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Eur Phys J C Part Fields
Eur Phys J C Part Fields
The European Physical Journal. C, Particles and Fields
1434-6044
1434-6052
Springer Berlin Heidelberg Berlin/Heidelberg

13232
10.1140/epjc/s10052-024-13232-4
Regular Article –Theoretical Physics
Carroll geodesics
Ciambelli Luca 1
Grumiller Daniel grumil@hep.itp.tuwien.ac.at

12
1 https://ror.org/013m0ej23 grid.420198.6 0000 0000 8658 0851 Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON N2L 2Y5 Canada
2 https://ror.org/04d836q62 grid.5329.d 0000 0004 1937 0669 Institute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8–10/136, 1040 Vienna, Austria
16 9 2024
16 9 2024
2024
84 9 9334 4 2024
11 8 2024
© The Author(s) 2024
2024
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Using effective field theory methods, we derive the Carrollian analog of the geodesic action. We find that it contains both “electric” and “magnetic” contributions that are in general coupled to each other. The equations of motion descending from this action are the Carrollian pendant of geodesics, allowing surprisingly rich dynamics. As an example, we derive Carrollian geodesics on a Carroll–Schwarzschild background and discover an effective potential similar to the one appearing in geodesics on Schwarzschild backgrounds. However, the Newton term in the potential turns out to depend on the Carroll particle’s energy. As a consequence, there is only one circular orbit localized at the Carroll extremal surface, and this orbit is unstable. For large impact parameters, the deflection angle is half the value of the general relativistic light-bending result. For impact parameters slightly bigger than the Schwarzschild radius, orbits wind around the Carroll extremal surface. For small impact parameters, geodesics get reflected by the Carroll black hole, which acts as a perfect mirror.

http://dx.doi.org/10.13039/501100002428 Austrian Science Fund P 32581 P 33789 P 36619 issue-copyright-statement© EDP Sciences, Societa Italiana di Fisica (SIF) and Springer-Verlag GmbH, DE, part of Springer Nature 2024
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pmcIntroduction

Geometry and physics have a long-lasting relationship that has evolved considerably over time. In the days of Plato’s school, whose entrance door allegedly had the engraving “Let no one ignorant of geometry enter,” space was Euclidean, and space and time were absolute. The only symmetries were space- and time-translations and spatial rotations. Nowadays, such spacetimes are referred to as “Aristotelian”.

The first substantial change in geometry came from physics: Galilei dropped absolute space and introduced relativity, using gedanken- and real experiments with ships. In modern jargon, the new symmetries introduced by Galilei are called boosts. Space was now relative, though time remained absolute in these Galilean spacetimes.

The second significant change came from mathematics: Riemann (after pioneering work by Lobachevsky, Bolyai, Gauss, and others) dropped Euclid’s fifth axiom and established curved spaces with signature (+,+,⋯,+). Symmetries were then formalized in terms of Killing vectors.

The third impactful change came from mathematics and physics in the wake of special (and later general) relativity: space and time were merged into a single entity, spacetime, requiring metrics with a pseudo-Riemannian signature (-,+,⋯,+). The Galilean boosts became Lorentzian, and neither space nor time remained absolute.

Half a century ago, in the authors’ opinion, the fourth major change in our understanding of geometry was initiated by Lévy-Leblond [1, 2], who introduced Carrollian spacetimes, which have a degenerate signature of (0,+,⋯,+).1 The simplest example of such a spacetime is the vanishing speed of light limit of the Minkowski metric ds2=limc→0(-c2dt2+δijdxidxj)=δijdxidxj. Carrollian time is relative, but Carrollian space is absolute. Moreover, to characterize a Carrollian spacetime, in addition to a Carroll metric hμν of signature (0,+,⋯,+), we need a Carroll vector vμ that lies in the kernel of the metric, vμhμν=0. In the example just discussed, the vector field is v=vμ∂μ=∂t and the Carroll metric is hμν=δijδμiδνj.

