General theory of swimming in curved spacetime

Swimming in curved spacetime is a phenomenon where small bodies evolving freely in curved spacetimes appear to be able to propel themselves, by performing fast cyclic internal motions, in a “non-dynamical” manner [1]. More precisely, the resulting motion would depend only on the sequence of conformational changes of the body and not on how fast they are performed, similarly to the motion of a swimmer in a fluid at low Reynolds numbers [2]. But a rigorous general treatment of this problem, consistent with general relativity, was still missing. The main goal of this paper is then to fill this gap.

The problem of interest consists of describing the trajectory of a light, small, articulated body, performing a given sequence of internal motions, in a curved spacetime. The body is light so that it can be regarded as a test body in a fixed background spacetime, it is small in comparison to the length scale over which the curvature varies, and it is articulated in the sense that it can adjust its shape in a controlled manner (say, it has a couple of arms which can be willfully stretched or waved). In order to solve this problem we employ the theory of dynamics developed by Dixon [4, 5, 6]. Dixon’s formalism has the valuable property of being automatically consistent with the fundamental conservation equation $\nabla_{a}T^{ab}=0$ of general relativity, which is directly implemented into the formalism from the very beginning. In addition, the equations of motion are given in a convenient multipole-expansion form which makes clear how the overall motion is affected by different orders in $l/r$, where $l$ is the (spatial) length scale of the body and $r$ is the characteristic (spatial) length scale that the gravitational field varies.

We will divide this paper into four sections, beginning with a succinct review of Dixon’s formalism, which will be the basis for the subsequent developments. Next, we deduce from Dixon’s equations an approximate equation for the trajectory of the center of mass of a general “small” body, in terms of the net gravitational force and torque acting on it. Basically we shall find expressions for the velocity of the center of mass of the body, correct up to order $(l/r)^{3}$, so that we may keep only terms up to octupole order in the expressions for the force and torque. Then, we proceed to discuss a convenient manner to prescribe the quadrupole and octupole moments, based on a local correspondence to a flat spacetime. Finally, considering the equations for the trajectory of the body, we are able to formally study the effective dynamics induced by fast cyclic internal motions. With the general formalism established, we proceed to analyze a regime suitable for “swimming at low Reynolds number”, showing that although the effect can take place, it is limited to specific scenarios (which does not mean that bodies are generally incapable of propelling themselves by performing swimming-like internal motions, but only that the resulting motion is typically not described by a geometric phase). In Appendix B we show how to apply the formalism to a concrete example, putting our theory to the test. This paper is supplementary to [3], where a summary of these results are presented.

We use Latin letters for abstract tensor indices and Greek letters for tensor components in a basis. The curvature tensor is defined by $\left(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a}\right)\omega_{c}=-R_{abc}^{\;\;\;\;\,\,d}\,\omega_{d}$, and the Ricci tensor and scalar curvature are respectively defined by $R_{ac}=-R_{abc}^{\;\;\;\;\,\,b}$ and $R=R_{a}^{\;\;a}$. The signature of the metric is $(+,-,-,-)$ and the system of units is such that $c=G=1$.

In a series of papers published in the 70’s [4, 5, 6], W. G. Dixon developed a formal theory to describe the dynamics of extended bodies in general relativity, in an explicitly covariant way. This section is dedicated to review (in a rather compressed way) the main results of his theory¹¹1Dixon’s theory is actually more general than this, allowing for the presence of external electromagnetic forces..

Let $T^{ab}$ be the stress-energy tensor of the body under consideration, assumed to be conserved (i.e., $\nabla_{a}T^{ab}=0$), and define the world-tube $W$ of the body as the support of $T^{ab}$. Then, under reasonable assumptions about $T^{ab}$ ²²2It is assumed that the spatial extension of the body is finite and such that the convex hull ${\cal B}$ of the intersection of its world-tube $W$ with any space-like hypersurface $\Sigma$ is contained in a normal neighborhood of any point of ${\cal B}$. This means, in particular, that there is only one geodesic in ${\cal B}$ connecting any two spatially-separated points of the body. Moreover, wherever $T^{ab}\neq 0$, the momentum density is assumed to be time-like and future-directed according to all observers; i.e., $T_{ab}n^{a}n^{b}>0$ and $T_{ab}T^{b}_{\,\;c}n^{a}n^{c}>0$ for all time-like vectors $n^{a}$. Finally, these conditions are supposed to hold also for the momentum density “propagated” from any point $x$ to any point $x^{\prime}$, both in ${\cal B}$, where the “propagation” is performed along the geodesic connecting $x$ and $x^{\prime}$, according to maps $K^{a^{\prime}}_{\;\;b}$ and $H^{a^{\prime}}_{\;\;b}$ which are properly defined in Appendix A. and somewhat weak conditions on the “strength” of the gravitational field [9], the following general results hold:

• (i)

  At any point $x\in W$, there is a unique time-like, future-pointing, unit vector $n^{a}=n^{a}(x)$ such that the linear momentum $p^{a}=p^{a}(x,n)$ of the body (properly defined [4] with respect to $x$ and $n^{a}$) is entirely in the direction of $n^{a}$. In essence, this means that there is a (unique) family of observers (those with four-velocity $u^{a}=n^{a}$) according to whom the total spatial momentum of the body is zero. In addition, at each point $x\in W$, there is a natural definition for the angular momentum $S^{kl}=S^{[kl]}=S^{kl}(x,n)$ of the body;

• (ii)

  There is a unique, covariantly defined, time-like curve $z(s)$, contained in the convex hull of $W$, which can be consistently identified as the world-line of the center of mass of the body (here $s$ is the proper time along this world-line). More specifically, it is defined as the set of points $z$ which satisfy $p_{k}(z,n(z))S^{kl}(z,n(z))=0$. It is worth mentioning that the tangent vector to this curve at $z(s)$, $v^{a}(s)$, is not, in general, parallel to $u^{a}(s)=n^{a}(z(s))$. We refer to $u^{a}$ and $v^{a}$ respectively as the dynamical and kinematical velocities of the body;

