geodesics $\gamma_x$, such that if $t(p) = 0$, $\gamma_x(p) = \iota(x, 0)$, and then the projection will be

\begin{equation} \pi_x(p) = \{ y | p \in \gamma_y \} \end{equation}

Our simultaneity bundle is a bundle $M \cong \mathbb{R} \times \Sigma$, of base space $\Sigma$ and typical fiber $\mathbb{R}$. Every point $x \in \Sigma$ has a corresponding fiber $\mathbb{R}_x$ representing an observer in our spacetime. Things will get more complex if we assume our main observer to be accelerated, so for now let's only consider a geodesic observer.

Our geodesics and foliation here allows us to relate points of distant fibers. In other words, they are a form of connection. To see this, let's try to consider thing in a more gauge approach. As our typical fiber is $\mathbb{R}$ here, we can try to promote it to a principal $\mathbb{R}_+$-bundle by considering the translation group. Our right-action on the bundle is

\begin{eqnarray} \phi_T : \mathbb{R}_+ \times E &\to& E\\ (T, \tau, x) &\mapsto& (\tau + T, x) \end{eqnarray}

So that the bundle action is simply to travel along our geodesic foliation further in time by $T$. This action indeed preserves the fiber.

We'll consider the effect of our synchronization by observing it in the typical fiber obtained by quotienting out the bundle group, ie

\begin{equation} \Sigma = M / G \end{equation}

In this case, any light-clock is simply a loop in $\Sigma$ with base point $\gamma(0)$. Unlike the case of Migunzzi[6], our spacetime isn't static, meaning that we cannot assign any induced metric here. We can't assign our synchronization simply given $\Sigma$, we have to work with every slice $\Sigma_t$.

The benefit here is that the tangent bundle of our bundle is simply the tangent bundle of our spacetime, ie

\begin{equation} TE = TM \end{equation}

As we'll

The clock synchronization can be modelled after holonomies : given a point $p$ for our observer,

???

From all these elements, let's consider a fairly simple case : the class of all conformally Minkowski two-dimensional spacetimes. We're making this choice as it is both simple, and by Kulkarni's theorem, covers most spacetimes with $\mathbb{R}^2$ topology under some fairly reasonable assumptions. We therefore have to find out a nowhere-vanishing function $\Omega$ such that

\begin{equation} ds^2 = \Omega^2(x,t) (-dt^2 + dx^2) \end{equation}

???

As we are working with infinitely many signals and infinite precision, it remains possible here to consider the convex normal neighbourhood without having to worry about its extent. We just need to first determine what behaviour we will see once leaving it.

For our preferred observer, we can consider the neighbourhood defined by, for some time interval $I = (\tau, \tau + \Delta \tau)$, its tubular neighbourhood. For every $\tau \in I$, the exponential map has a certain injectivity radius $\rho_\tau$. If we take the radius

\begin{equation} \rho = \min_{\tau \in I} \rho_\tau \end{equation}

Then all of our neighbourhood is appropriately normal. In our entire neighbourhood, we can simply map behaviour from the tangent space to the manifold itself.

2. Measuring distances in general relativity

Despite a few issues, measuring time in general relativity remains simple enough to work out with minimal assumptions. Distances, on the other hand, being rather non-local, are much more difficult to establish.

The two fundamental process for establishing distances in general relativity are rulers and light signals. Rulers are, generally speaking, objects that we suppose to be of fixed length, while light signals are done by an exchange of light signals between two objects, as we can assume such signals to propagate at a constant speed.

Distances in special relativity

For a bit of a simpler case, let's consider distances in special relativity first, where the metric is everywhere flat as

\begin{equation} ds^2 = -dt^2 + dx^2 \end{equation}

The obvious theoretical distance between two objects, given the canonical foliation, is

\begin{equation} d(\gamma_1, \gamma_2) = \| x_2(t) - x_1(t) \| \end{equation}

This is entirely well-defined (although requires the gauge $t(\tau) = \tau$ here and depends on the foliation), but it is not measurable directly. Instead what we need to have is an exchange of signals, so that every process will remain local.