A remarkable aspect of Carrollian spacetimes is that they can possess infinite-dimensional isometries, unlike (pseudo-)Riemannian spacetimes. Indeed, for the example above, solving the Killing equations ξμ∂μhαβ+hαμ∂βξμ+hμβ∂αξμ=0=vμ∂μξν-ξμ∂μvν yields Euclidean translations and rotations, and additionally an infinite set of Killing vectors ξ=f(xi)∂t that preserve the Carroll metric hμν and the Carroll vector vμ.

It took the physics community a while to figure out that Carrollian spacetimes are applicable to anything – contrary to the Galilean limit, where the speed of light tends to infinity, there is no everyday intuition associated with the Carrollian limit. However, in the past decade, several physics applications of Carrollian symmetries have emerged.

Perhaps the clearest one is associated with the structure of null infinity, see e.g. [4–8]. Formally, the metric at null infinity is 0·du2+dΩ2, where u is retarded (or advanced) time and dΩ2 the metric of the celestial sphere. The infinite-dimensional Carrollian symmetries are the supertranslations discovered by Bondi, van der Burgh, Metzner, and Sachs (BMS) [9, 10]. So null infinity in asymptotically flat spacetimes naturally carries a Carrollian structure, and the BMS symmetries are equivalent to (conformal) Carrollian symmetries [11–13]. The relation between BMS and conformal Carrollian symmetries paved the way for the Carrollian approach to flat space holography, which so far has been studied mostly in three spacetime dimensions [14–28] and more recently in four spacetime dimensions [4–8, 29–35]. Other physics applications of Carrollian symmetries include the description of null hypersurfaces [36–44], the fluid/gravity correspondence [45–51], Carrollian algebra, scalar fields and particles [52–64], tensionless strings [65–69], cosmology [70], Hall effects [71], fractons [72–75], flat bands [76], Bjorken flow [77], supersymmetry and supergravity [78–81]. See [82, 83] for more references.

Given the success of general relativity, which geometrically can be understood as emerging from gauging the Poincaré-algebra, and the ubiquitousness of Carroll symmetries, it is natural to gauge the Carroll algebra [84] and establish Carroll gravity theories [70, 85–96], which may exhibit Carroll black hole solutions [97].

So far, the only discussion in the literature on how to probe Carrollian spacetimes with test particles is in [82], where the approach is to take the Carrollian limit of the relativistic geodesic equation. There, it was found that in the limit particles either cannot move or have zero energy. As we will show, however, an intrinsic Carrollian analysis gives rise to more possibilities.

To paraphrase Wheeler, we know already the way in which “matter tells Carroll geometry how to curve”. The main purpose of our Letter is to establish how “Carroll geometry tells matter how to move”. In other words, our goal is to derive the intrinsic Carrollian version of geodesics.

Our main conclusion is that our intrinsic analysis shows that Carroll particles following Carrollian geodesics can move, with arbitrary energy. We shall demonstrate this result first in full generality and then by means of a pertinent example, a Carroll test particle moving on the background of a Carroll–Schwarzschild black hole. Our conclusions are supported by the analysis of a companion paper [83] that analyzes Carroll scalar fields and also finds non-trivial dynamics.

Carrollian geometry basics

We review salient features of ruled Carrollian geometry, see [39, 84, 92, 97] for more details. In addition to the Carroll metric hμν and the Carroll vector vμ, we introduce the dual of the Carroll vector, Eμ, known in the literature as “clock 1-form” or “Ehresmann connection”, as part of our geometric data. This allows defining the projector hμν=δμν-Eμvν. The main properties of these quantities are1 hμνvν=vνhνμ=hνμEμ=0Eμvμ=1hμαhαν=hμν.

We split the coordinates into perpendicular and parallel components (Ω is a positive scale factor to be fixed),2 xμ=x⊥μ+Ωvμx‖x⊥μ:=xνhνμx‖:=EμxμΩ.