• (iii)

  At each point $z(s)$ on the world-line of the center of mass, there are also natural definitions for the (higher) multipole moments $J^{k_{1}\cdots k_{n}pqlr}(s)$, $n\geq 0$, of the body ³³3We are referring to these as higher multipole moments because the linear and angular momenta, $p^{k}$ and $S^{kl}$, can also be regarded as multipole moments – respectively, the monopole and dipole moments.. The set of tensors $\{p^{k},S^{kl}\}\cup\{J^{k_{1}\cdots k_{n}pqlr};n\geq 0\}$ depends, at each $s$, only on the restriction of $T^{ab}$ to $W\cap\Sigma(s)$, where $\Sigma(s)$ is the space-like surface generated by all geodesics starting at $z(s)$ orthogonal to $u^{k}(s)$. Conversely, if the tensors in this set are given for all $s$, then $T^{ab}$ can be completely reconstructed;

• (iv)

  The conservation equation $\nabla_{a}T^{ab}=0$ can be translated into only two dynamical equations for the linear and angular momenta

  	$\displaystyle\frac{\delta p^{k}}{ds}$	$\displaystyle=$	$\displaystyle F^{k}$
  	$\displaystyle\frac{\delta S^{kl}}{ds}$	$\displaystyle=$	$\displaystyle 2p^{[k}v^{l]}+Q^{kl}\,,$		(1)

  where $\delta/ds:=v^{k}\nabla_{k}$ is the covariant derivative operator along $z(s)$, and $F^{k}$ and $Q^{kl}$ are respectively called the gravitational force and torque acting on the body (and they can be expressed in terms of couplings between the Riemann curvature tensor or its derivatives and the multipole moments);

• (v)

  The proper mass of the body, $M$, is defined as the scalar in the expression $p^{k}=Mu^{k}$. If $\vartheta_{klqr}$ is the volume element of the spacetime (associated with the metric), then we define the spin vector of the body as $S_{k}=\frac{1}{2}\vartheta_{klqr}u^{l}S^{qr}$, and it satisfies

  $\displaystyle\frac{\delta S^{k}}{ds}=-u^{k}\dot{u}_{l}S^{l}+Q^{k}\,,$

  (2)

  where ${\dot{u}}^{k}=\delta u^{k}/ds$ and the torque vector $Q^{k}$ is defined from $Q^{kl}$ similarly to the spin vector. Then, the misalignment between $u^{k}$ and $v^{k}$ is given by

  $\displaystyle u^{[k}v^{l]}=\frac{1}{2M^{2}}\vartheta^{klqr}\bar{F}_{q}S_{r}+\frac{1}{M}u^{[k}Q^{l]q}u_{q}\,\,,$

  (3)

  where $\bar{F}^{k}=({\delta}^{k}_{\>\>{l}}-u^{k}u_{l})F^{l}$. Note that this provides an equation for $v^{k}$ in terms of the linear momentum, spin, force and torque.

• (vi)

  For each (if any) symmetry of the spacetime with generator $\xi^{a}$, there is a conserved dynamical quantity $\chi_{\xi}$ given by

  $\displaystyle\chi_{\xi}=p^{k}\xi_{k}+\frac{1}{2}S^{kl}\nabla_{k}\xi_{l}\,\,,$

  (4)

  which can be seen as a version of Noether’s theorem.

• (vii)

  For a spatially small test-body, one can keep only a few multipole terms in the expressions for the force and torque. Up to octupole order, they are

  	$\displaystyle F^{k}=$	$\displaystyle\frac{1}{2}v^{l}S^{qr}{R^{k}}_{lqr}+\frac{1}{6}J^{qrlw}\nabla^{k}R_{qrlw}+$
  		$\displaystyle+\frac{1}{18}J^{qrwtl}\nabla^{k}\left(\nabla_{q}R_{rwtl}-2\nabla_{t}R_{qrwl}\right)\,,$
  	$\displaystyle Q^{kl}=$	$\displaystyle\frac{4}{3}J^{qrw[k}{R_{qrw}}^{l]}+$
  		$\displaystyle+\frac{1}{3}J^{qrwt[k}\left(3\nabla_{q}{R_{rwt}}^{l]}+2\nabla_{t}{R_{qrw}}^{l]}\right)\,,$		(5)

  and we shall refer to the three terms in the expression for the force as, respectively, $F^{k}_{spin}$, $F^{k}_{quad}$ and $F^{k}_{octu}$; and to the two terms in the expression for the torque as, respectively, $Q^{kl}_{quad}$ and $Q^{kl}_{octu}$.

The problem of solving for the dynamical evolution of a finite body in general relativity can be formulated as an initial-value problem with Dixon’s equations. Hence, we shall assume that the initial position of the center of mass of the body $z(s_{0})$ is known, as well as the initial linear momentum $p^{k}(s_{0})$, the initial spin $S^{k}(s_{0})$, and the initial multipole moments $J(s_{0})$ (the indices are omitted). The worldline of the center of mass $\ell$ is parameterized as $z(s)$, where $s$ is the proper time along $\ell$, so that the tangent vector $v^{k}$ has unit modulus.

In a coordinate system $\{x^{\mu}\}$ around $z(s_{0})$, the trajectory $z^{\mu}(s)$ of the center of mass of the body satisfies

$$v^{\mu}(s)=\frac{dz^{\mu}}{ds}\,,$$

(6)

where $v^{\mu}$ can be obtained algebraically from (3), in terms of other dynamical quantities like $p^{\mu}$, $S^{\mu}$ and the $J$’s. This equation can then be solved in conjunction with (1), as long as we know how to evolve the $J$’s. The most direct approach for determining the moments is to translate the internal laws governing the substances constituting the body into differential equations for the $J$’s. In this case, depending on the order of the system of differential equations, the values of the $J$’s and the appropriate set of their time derivatives $dJ/ds$, $d^{2}\!J/ds^{2}$, etc must be provided at $s=s_{0}$. We shall, nonetheless, adopt a different approach to specify the moments (see next section), significantly more appropriate for our purposes. In our description, we will find a way to evaluate the multipole moments as functions of the internal motions performed by the body. Thus, the internal motions (and the $J$’s) should be seen as part of the input of the problem, not the output.