The first observer $\gamma_1$ emits a light signal $\ell_1$ at $\gamma_1(\tau_1)$, which reaches the observer $\gamma_2$ at $\gamma_2(\tau_2)$ before re-emitting it towards $\gamma_1$, which receives it at $\gamma_1(\tau_3)$.

From the point of view of $\gamma_1$, the number of informations we can extract from the light signal are somewhat limited.

• We know (up to some uncertainty) the proper time it took the light signal to bounce back, $\tau_3 - \tau_1$
• If properly encoded, the light signal can encode the proper time at which the signal was received by $\gamma_2$, $\tau_2$, although relating it to $\gamma_1$'s own proper time is delicate.
• Given a standard of the emission, we can deduce the redshift of the signal, giving us the speed (and, for a sufficiently long signal emitted, the acceleration) that both observers are relative to one another (ie the speed of $\gamma_1$ and $\gamma_2$ affect the redshift of the first signal, and the same is true for the return signal).

The common way to measure the distance of two observers with this method is the Synge formula :

\begin{equation} d(\gamma_1, \gamma_2) = c \sqrt{(t - t_1)(t_2 - t)} \end{equation}

...

3. Measuring angles in general relativity

Comparatively to times and distances, angles are fairly easy to do as they are very much a local measurement.

4. Determination of the metric tensor

Once we have all these elements, all that remains for us is to attempt to determine the components of the metric tensor in the coordinates we have defined for our observer. If we have the full measurements of every observer in our grid, this should be doable.

Measurements in the ADM formalism

For later use, it will be practical to work out some of those measurement issues in the ADM formalism, where we have the decomposition of our metric as

\begin{equation} ds^2 = -N^2 dt^2 + \gamma_{ij} (dx^i + \beta^i dt) (dx^j + \beta^j dt) \end{equation}

So that

\begin{eqnarray} g_{00} &=& -N^2 + \beta_k \beta^k\\ g_{0i} &=& \beta_i\\ g_{ij} &=& \gamma_{ij} \end{eqnarray}

with the three functions $N$, $\beta$ and $\gamma$. $N$, the lapse function, and $\beta$, the shift function

What we want is to have $N$ and $\beta$ to correspond to our family of observers. In the formalism, our observers $\gamma$ map

\begin{eqnarray} \gamma : \mathbb{I} &\to& \mathbb{R} \times \mathbb{R}^3 \\ \tau &\mapsto& (t(\tau), x^i(\tau)) \end{eqnarray}

If we want the time coordinate to be such that $\Delta t = \Delta \tau$ for our curves, that will be

\begin{eqnarray} \Delta \tau = \int_{\tau_1}^{\tau_2} \sqrt{- g(\gamma', \gamma')} d\tau\\ &=& \int_{\tau_1}^{\tau_2} \sqrt{- g(\gamma', \gamma')} d\tau \end{eqnarray}

...

This implies that

\begin{equation} \beta^k \beta_k = N^2 \end{equation}

...

\begin{eqnarray} a &=& \ddot{\gamma}\\ &=& \nabla_\dot{\gamma} \dot{\gamma} \\ &=& u^\mu (\partial_\mu u^\nu + {\Gamma^\nu_{\mu\sigma}} u^\sigma) \end{eqnarray}

Foliations from initial conditions

As we will not know what the future history of our manifold looks like, let's consider the following problem : considering our initial Cauchy surface $\Sigma$ equipped with a nowhere-zero vector field $v \in \Gamma(TM|_{\Sigma})$, corresponding to the initial positions and velocities of our fleet of observer, and a general vector field $a \in \Gamma(TM)$, corresponding to the acceleration of those curves throughout their history, such that $\gamma_p(0) = v(p)$ and $\nabla_{\dot{\gamma}}\dot{\gamma} = a$, is there a foliation of the manifold for each possible configuration?