We demand an analogous split for the coordinate differentials,3 dxμ=δνμdxν=dxν(hνμ+Eνvμ)=!dx⊥νhνμ+Ωvμdx‖

where the first two equalities hold by definition, and the last one imposes non-trivial requirements: vμ changes only parallel to itself, dvμ=vμdα with some scalar α, and we exploit local Carroll boosts Eμ→Eμ-λμ to impose dEμ=-Eμdα, consistent with d(Eμvμ)=0. Together, these conditions imply dhνμ=0 and thus dx⊥μ=dx⊥νhνμ. The remaining term in (3) is compatible, provided we fix the scale factor Ω=exp(-α). These requirements are invariant under the Carroll diffeomorphisms introduced in [4], see [39]. From these data, we extract two Carroll-diffeomorphism invariant quantities, hμνdx⊥μdx⊥ν and dx‖. Both quantities are also invariant under local Carroll boosts, preserving therefore the local Carroll symmetries of the background.

Carroll geodesics

The philosophy of effective field theories is to write down the most general action compatible with the field content, the symmetries, and possibly further consistency requirements and then perform a derivative expansion, keeping only the terms at most quadratic in time derivatives since higher order terms generically lead to an Ostrogradsky instability [98]. We apply this scheme to derive an action for Carrollian test particles moving in the background of an arbitrary Carrollian spacetime.

Let us start with symmetries. We require the Carrollian geodesic action to be invariant under worldline reparametrizations and Carroll diffeomorphisms. The field content is x‖, x⊥μ, and the worldline einbein e. Because of translation invariance, the action only should depend explicitly on x˙‖ and x˙⊥μ, where dots denote derivatives with respect to the worldline parameter τ. Keeping only terms with up to two derivatives (and dropping total derivative terms) yields the unique action24 I[x‖,x⊥μ,e]=∫dτe(g0+g1e-2hμνx˙⊥μx˙⊥ν+g2e-2x˙‖2)

where the remaining freedom are the coupling constants gi. The action (4) is the Carrollian analog of the geodesic action and our first key result.

Some remarks are in order. In Carrollian jargon (see, e.g., [82]), the action (4) contains both an “electric term” (proportional to g2) and a “magnetic term” (proportional to g1), that can be isolated by setting to zero the other coupling constant. For instance, the term proportional to g1 emerges from the relativistic geodesic Lagrangian L=e-1g2gμνx˙μx˙ν in the Carroll limit x0=ct, c→0. Similarly, the term proportional to g2 emerges from the relativistic geodesic Hamiltonian H=e2p2 in the c→0 limit after rescaling e→ec2 and p0→E/c. The scaling in c of the coupling constants gi is different in the c→0 limit, making the electric and magnetic pieces appear at different orders [85]. It is a key aspect of our Carrollian geodesic action (4) that we both have electric and magnetic terms. Tuning the coupling constants, we can reach the relativistic geodesic action, as discussed in [83]. For typical Carroll spacetimes, such as in our example below, the quantity x‖ is (minus) the coordinate time t; it transforms as a scalar under Lie variations, Lξx‖=ξμ∂μx‖.

Variation of the Carroll geodesic action (4) with respect to the einbein yields the constraint5 e=±g1g0hμνx˙⊥μx˙⊥ν+g2g0x˙‖2.

Like for standard geodesics, we assume from now on an affine parametrization where e=const. Variation with respect to x‖ yields the equation of motion6 x¨‖+Γμν‖x˙⊥μx˙⊥ν=0

with the Christoffel-like quantity Γμν‖=-g12g2∂‖hμν, where we assumed g2≠0 and defined ∂‖:=∂/∂x‖. Similarly, variation with respect to x⊥μ yields the equation of motion7 hσμx¨⊥μ+Γμνσ⊥x˙⊥μx˙⊥ν=0

with the Christoffel-like quantity Γμνσ⊥=12(∂μ⊥hνσ+∂ν⊥hμσ-∂σ⊥hμν), where we defined ∂μ⊥:=∂/∂x⊥μ. The Eqs. (6)–(7) are the Carrollian geodesic equations, our second main result.