Our task now is to find an explicit formula for the kinematical velocity $v^{k}$ in terms of dynamical quantities like $u^{k}$, the spin and the force and torque. This formula does not need to be exact, but it must be accurate up to order $(l/r)^{3}$. The motivation for assuming this degree of accuracy comes from $(i)$ the fact that the swimming effect is expected to be observed at this order⁴⁴4The Newtonian analysis suggests that the swimming effect should be observed at order $(l/r)^{2}$. The analysis of the motion of a bipod moving freely on a spacetime with topology $\mathbb{R}\times S^{3}$ shows that the swimming propulsion can indeed be at this same order. The study of a tripod falling aligned with the radial direction of Schwarzschild spacetime, in a specific setup, yields that the propulsion caused by the internal motions takes place at order $(l/r)^{3}$ only. and that $(ii)$ the equations are still sufficiently simple and manageable (for instance, we can only keep terms up to octupole order in the equations of motion).⁵⁵5Dixon’s equations have been used to study the motion of extended bodies in curved spacetimes, for example in [7, 8], but here we focus on the application to the swimming effect.

Let us briefly study the magnitudes, in orders of $l/r$, of the various terms in the equations of motion. Typically, in a (normal) coordinate system centered at $z(s)$, aligned with $u^{k}(s)$, the components of the energy-momentum tensor $T^{\mu\nu}$ have comparable magnitudes. The components of the multipole moments, in the same coordinate system, are usually ⁶⁶6In the low “internal” velocities regime, some of these components can be much smaller than implied, so we can think of these magnitudes as upper bounds in the subsequent analysis. of the following orders of magnitude

$\displaystyle p^{\mu}\sim M,\,\,S^{\mu\nu}\sim Ml,\,\,J^{\mu\nu\lambda\sigma}\sim Ml^{2},\,\,J^{\mu\nu\lambda\sigma\rho}\sim Ml^{3}\,\,,$

where $M=\sqrt{p^{k}p_{k}}$ is the total mass of the body and $l$ is its characteristic (spatial) length scale (as measured over the hypersurfaces $\Sigma(s)$ orthogonal to $u^{k}$). If it is possible to write the components of the curvature tensor and its first derivative as

$\displaystyle R_{\mu\nu\lambda\sigma}\sim\frac{k}{r^{2}}\,,\quad\nabla_{\rho}R_{\mu\nu\lambda\sigma}\sim\frac{k}{r^{3}}\,,$

(7)

where $k$ is some dimensionless factor (not necessarily constant) and $r$ is some parameter with dimension of length (to be interpreted as the local radius of curvature), then each term in (5) has the respective order of magnitude

	$\displaystyle F^{\mu}\sim\frac{kM}{r}{\left(\frac{l}{r}\right)}+\frac{kM}{r}{\left(\frac{l}{r}\right)\!}^{2}+\frac{kM}{r}{\left(\frac{l}{r}\right)\!}^{3}\,\,,$
	$\displaystyle Q^{\mu\nu}\sim kM{\left(\frac{l}{r}\right)\!}^{2}+kM{\left(\frac{l}{r}\right)\!}^{3}\,\,,$		(8)

where the order-to-order separation, in the parameter $l/r$, is evident.

In order to find the desired formula for $v^{k}$, we first write

$$u^{[k}v^{l]}=\epsilon^{kl}\,,$$

(9)

where $\epsilon^{kl}$ is an anti-symmetric tensor corresponding to the right-hand side of equation (3). We see that it has order

$$\epsilon^{\mu\nu}\sim k\left[{\left(\frac{l}{r}\right)\!}^{2}+{\left(\frac{l}{r}\right)\!}^{3}+{\left(\frac{l}{r}\right)\!}^{4}\right]+k\left[{\left(\frac{l}{r}\right)\!}^{2}+{\left(\frac{l}{r}\right)\!}^{3}\right]\,\,,$$

(10)

where the first three terms come from the contraction of the force and spin (inside the square brackets we have, respectively, the spin-order, the quadrupole-order and the octupole-order force terms); and the last two terms come from the torque contracted with the dynamical velocity (inside the square brackets we have, respectively, the quadrupole-order and the octupole-order torque terms). Thus, the tensor $\epsilon^{\mu\nu}$ is of order $(l/r)^{2}$ or higher.

Contracting $u^{[k}v^{l]}$ with $u_{k}$, and doing some approximations, we get

$$v^{k}u_{k}=1-2\epsilon^{kl}\epsilon_{rl}u_{k}u^{r}+\mathcal{O}(\epsilon^{4})\,\,,$$

(11)

where $\mathcal{O}(\epsilon^{4})$ represent terms that contain at least four factors of $\epsilon^{kl}$. Since we are only keeping terms up to order $(l/r)^{3}$, we can even neglect terms with two factors of $\epsilon^{kl}$, i.e., $\mathcal{O}(\epsilon^{2})$, obtaining

$$v^{k}\approx u^{k}+2\epsilon^{lk}u_{l}=u^{k}+\frac{1}{M^{2}}S^{kl}F_{l}+\frac{1}{M}Q^{kl}u_{l}\,,$$

(12)

which is correct up to order $(l/r)^{3}$. It is interesting to note that if we evaluate $u^{[k}v^{l]}$ from this approximate formula for $v^{l}$, we obtain $\epsilon^{kl}$ exactly.