The case $a = 0$ is the geodesic case, where every observer is a geodesic

The test field limit, matter approximation and far-off sources

There are a number of approximations that are essential to experimental general relativity :

1. The various objects we use for measurements do not contribute significantly to the gravitational field we want to study. This is the test field limit.
2. If we can approximate the sources to some degree of error, the deviations from this approximation will have negligible effects on the gravitational field.
3. The influence of very far-off sources can be neglected

All three of those assumptions basically boil down to an estimation problem of the Einstein field equation.

Error estimation in general relativity

For all those approximations, we will essentially get some kind of uncertainty on the metric tensor. Given the space of Lorentz metrics on our spacetime, $\mathrm{Conf}_g = \mathrm{Lor(M)} / \mathrm{Diff}(M)$, our various approximations will give us some representative metric $g$ (corresponding to the case where our approximations are in fact exact) and a subset $\mathrm{Conf}_g^{\mathrm{approx}}$ where our metric could be within the approximations we constructed. Hopefully, within some function space, we have some estimate such that for $g' \in \mathrm{Conf}_g^{\mathrm{approx}}$, we have

\begin{equation} \| g - g' \| \leq C \end{equation}

leading us, in turn, to some bounds for the error of our measurements.

The test field limit

In the test field limit, we presume that the real stress-energy tensor differs from the one we're using by a small amount in a compact region (the region isn't necessarily compact of course, as a charge will for instance have influences reaching potentially to infinity, but such considerations will be left for the following approximations). In other words, we have

\begin{equation} T = T_{A} + \Pi_S \Delta T \end{equation}

Inside a top-hat function $\Pi$ of a region (generally small compared to the system we study), there is a stress-energy tensor which is small in some sense. This will generally be small compared to the source we're considering.

What is the error we get from such a term? If we consider a known stress-energy tensor $T_A$ with known solution $g_A$, and the actual stress-energy tensor $T$ with solution $g$, we are looking for the difference between the two, and the consequences on actual measurements. With the Einstein tensor $G$, we

Source approximation

The problem of the source approximation is fairly close to the test field limit problem, but here we have that, rather than studying a system with a stress-energy tensor $T$, we have a very slight deviation $T + \Delta T$. This deviation does not need to be localized within a small region, but relevant quantities should hopefully be of a scale much larger than the error on it.

Far-off sources

The last issue is that of possibly very large sources of the stress-energy tensor that are far-off enough that any effect on the metric tensor will be negligible.

Error estimation

\begin{equation} G[g + \Delta g] = \kappa (T + \Delta T) \end{equation}

Things are not simplified by the appearance of the non-linear in $g$ Einstein tensor, so for now, let's consider the linearized gravity equation in the de Donder gauge

\begin{equation} \Box (h + \Delta h) = \kappa (T + \Delta T) \end{equation}

Here $\Delta h$ is the error on the calculated deviation from Minkowski space.

\begin{equation} \Box (h + \Delta h) - \Box (h) = \kappa (T[h + \Delta h] + \Delta T[h + \Delta h]) - \kappa (T[h]) \end{equation}

For most sources of matter, $T$ is generally linear in the metric [CHECK WRT INVERSE METRIC], so that we can use

\begin{equation} \Box (\Delta h) = \kappa (T[\Delta h] + \Delta T[h + \Delta h]) \end{equation}

Green's function :

\begin{equation} \Delta h = \kappa \int G(x, s) \left[(T[\Delta h](s) + \Delta T[h](s) + \Delta T[\Delta h](s)) \right] ds \end{equation}

...

ADM :

As we want to predict the evolution of a system from some initial values, the ADM formalism is advised here.

\begin{eqnarray} \dot{\gamma_{ij}} - \mathcal{L}_\beta \gamma_{ij} &=& -2N K_{ij}\\ \dot{K_{ij}} - \mathcal{L}_\beta K_{ij} &=& -D_i D_j N + N \left[ R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j + 4 \pi ((S - E) \gamma_ij - 2 S_{ij}) \right]\\ R + K^2 - K_{ij} K^{ij} &=& 16\pi E\\ D_j {K^j}_i - D_i K &=& 8 \pi p_i \end{eqnarray}

Now let's suppose that, given some initial conditions $(\Sigma_0, N, \beta, \gamma, K, E, p_i, S)$, our measured values are, for actual dynamical variables, $(\bar{\gamma}, \bar{K}, \bar{E}, \bar{p_i}, \bar{S})$, with for each variable the relation $q = \bar{q} + \delta q$. We also know the behaviour of $E, p_i, S$ given by some equation of motion of the individual fields or an equation of state. What is the error of the prediction on those measured values, given some bounds on $\delta q$?