Carroll–Schwarzschild black hole

In this section, we study in detail Carroll geodesics on a Carroll–Schwarzschild background.

Background structure and Carroll geodesic action

Rather than directly evaluating the Carrollian geodesic equations (6)–(7), we insert the Carroll–Schwarzschild black hole (xμ=(t,r,θ,φ)) [91, 92, 97]8 hμνdxμdxν=dr21-2Mr+r2(dθ2+sin2θdφ2)

9 vμ∂μ=-11-2Mr∂tα=-12ln(1-2Mr)

10 Eμdxμ=-1-2MrdtΩ=1-2Mr

as background into the Carroll geodesic action (4), obtaining11 I=∫dτe[g0+g1e-2(r˙21-2Mr+r2φ˙2)+g2e-2x˙‖2]

where we assumed, without loss of generality, motion in the equatorial plane θ=π/2 and θ˙=0. Since on this background one has hμν=diag(0,1,1,1), we used the identities x⊥t=0, x⊥r=r, x⊥θ=θ, x⊥φ=φ, and x‖=-t.

Geodesic equations and effective potential

Varying the action (11) with respect to x‖ yields12 x¨‖=0⇒x‖=Fτ+x‖0

with some integration constants F and x‖0. Variation with respect to φ produces another constant of motion, the angular momentum ℓ,13 ∂τ(r2φ˙)=0⇒φ˙=ℓr2

Like for standard geodesics on a Schwarzschild background, it is efficient to avoid varying the geodesic action with respect to r and instead exploit the affine parametrization using (5)14 g1g0(r˙21-2Mr+ℓ2r2)+g2g0F2=e2=const.

Defining the test particle energy (per mass unit)15 E:=g02g1e2-g22g1F2

allows rewriting (14) as an energy conservation equation16 r˙22+Veff(r)=E

with the effective potential17 Veff(r)=2MEr+ℓ22r2-Mℓ2r3.

Remarkably, there is only one small but significant difference to the effective potential for standard geodesics in a Schwarzschild background: the first (Newtonian) term in (17) depends linearly on the test particle’s energy and can have either sign. As we shall see, this difference has drastic consequences for the orbits of test particles. The technical key aspect is that the energy conservation equation (16) factorizes,18 r˙2=(1-2Mr)(1b2-1r2)ℓ2

and the first factor on the right-hand side vanishes at the locus r=2M corresponding to a Carroll extremal surface (CES) [97]. Here, we introduced the impact parameter19 b=ℓ2E

that determines the asymptotic radial velocity |r˙|r→∞=ℓ/b and plays a prominent role in the forthcoming orbit analysis.3

Circular orbits

We consider next circular orbits, defined by the circularity conditions20 Veff=EdVeffdr=0.

Evaluating (20) yields two algebraic equations.21 (1-2Mr)(1b2-1r2)ℓ2=0=(3Mr2-Mb2-1r)ℓ2r2

If the test particle is at the CES, r=2M, then the first identity above holds automatically, and the second one yields the constraint bcirc=2M. Thus, at the CES circular orbits are possible for any non-negative energy. If we assume instead r≠2M we get a contradiction (except for the trivial case E=ℓ=0, which we disregard4): The first circularity condition requires b=r, the second circularity condition demands (2M-r)ℓ2/r4=0, and together they imply r=2M, negating our original assumption.

We have just proven that there are no circular orbits except at the CES. They are unstable with respect to radial perturbations because the second derivative of the effective potential is negative, d2Veffdr2|r=b=2M=-ℓ28M4<0.

Thus, the linear dependence on energy in the Newton term in the effective potential (17) completely changes the spectrum of circular orbits compared to standard Schwarzschild geodesics. There are no stable circular orbits (and hence no innermost stable circular orbits), and the only unstable circular orbit is located at the CES, r=2M. Outside of the CES, no circular motion is possible. Hence, there is no planetary motion, no Kepler’s laws, and no perihelion that can be shifted.