If the body were point-like and free, its worldline would be a geodesic of the spacetime and thus satisfy the geodesic equation $v^{r}\nabla_{r}v^{k}=0$. However, since the body is not point-like, its worldline $\ell$ should satisfy a different equation, namely

$$v^{r}\nabla_{r}v^{k}=v^{r}\nabla_{r}u^{k}+v^{r}\nabla_{r}\left[\frac{1}{M^{2}}S^{kl}F_{l}+\frac{1}{M}Q^{kl}u_{l}\right]\,\,.$$

(13)

From equation (1) we obtain

	$\displaystyle v^{l}\nabla_{l}v^{k}$	$\displaystyle=\frac{1}{M}(F^{k}-F^{l}u_{l}u^{k})-\frac{3}{M^{3}}F^{r}u_{r}S^{kl}F_{l}$
		$\displaystyle+\frac{1}{M^{2}}Q^{kl}(2F_{l}-3F_{r}u^{r}u_{l})+\frac{1}{M^{2}}u^{k}Q^{lr}F_{l}u_{r}$
		$\displaystyle+\frac{1}{M^{2}}S^{kl}v^{r}\nabla_{r}F_{l}+\frac{1}{M}v^{r}\nabla_{r}Q^{kl}u_{l}\,,$		(14)

which is the desired equation for the worldline of the body. Note that the terms involving two factors of force or a product of a force and a torque implicitly contain two factors of curvature, which contributes with the small factor of order $k^{2}/r^{4}$. Since we are dropping terms of this order, this equation can be simplified to

$$v^{l}\nabla_{l}v^{k}=\frac{1}{M}(F^{k}-F^{l}u_{l}u^{k})+\frac{1}{M^{2}}S^{kl}v^{r}\nabla_{r}F_{l}+\frac{1}{M}v^{r}\nabla_{r}Q^{kl}u_{l}$$

(15)

which contains a force term, a term with the spin and the time derivative of the force, and a term involving the time derivative of the torque. The order of magnitude of the last two terms, containing the time derivatives of the force and the torque, cannot be precisely evaluated yet as we still do not have a prescription to explicitly calculate the multipole moments. Therefore, before going further with the analysis of this equation, let us discuss a proposal to specify the multipole moments.

This section is devoted to introduce the core element of the covariant theory of swimming: the special manner to prescribe the multipole moments of the body. Considering equation (15) for the worldline of the center of mass of the body, we can see that all the information concerning its geometric shape and internal motions have to be fully encoded in the multipole moments.

The most straightforward manner to prescribe the multipole moments, as noted by Dixon in his papers, is to translate the internal differential equations governing the substances compounding the body into equations relating the multipole moments. For instance, Dixon discusses a reasonable definition for a rigid body in general relativity, where the components of the moments are assumed to be constant in a tetrad rotating in a way determined by the total spin of the body. In the general case, one could use the correspondence between the stress-energy tensor and the set of multipole moments, mentioned in item $(\text{iii})$ of section (II), to translate the dynamical differential equations from one description to the other. Since we are going to this correspondence in this section, let us write it explicitly below. The (higher) multipole moments are given, for $n\geq 0$, by

$$J^{k_{1}\cdots k_{n}lpqr}=\Upsilon^{k_{1}\cdots k_{n}[l[qp]r]}+\frac{1}{n+1}\Pi^{k_{1}\cdots k_{n}[l[qp]r]t}v_{t}\,,$$

(16)

where $\Upsilon$ and $\Pi$ are respectively given, for $n\geq 2$, by

	$\displaystyle\Upsilon^{k_{1}\cdots k_{n}lq}(s)=(-1)^{n}\!\int_{\Sigma(s)}\sigma^{k_{1}}\ldots\sigma^{k_{n}}{\sigma}^{l}_{\>\>{a}}{\sigma}^{q}_{\>\>{b}}T^{ab}w^{c}d\Sigma_{c}$
	$\displaystyle\Pi^{k_{1}\cdots k_{n}lqr}(s)=2(-1)^{n}\!\int_{\Sigma(s)}\sigma^{k_{1}}\ldots\sigma^{k_{n}}{\langle n\rangle}^{prlq}H_{ap}T^{ab}d\Sigma_{b}$		(17)

where $\Upsilon^{k_{1}\cdots k_{n}lq}(s)=\Upsilon^{k_{1}\cdots k_{n}lq}(z(s))$ and $\Pi^{k_{1}\cdots k_{n}lqr}(s)$ are tensors at $z(s)$. The integral run over $x\in\Sigma(s)$, with $d\Sigma^{c}$ being the vector-valued element of volume induced on $\Sigma(s)$. The stress-energy tensor $T^{ab}$ is based at $x$, and $w^{a}$ is any vector field such that $w^{a}\delta s$ carries $\Sigma(s)$ to $\Sigma(s+\delta s)$. The quantities $\sigma^{k}$, ${\sigma}^{l}_{\>\>{a}}$, ${\sigma}^{q}_{\>\>{b}}$, $H_{ap}$ and ${\langle n\rangle}^{prlq}$ are bitensors, based at $(x,z(s))$, defined in Appendix A. Note that we use a similar index convention for bitensors as for tensors, with the difference that for bitensors based at $(x,z)$ the first half of the alphabet (“$a,b,c,\ldots$” or “$\alpha,\beta,\gamma,\ldots$”) is reserved for indices at $x$ (first entry) and the second half of the alphabet (“$k,l,m,\ldots$” or “$\kappa,\lambda,\mu,\ldots$”) for indices at $z$ (second entry).

So, how exactly one should proceed to specify these moments in a particular situation? In the case of the swimming bipod, for example, we could consider that a certain physical device is responsible for producing the desired internal movements. A simple realization is to consider a set of springs interconnecting the legs, so as to make them open or close as the springs oscillate; also, the legs could be considered to be springs themselves, so that they would be stretching and shrinking in an oscillatory fashion. We would then have to provide a local, covariant description for these springs. However, even the simplest model (of locally Hookean springs⁷⁷7By locally Hookean spring we mean a relativistic model for a spring that obeys the Hooke’s law in the local instantaneous rest frame of any infinitesimal piece of it. See [10].) is described by highly involved differential equations. Moreover, an important drawback of this type of approach is that the internal motions are not easily controllable. That is, it would be a considerable effort to find which structural modifications (such as changing the position or stiffness of the springs) would have to be implemented, at each time, in order to make the body perform the desired sequence of internal motions.