...

Our computed solution $\bar{\gamma}$ is such that its restriction on $\Sigma_0$ gives us our initial conditions, and it obeys the ADM equations. Our real solution will be of the form

\begin{eqnarray} \dot{\bar{\gamma}}_{ij} + \delta\dot{\gamma}_{ij} - \mathcal{L}_\beta \bar{\gamma}_{ij} - \mathcal{L}_\beta \delta \gamma_{ij} &=& -2N K_{ij}\\ \dot{K_{ij}} - \mathcal{L}_\beta K_{ij} &=& -D_i D_j N + N \left[ R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j + 4 \pi ((S - E) \gamma_ij - 2 S_{ij}) \right]\\ R + K^2 - K_{ij} K^{ij} &=& 16\pi E\\ D_j {K^j}_i - D_i K &=& 8 \pi p_i \end{eqnarray}

We want to know what the measurement at $\gamma(t + \Delta t)$ will be.

7. Real measurements of the spacetime metric

Now that we have a theory for the measurement in our extremely idealized case, let's consider a more realistic scenario. There are many simplifying assumptions that we have taken throughout this paper and it is time to evaluate what error these may induce in our process.

The continuous approximation

The first obvious limitation we have are the number of measuring apparatus and their precision. Instead of our apparatus foliating the spacetime, we have a finite number $N$ of them, each one with some index $n\in [1 \ldots N]$. As we do not know the geometry of spacetime quite yet, it's a bit early to talk about our measuring apparatus on a grid of appropriate spacing, although this may be done later on. From what we've seen on clocks, there is also only so much we can do for the precision of our clocks. Not only that, but as we are encoding data in those light signals, those are not instantaneous either : each train of data takes a certain time to be emitted, as well as being decoded.

What we are basically getting from all our data (ignoring for now the other uncertainties related to instrumentation) is a triangulation of our spacetime manifold into a simplicial complex, similar to the one we'd get in Regge calculus. If we had for instance two observers in Minkowski space exchanging one light signal, from the point of view of the emitting observer, the simplicial complex would be comprised of four points : $t_1$, the sending of our light signal, $t_2$, the receiving of the response, $t$ the bounding from the other observer and $t'$ the synchronized point on its own trajectory. There are a total of $5$ edges here : $t_1t$ and $t_2$ are both null paths of lengths $0$, $t_1t'$ and $t'_2$ are both timelike and of known proper time, and finally $t't$ is spacelike and of known distance.

A metric is essentially infinite-dimensional and here our measurements are not even directly related directly related to it. So first, let's consider the equivalence class involved here. We'll say that two metrics $g_1$, $g_2$ fit the measurements if the resulting spacetime can be skeletonized

Objects as causal curves

Another rather large simplification is assuming that the objects we consider is that they can be approximated as points, with the assumption that the error induced by such an approximation will be below the margin of error of measurements in any likely case.

Let's consider a simple example of a non-point like apparatus and see what effects the local spacetime curvature is likely to have.