Deflection angle

We now derive the deflection angle for a Carroll test particle incoming from infinity and scattered off the Carroll black hole. Combining the equations of motion (13)–(17) yields22 dφdr=±br(r-2M)(r2-b2)

Introducing the integration variable u=1/r, we can represent the deflection angle as23 φ∞=2∫0u0bdu(1-2Mu)(1-b2u2)

where u0 is the turning point of the trajectory, defined by the smallest positive root of the condition du/dφ=0. For impact parameter b>2M, we have u0=1/b.

For simplicity, consider first the massless case M=0. Then the integral (23) is elementary and yields24 φ∞=2arcsin(bu0)→u0=1/bπ.

As expected on physical grounds, in the massless case, the deflection angle is π, i.e., the outgoing trajectory is antipodal to the ingoing trajectory.

To obtain the deflection angle to first order in the mass M, we differentiate the integrand in (23) with respect to M and set M to zro before integrating. The deflection angle relative to the case without a Carroll black hole,25 φ∞-π≈M∂φ∞∂M|M=0=2Mb(1-1-b2u02)→u0=1/b2Mb

is half the deflection angle of lightlike geodesics on a Schwarzschild background, see e.g. [99].

To all orders in the mass, assuming u0=1/b, the result for the deflection angle is given in terms of elliptic integrals,526 φ∞=41+x(K(2x1+x)-F(π4,2x1+x))

where x:=2M/b∈(0,1). The small-x expansion (b≫2M) is φ∞=π+x+3πx2/16+⋯ and recovers (25), while the expansion when x approaches 1 from below yields φ∞=-2ln(1-x)+Δ+⋯, with Δ=2ln32(1+2)2=2.408⋯

The last result means that the orbit winds around the azimuthal direction repeatedly as the limiting case b→2M is approached (from above). The approximate value for the impact parameter associated with n windings of the Carroll geodesics around the CES for large n is given by27 b(n)≈2M1-e-(2πn-Δ)/2.

We have collected some Carroll geodesic orbits in Fig. 1.

Note that reality of the velocity (18) requires r2≥b2≥0 outside the CES. Thus, Carroll particles cannot be placed in the forbidden zone 2M<r<b.Fig. 1 For b>2M, Carroll geodesics bend and turn at u0=1/b

Perfect mirror

Finally, we consider small impact parameters, 0≤b≤2M. At the upper end of the interval, b=2M, we have two different types of orbits: the (unstable) circular orbit at r=2M studied above and orbits that arrive tangential to the CES but need infinite worldline time to reach it, as can be seen inserting b=2M into Eq. (18),28 2Mdr(1-2M/r)1+2M/r=±ℓdτ.

The case b=ℓ=0 with r(0)>2M yields a radial geodesic that solves r¨=2ME/r2 and bounces back from the CES.Fig. 2 For b<2M, the geodesics touch the CES and are reflected

Within the interval, 0<b<2M, geodesics reach the CES with finite affine parameter and zero coordinate velocity, r˙=0, and then get reflected at the CES. Integrating numerically the radial Carroll geodesic equation29 r¨ℓ2=Mb2r2+1r3-3Mr4

with the initial data r(0)=2M, r˙(0)=0 and the angular equation (13) with φ(0)=0 yields the orbits shown in Fig. 2. We also created a nMathematica notebook that displays geodesics as function of the impact parameter [100].

In conclusion, Carroll black holes act like a perfect mirror for small impact parameters, unlike their Lorentzian counterparts, which in that regime absorb everything thrown at them. The reflection happens at the locus where the geodesics kiss the CES, providing congenial context to Lévy-Leblond’s allusion to Lewis Carroll’s book “Through the Looking Glass”.