Therefore, we shall look for an alternative approach to the prescription of the multipole moments, particularly one that allows us to directly provide the sequence of internal motions rather than the underlying mechanisms necessary to produce them. As proposed by Wisdom, we might assume that there is some ingeniously-designed device, continuously coupled along the whole extension of the body, that applies the exact amount of tension to each of its points such as to produce a formerly specified sequence of internal motions. One way to conceive this is to imagine that the problem is first solved theoretically to determine the required internal forces that should be applied to each point, then all the information is stored in the memory of microcomputers spread all over the body, which will tell the local mechanisms of the device how to act in order to apply the correct tensions once the body is actually released to swim. We shall not be too concerned with the engineering details of the swimming mechanisms, as we are only interested in the resulting dynamics of a body going through a sequence of shape changes.

The basis of the covariant theory of swimming is the realization that, if only $(l/r)^{3}$ order of precision is required in the equations of motion, then the quadrupole and octupole moments can be evaluated as if the neighboring spacetime were flat. This follows from the fact that curvature only affects the quadrupole and octupole moments at order $(l/r)^{2}$. Thus, since the lowest-order contribution to the equations of motion from these moments (that would come from their evaluation in a flat background) are already at order $(l/r)^{2}$, then curvature would only produce terms of order $(l/r)^{4}$ or higher in the equations of motion. Basically, we will treat the equations of motions perturbatively in the curvature, so that we first study the internal motions as if the spacetime were flat, computing the multipole moments accordingly, and then insert them into the Dixon’s equations of motion. It should be stressed that this procedure plays a role only in the translation from the description of the internal states of a body in terms of an energy-momentum tensor to the corresponding description in terms of the multipole moments.

Before going through the justification of this claim, let us consider the case where the spacetime is actually flat. In Minkowski spacetime the equations of motion reduce to

	$\displaystyle\frac{\delta p^{k}}{ds}=0$
	$\displaystyle\frac{\delta S^{kl}}{ds}=0$
	$\displaystyle u^{[k}v^{l]}=0\,,$		(18)

which imply that the mass, the dynamical velocity and the spin are conserved. Also, since $v^{k}=u^{k}$, the worldline of the center of mass $\ell$ is a straight line (i.e., geodesic). The internal motions have only to satisfy the conservation equation

$$\partial_{a}T^{ab}=0\,,$$

(19)

where $\partial_{a}$ is the derivative operator associated with any inertial (Cartesian) coordinate system.

In inertial (Cartesian) coordinates, the bitensors appearing in (17) become

	$\displaystyle\sigma^{\kappa}=-(x^{\kappa}-z^{\kappa})$
	$\displaystyle\sigma^{\alpha}=-(z^{\alpha}-x^{\alpha})$		(20)

	$\displaystyle{\sigma}^{\alpha}_{\>\>{\beta}}={\delta}^{\alpha}_{\>\>{\beta}}$
	$\displaystyle{\sigma}^{\alpha}_{\>\>{\lambda}}=-{\delta}^{\alpha}_{\>\>{\lambda}}$
	$\displaystyle{\sigma}^{\kappa}_{\>\>{\lambda}}={\delta}^{\kappa}_{\>\>{\lambda}}$
	$\displaystyle{\sigma}^{\kappa}_{\>\>{\beta}}=-{\delta}^{\kappa}_{\>\>{\beta}}$		(21)

	$\displaystyle{H}^{\alpha}_{\>\>{\kappa}}={\delta}^{\alpha}_{\>\>{\kappa}}$
	$\displaystyle{K}^{\alpha}_{\>\>{\kappa}}={\delta}^{\alpha}_{\>\>{\kappa}}$		(22)

$$\langle n\rangle^{\mu\nu\kappa\lambda}=\eta^{\mu(\kappa}\eta^{\lambda)\nu}\quad\text{for {\hbox{n \geq 2}}}\,,$$

(23)

where $\delta$ is the Kronecker delta and $\eta$ is the flat (Minkowski) metric.

Since the worldline of the center of mass is straight, the spatial hypersurfaces $\Sigma(s)$ are all parallel to each other, so that we may simply take the $w$-vector as

$$w^{\mu}=u^{\mu}=v^{\mu}\,\,.$$

(24)

Consider the particular inertial frame that is at rest with respect to the center of mass. Further, let it be located at the spatial origin of the coordinates. Thus, the worldline of the center of mass $\ell$ is described by the vertical line

$$z^{\mu}(s)=(s,0,0,0)\,\,,$$

(25)

where $s$ is the proper time along it. The momentum-energy tensor can be decomposed⁸⁸8In this section, the indices $i,j,k,l,m$ may be used to indicate spatial indices of tensor components, so that they assume values from $1$ to $3$. Hopefully, it will be sufficiently clear where they are being used with this meaning and where the standard convention (abstract indices) is being used. as

$$T^{\mu\nu}=\left(\begin{array}[]{cccc}\rho&g^{1}&g^{2}&g^{3}\\ g^{1}&h^{11}&h^{12}&h^{13}\\ g^{2}&h^{12}&h^{22}&h^{23}\\ g^{3}&h^{31}&h^{32}&h^{33}\end{array}\right)\,\,,$$

(26)

where $\rho$ can be interpreted as the mass density of the matter distribution, $g^{i}=(g^{1},g^{2},g^{3})$ can be seen as the density of linear 3-momentum, and the 3-tensor $h^{ij}$ is called the total stress 3-tensor. If $u^{i}$ is the 3-velocity of a piece of matter at a given point, then the total stress 3-tensor $h^{ij}$ is related with the stress 3-tensor $\tau^{ij}$ by $h^{ij}=\tau^{ij}+g^{i}u^{j}$, and $\tau^{ij}$ is such that, for an infinitesimal oriented surface of area $\delta A$ and a direction defined by the unit 3-vector $\hat{n}=(n^{1},n^{2},n^{3})$, then the 3-vector $\tau^{ij}n_{j}\delta A$ yields the force that the matter on the negative side of the surface applies to the matter on the positive side of it.