Real clocks

One important aspect of the chronometric measurement scheme, or the use of light clocks in other schemes, is the assumption that our clocks will be roughly measuring the proper time

No matter what happens, if our spacetime is globally hyperbolic anyway, our clocks will define some subset of a time parametrization for our observer, ie, $\gamma(t_0 + k T)$ is such that there exists a parametrization where this is the correct measure of time. What we would like in the ideal case as well is that the clock also measures the proper time, as this would simplify a lot of our issues. It's quite easy to see that for instance, if our clock only ticked for every $\tau^3$

All of our clocks fundamentally rely on electromagnetism and the Dirac equation to work : either via the transition frequency of electron in an atom or via the length of a rod, which depends roughly on the atomic radius. But the Dirac equation is something of the form

\begin{equation} i \gamma^a e^\mu_a (\partial_\mu - \frac{i}{4} \omega^{ab}_\mu) \sigma_{ab} - i Ze A^\mu) \psi - m \psi = 0 \end{equation}

Which leads us to the question : does the influence of gravity on our measuring apparatus works out in the end, or does it induce some fundamental uncertainty? It's entirely possible that an atomic clock will still measure the proper time, but that is certainly not a trivial statement.

There are two main drawbacks to all of this. The first one being its discrete nature (we can't really get much better than the time between two crests), and it is itself sensible to the spacetime geometry (among other things, as it will also be sensible to a variety of other circumstances). Indeed, let's consider what happens here. Without getting too deeply into quantum field theory, let's consider some Riemann normal coordinates around the nucleus, which is considered fixed and classical, and look at the Dirac equation in curved spacetime (the Einstein-Maxwell-Dirac equation) for an electron :

\begin{eqnarray} e^\mu_a &=& \delta^\mu_a + h^\mu_a\\ \omega^{ab}_\mu &=& \end{eqnarray}

The inverse problem in general relativity

Synchronization of Regge manifold

Net of synchronization

Foliations adapted to an observer

Given a set of $N$ observers $\gamma_i$, we would like to be able to define a foliation of the region they span, which will roughly correspond to the union of their causal diamonds :

\begin{equation} \bigcup_{i \in N} J^+(\gamma_i(t_0)) \cap J^-(\gamma_i) \end{equation}

We would like to see if there exists at least a spacelike foliation for this set of curves, and ideally adapted coordinates such as some compromise between their respective radar coordinates.

Injectivity radius in spacetimes

One of our big obstacle for measurement is the cut locus. Without knowing how large a normal neighbourhood around a point is, we won't be able to determine what's an appropriate scale for our measurements. Minkowski space's normal neighbourhoods may cover the entire spacetime, while some kind of Wheeler-style quantum foam could have one below experimental reach, given small enough wormholes riddling the spacetime.

What we need is some bounds on the injectivity radius of the exponential map linked to reasonably measurable quantities. This will not be entirely satisfying since any measurement we can perform will have to rely on the spacetime geometry, giving us some kind of bootstrapping problem, and as mentionned, we could have some quantum foam situation, where the very reasonable measurable quantities are just averages of extremely variable densities. But we need to draw the line somewhere, and for now we'll assume that matter, at least in our vicinity, has some reasonable bounds on the stress energy tensor.

There are many ways to deal with the injectivity radius in spacetimes, as we can't use directly the Riemannian definition, lacking a distance function.

As we will be primarily dealing with observers, another useful notion will be the injectivity radius of a timelike curve. Given a timelike curve [Find some Cauchy neighbourhood]

Extended objects

Real measuring apparatus and test objects are not by any stretch point-like, so that it may be of interest to find out how such things play out for actual extended objects.

We can model our apparatus by some kind of congruence of timelike curves of limited span. that is, if we assume that we have some initial Cauchy surface $\Sigma$, our apparatus will occupy some subset $U \subset \Sigma$, and from then on follow the path of each particle composing it. While this will span a submanifold, we cannot rely entirely on its description as a submanifold, as this would lose some informations regarding its rotation.

Rigid objects

One issue coming up if we try to model measuring apparatus is that if we want to define them as rigid objects, or objects with a very high rigidity at least, we need to define this notion in terms of general relativity, which is difficult when space is dynamic.

Born rigidity

Fields for measurements

For some measurement processes, such as radar, the approximation as a field may be more useful than as a point particle.

Most likely field measurements done for general relativity will be done with the electromagnetic field

Quantum measurements

Splitting of the Hilbert space into orthogonal subspaces corresponding to eigenvalues as read on the instruments

Bibliography

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Last updated : 2022-09-06 10:05:05