Conclusions

Working intrinsically on a Carrollian geometry, we have constructed the Carrollian analog of the geodesic action (4). Remarkably, the equations of motion (6)–(7) lead to non-trivial dynamics: the Carrollian universe admits non-trivial orbits for test particles.Table 1 Properties of Carroll geodesics for various values of b

Value of b	Key property of Carroll geodesics	
b≫2M	Small deflection angle (25) with u0=1/b	
b>2M	Finite deflection angle (26) with x=2M/b	
b≳2M	Winding orbits (27), see purple/dashed orbit in Fig. 1	
b=2M	Unstable circular orbit at CES or tangential orbit (28)	
0≤b<2M	Particle reflected by CES, see Eq. (29) and Fig. 2	
b2<0	Impossible unless r<2M	

Focusing on the Carroll–Schwarzschild background, we have seen that the radial geodesics are strikingly similar to the relativistic case, except that the energy enters the effective potential (17). This feature prevents circular orbits anywhere except at the CES, where such an orbit is unstable. We then computed the deflection angle, winding numbers, and showed that for small impact parameters b (19), the Carroll black hole acts as a perfect mirror. Table 1 summarizes properties of Carroll geodesics with various ranges of the impact parameter b, displayed in Figs. 1 and 2 (see also [100]).

There are many outlooks of this paper, such as computing geodesics on different Carroll backgrounds, revisiting our results in the first order formulation, studying the phase space near the CES, applying our results to cosmology, exploring quantum effects like the Hawking effect and evaporation, etc. In conclusion, after many years and efforts in Carrollian physics, finding non-trivial motion could open new fascinating avenues to explore, and one cannot avoid thinking of the famous expression by Galileo, “eppur si muove”.

Acknowledgements

L.C. thanks Etera Livine for stimulating him to study simple Carroll questions. D.G. thanks Céline Zwikel for hosting his research stay at Perimeter Institute, where this project started. We thank Ankit Aggarwal and Florian Ecker for comments on a nearly final version of this work. We are grateful to the organizers and participants of the kickoff event for the Simons Collaboration on Celestial Holography, where some discussions were finalized. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. This work was supported by the Austrian Science Fund (FWF), projects P 32581, P 33789, and P 36619.

Data Availability

This manuscript has associated data in a data repository http://quark.itp.tuwien.ac.at/~grumil/Carroll geodesics.nb.

Code Availability

This manuscript has associated code/software in a data repository http://quark.itp.tuwien.ac.at/~grumil/Carroll geodesics.nb.

1 It may be too early to reach this verdict since we are still in the discovery phase of Carrollian geometries and physics applications thereof. However, we believe that a change from Euclidean to a degenerate signature is more radical than a change from Euclidean to a pseudo-Euclidean signature. Note that there is a dual version of this geometric construction for Galilean spacetimes, where instead the inverse metric has signature (0,+,⋯,+) and it is supplemented by a clock 1-form associated with the time direction, see e.g. [3].

2 In lower dimensions, additional parity-odd terms can appear. In two dimensions the term e-1ϵμνx˙⊥μvν can be added to (4), and in three dimensions the term e-2ϵμνσx˙⊥μx˙⊥νvσ. The unique (parity even) term with only one derivative, e-1x˙‖, is a total derivative term and hence was dropped.

3 The constant of motion ℓ is irrelevant as long as ℓ≠0. This can be verified by rescaling the affine parameter τ by a positive constant, which effectively rescales ℓ by the inverse of this constant. Thus, only b is relevant. Whenever b is real we assume it is non-negative, without loss of generality.

4 On trivial orbits, E=ℓ=0, the azimuthal angle and the radius are fixed, φ˙=0=r˙. Thus, the only “dynamics” of such orbits consists of time flow, given by (12). While formally such orbits obey the circularity conditions (20) we refrain from referring to them as “circular” due to their triviality.

5 The functions K and F correspond to the Mathematica functions EllipticK and EllpiticF, respectively.
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