Inserting all these results in the definitions of $\Upsilon$ and $\Pi$, we get the following expressions for the quadrupole and octupole moments in flat spacetime

	$\displaystyle J^{\mu\nu\rho\sigma}=\int_{\Sigma}d\Sigma\,r^{[\mu}r^{[\rho}\left(T^{\nu]\sigma]}+v^{\nu]}T^{\sigma]0}+v^{\sigma]}T^{\nu]0}\right)$
	$\displaystyle J^{\lambda\mu\nu\rho\sigma}=\int_{\Sigma}d\Sigma\,r^{\lambda}r^{[\mu}r^{[\rho}\left(T^{\nu]\sigma]}+\frac{1}{2}v^{\nu]}T^{\sigma]0}+\frac{1}{2}v^{\sigma]}T^{\nu]0}\right)\,\,.$		(27)

where $r^{\mu}=x^{\mu}-z^{\mu}(s)=(0,x^{1},x^{2},x^{3})$, with $x\in\Sigma(s)$, are the “position vectors” across the spatial surfaces $\Sigma(s)$. It will be useful to write down explicitly, in terms of $\rho$, $g^{i}$ and $h^{ij}$, each non-vanishing component of these moments. For the quadrupole,

	$\displaystyle J^{0i0j}=\frac{3}{4}\int d^{3}\!x\,x^{i}x^{j}\rho$
	$\displaystyle J^{0ijk}=\frac{1}{4}\int d^{3}\!x\left(3x^{i}x^{k}g^{j}-2x^{i}x^{j}g^{k}-x^{j}x^{k}g^{i}\right)$
	$\displaystyle J^{ijkl}=\int d^{3}\!x\,x^{[i}x^{[k}h^{j]l]}\,,$		(28)

and for the octupole,

	$\displaystyle J^{m0i0j}=\frac{1}{2}\int d^{3}\!x\,x^{m}x^{i}x^{j}\rho$
	$\displaystyle J^{m0ijk}=\frac{1}{8}\int d^{3}\!x\,x^{m}\left(4x^{i}x^{k}g^{j}-3x^{i}x^{j}g^{k}-x^{j}x^{k}g^{i}\right)$
	$\displaystyle J^{mijkl}=\int d^{3}\!x\,x^{m}x^{[i}x^{[k}h^{j]l]}\,,$		(29)

where the functions $\rho$, $g^{i}$ and $h^{ij}$ are all evaluated at the point $(s,x^{1},x^{2},x^{3})$. This makes clear how the different components of the moments depend on the dynamical quantities associated with the matter distribution: when there are two time-indices (i.e., $\mu=0$) among the last four indices, then the component depends on the mass distribution of the matter; when there is only one time-index among the last four, then the component depends on the linear 3-momentum distribution; and when there is no time-index among the last four, then the component depends on the total stress distribution (which includes the 3-velocity, linear 3-momentum and the stress 3-tensor distributions).

Now we show that the “correct” expressions for the multipole moments, when curvature effects are included, differ from the flat spacetime expressions (27) only at order $(l/r)^{2}$. First let us argue that, in the suitable sense, the geometry of a bounded object⁹⁹9An object here may be seen as a subset of the space defined by a set of intrinsic geometric rules. of size (length) $l$ is affected by curvature only at order $Rl^{2}\times l\sim k(l/r)^{2}\times l$, if $Rl^{2}\ll 1$. It is clear that we need to define precisely what it means to “turn on the curvature” in the neighborhood of the object. Although there is no absolute way to define this operation, the exponential map provides a very convenient manner to relate the curved with the flat space – we just let the tangent space (say, at the center of mass) be the “flattened” projection of the real (curved) space.

Assume that there is some point $p\in\mathcal{V}\subset\mathcal{M}$, where $\mathcal{V}$ is any open normal neighborhood in $\mathcal{M}$ which contains the convex envelope of the geometrical object. Since $\mathcal{V}$ is a normal neighborhood of $p$, we have that, for each point $x\in\mathcal{V}$, there is a unique vector $v$ in $T_{p}\mathcal{M}$ such that $\exp_{p}v=x$. In other words, there is a diffeomorphism of a neighborhood of $T_{p}\mathcal{M}$ (containing the $0$ vector) onto $\mathcal{V}$ given by the exponential map at $p$. Also, there is a natural coordinate system on $\mathcal{V}$ induced from a Cartesian coordinate system on $T_{p}\mathcal{M}$: if $v$ is decomposed in an orthonormal vector basis $\{e_{\mu}\}$ at $p$ as $v=v^{\mu}e_{\mu}$, then the coordinates $v^{\mu}$ can be assigned to the point $x$, i.e., we define $x^{\mu}=v^{\mu}$ (called Riemann normal coordinates). On top of being a geometrically natural coordinate system, it has the interesting property that the coefficients of the Taylor expansion of the metric $g_{\mu\nu}(x)$, with respect to $x^{\mu}$ about $0$, are determined by the curvature and its (covariant) derivatives. For instance, up to second order we have

$$g_{\mu\nu}(x)=\delta_{\mu\nu}+\frac{1}{3}R_{\mu\rho\nu\sigma}(0)x^{\rho}x^{\sigma}+\ldots\,,$$

(30)

which suggests a very natural way to implement the “turning on” of the curvature. We simply define a one-parameter family of metrics $g_{\lambda}$ by

$$[g_{\lambda}]_{\mu\nu}(x)=\delta_{\mu\nu}+\frac{1}{3}\lambda R_{\mu\rho\nu\sigma}(0)x^{\rho}x^{\sigma}+\ldots\,\,,$$

(31)

so that $\lambda=0$ corresponds to a flat metric and $\lambda=1$ corresponds to the actual metric of the space.

Let us now try to convince ourselves that curvature affects the basic laws of flat geometry at order $Rl^{2}$ only (note that this is not intended to be a rigorous argument, in part because the proposition itself lacks the necessary precision). We shall consider a few basic geometrical constructions and try to identify the effect of curvature. For instance, note that the trajectory of a geodesic starting at $x_{0}^{\mu}$ with (unit) tangent vector $v^{\mu}$ is described by

$$x^{\mu}(s)=x_{0}^{\mu}+v^{\mu}s-\frac{1}{3}s^{2}{R_{\nu\rho\sigma}}^{\mu}x_{0}^{\nu}v^{\rho}v^{\sigma}+\ldots\,\,,$$

(32)

where $s$ is the proper length along it and the curvature tensor is evaluated at the origin, $p$. If a geodesic defined by the same prescription lived in a flat space, then its trajectory $\widetilde{x}^{\mu}(s)$ would be given by

$$\widetilde{x}^{\mu}(s)=x_{0}^{\mu}+v^{\mu}s\,\,,$$

(33)

which indicates that they deviate from each other by

$$\Delta x^{\mu}(s)=x^{\mu}(s)-\widetilde{x}^{\mu}(s)=-\frac{1}{3}s^{2}{R_{\nu\rho\sigma}}^{\mu}x_{0}^{\nu}v^{\rho}v^{\sigma}+\ldots\,\,,$$

(34)

We then see that the effect of the curvature $R\sim k/r^{2}$, in a geodesic of length $l$ within $\mathcal{V}$ (so that $x_{0}^{\mu}\lesssim l$), is of order $l\times k(l/r)^{2}$, supporting the claim.

For a few more examples [12], consider the $(i)$ geodesic distance $L(x_{1},x_{2})$ between two points $x_{1}^{\mu}$ and $x_{2}^{\mu}$ in $\mathcal{V}$,

$$L(x_{1},x_{2})^{2}=\delta_{\mu\nu}\Delta x_{12}^{\mu}\Delta x_{12}^{\nu}+\frac{1}{3}R_{\mu\rho\nu\sigma}x_{1}^{\rho}x_{1}^{\sigma}x_{2}^{\mu}x_{2}^{\nu}+\ldots\,\,,$$

(35)

where $\Delta x_{ij}^{\mu}=x_{j}^{\mu}-x_{i}^{\mu}$; the $(ii)$ cosine law for a geodesic triangle with vertices $x_{0}$, $x_{1}$ and $x_{2}$,

	$\displaystyle 2L_{01}L_{02}\cos\!\theta_{0}$	$\displaystyle=L_{01}^{2}+L_{02}^{2}-L_{12}^{2}$
		$\displaystyle+\frac{1}{3}R_{\mu\rho\nu\sigma}\Delta x_{01}^{\rho}\Delta x_{01}^{\sigma}\Delta x_{02}^{\mu}\Delta x_{02}^{\nu}+\ldots\,,$		(36)

where $\theta_{0}$ is the angle subtended at the vertex $x_{0}$ and $L_{ij}=L(x_{i},x_{j})$; and the $(iii)$ parallel transport of a vector $v_{0}^{\mu}$ at $x_{0}$, along an arbitrary curve $x^{\mu}(s)$ starting at $x_{0}$,

$$v^{\mu}(s)=v_{0}^{\mu}+\frac{1}{3}\left({R_{\nu\rho\sigma}}^{\mu}+{R_{\nu\sigma\rho}}^{\mu}\right)v_{0}^{\rho}a_{1}^{\sigma}\left(sa_{0}^{\nu}+\frac{1}{2}s^{2}a_{1}^{\nu}\right)+\ldots$$

(37)

where $a_{0}^{\mu}=x_{0}^{\mu}$ and $a_{1}^{\mu}=\left.dx^{\mu}/ds\right|_{s=0}$. This suggests that the geometry of the body can be well-represented in a flat space, given our approximations.

Let us now examine other aspects of this approximation. Observe that the fundamental conservation equation, $\nabla_{a}T^{ab}=0$, is only affected by the curvature at an order assumed to be negligible in Dixon’s equations

	$\displaystyle 0$	$\displaystyle=\nabla_{\mu}T^{\mu\nu}(x)=\partial_{\mu}T^{\mu\nu}+\Gamma^{\mu}_{\,\,\mu\sigma}T^{\sigma\nu}+\Gamma^{\nu}_{\,\,\mu\sigma}T^{\mu\sigma}\approx$
		$\displaystyle\approx\partial_{\mu}T^{\mu\nu}-\frac{1}{3}R_{\rho\sigma}x^{\rho}T^{\sigma\nu}+\frac{2}{3}{R_{\rho\mu\sigma}}^{\nu}x^{\rho}T^{\mu\sigma}\,,$		(38)

which implies that $\partial_{\mu}T^{\mu\nu}\approx 0$ as the terms containing the curvature can be dropped (for they contain a factor of order $k/r^{2}$). Hence, the dynamics of the internal motions of the body can be described by special relativity.

The first curvature-related corrections to the flat spacetime expressions for the bitensors in (17), if we particularize to $z=p$, are

	$\displaystyle\sigma^{\alpha}=x^{\alpha}$
	$\displaystyle\sigma^{\kappa}=-x^{\kappa}\,\,,$		(39)

	$\displaystyle{\sigma}^{\alpha}_{\>\>{\kappa}}=-\delta^{\alpha}_{\,\,\kappa}+\frac{1}{3}{R_{\beta\kappa\gamma}}^{\alpha}x^{\beta}x^{\gamma}$
	$\displaystyle{\sigma}^{\alpha}_{\>\>{\beta}}=\delta^{\alpha}_{\,\,\beta}+\frac{1}{3}{R_{\epsilon\beta\gamma}}^{\alpha}x^{\epsilon}x^{\gamma}$
	$\displaystyle{\sigma}^{\kappa}_{\>\>{\alpha}}=-\delta^{\kappa}_{\,\,\alpha}$
	$\displaystyle{\sigma}^{\kappa}_{\>\>{\lambda}}=\delta^{\kappa}_{\,\,\lambda}+\frac{1}{3}{R_{\alpha\lambda\beta}}^{\kappa}x^{\alpha}x^{\beta}\,\,,$		(40)

	$\displaystyle{H}^{\alpha}_{\>\>{\kappa}}=\delta^{\alpha}_{\,\,\kappa}$
	$\displaystyle{K}^{\alpha}_{\>\>{\kappa}}=\delta^{\alpha}_{\,\,\kappa}+\frac{1}{3}{R_{\beta\kappa\gamma}}^{\alpha}x^{\beta}x^{\gamma}\,,$		(41)

showing that all these quantities only deviate from their flat spacetime counterparts, (20-22), at order $k(l/r)^{2}$. Other geometrical quantities in (17) are the $n$-potential function $\langle n\rangle$, the volume element and the $w$-vector. The first two can be similarly shown to deviate from their flat spacetime counterparts by terms proportional to the curvature tensor.

The $w$-vector, on the other hand, deserves a special consideration for it is not exclusively related to the curvature. For a straight line $\ell$ in the flat spacetime, it is clear that $w^{a}$ can be chosen as the covariant extension of $u^{k}=v^{k}$, that is, the vector field that coincides with $u^{k}$ on $\ell$ and satisfies $\nabla_{a}w^{b}=0$ everywhere. If $\ell$ is taken as a general (non-straight) line, so that the orthogonal hypersurfaces $\Sigma(s)$ are not parallel to each other, then $w^{a}$ will not be a constant vector field. Let $z^{\mu}(s)$ be the trajectory of $\ell$ in the Riemann normal coordinate system centered at $z^{\mu}(0)=0$, with temporal vector $(e_{0})^{\mu}=v^{\mu}(0)$. Using the Taylor expansion in the proper length $s$, we can write

$$z^{\mu}(s)=(e_{0})^{\mu}s+\frac{1}{2}a^{\mu}(0)s^{2}\,\,,$$

(42)

where $a^{\mu}$ denotes the covariant acceleration $v^{\nu}\nabla_{\nu}v^{\mu}$. The hypersurface $\Sigma_{\delta s}$ at the infinitesimal proper time $\delta s$ is formed by all geodesics starting at $z^{\mu}(\delta s)$ orthogonal to $u^{\mu}(\delta s)$. Since $u^{\mu}$ only differs from $v^{\mu}$ by terms involving the curvature tensor, we may consider that $\Sigma_{\delta s}$ is actually formed by geodesics orthogonal to $v^{\mu}(\delta s)$ instead. Let $q^{\mu}(s)$ be a general vector at $z^{\mu}(s)$ orthogonal to $v^{\mu}(s)$, so that the set of all $q^{\mu}(s)$ form a 3-dimensional vector subspace at $z^{\mu}(s)$. At $\delta s$, we have $0=q^{\mu}(\delta s)v_{\mu}(\delta s)\approx\eta_{\mu\nu}q^{\mu}v_{\nu}=q^{0}-\vec{q}\cdot\vec{a}\delta s$, where the curvature term in $g_{\mu\nu}(z(\delta s))\approx\eta_{\mu\nu}$ was dropped. In this equation we are writing $q^{\mu}=(q^{0},\vec{q})$, in which $\vec{q}=(q^{1},q^{2},q^{3})$, and $a^{\mu}=a^{\mu}(0)=(0,\vec{a})$. The dot denotes the trivial Euclidean scalar product, $\vec{q}\cdot\vec{a}=\sum_{i=1}^{3}q^{i}a^{i}$. Thus, we have $q^{\mu}(\delta s)=(\vec{q}\cdot\vec{a}\delta s,\vec{q})$. A geodesic starting at $z^{\mu}(\delta s)$ and tangent to $q^{\mu}(\delta s)$, denoted by $x^{\mu}(\lambda)$, where $\lambda$ is any affine parameter, can be given approximately by

$$x^{\mu}(\lambda)=z^{\mu}(\delta s)+q^{\mu}(\delta s)\lambda-\frac{1}{3}\lambda^{2}{R_{\nu\rho\sigma}}^{\mu}z^{\nu}(\delta s)q^{\rho}(\delta s)q^{\sigma}(\delta s)\,,$$

(43)

in which the last term, containing the curvature tensor, can be dropped. The $w$-vector can then be chosen at $\Sigma_{0}$ as the vector field satisfying

$$w^{\alpha}(0,\vec{x}(\lambda))\delta s=(e_{0})_{\mu}x^{\mu}(\lambda)(e_{0})^{\alpha}$$

(44)

for infinitesimal $\delta s$. The $x^{\mu}(\lambda)$ in the right-hand side of the equation has to be evaluated up to first order in $\delta s$, yielding $(e_{0})_{\mu}x^{\mu}(\lambda)(e_{0})^{\alpha}=(\delta s+\lambda\vec{q}\cdot\vec{a}\delta s)(e_{0})^{\alpha}$; while the argument of $w^{\alpha}$ in the left-hand side can be evaluated to the lowest order, $\vec{x}(\lambda)=\lambda\vec{q}$. It then follows that

$$w^{\alpha}(0,\vec{x})=(1+\vec{x}\cdot\vec{a})(e_{0})^{\alpha}\,\,,$$

(45)

showing that, if $\ell$ could be prescribed arbitrarily, then the $w$-factor would not necessarily be a constant vector field, even if the spacetime were flat. In our case, nonetheless, $\ell$ is the worldline of the center of mass and its 4-acceleration is given by equation (15), where it is clear that $\vec{a}$ necessarily contains a curvature factor (from the gravitational force and torque), that implies that its order of magnitude has a factor $\sim k/r^{2}$. Therefore the $w$-factor can be approximated as

$$w^{\alpha}\approx(e_{0})^{\alpha}=\delta^{\alpha}_{\,\,0}\,\,,$$

(46)

as if they were evaluated in a exactly flat spacetime (where $\ell$ would be exactly straight).