Colliding scalar pulses in the Einstein-Gauss-Bonnet gravity
Hisaaki Shinkai1 ,⋆ and Takashi Torii2 ,⋆⋆12

Department of Information Systems, Osaka Institute of Technology, Hirakata City, Osaka 573-0196, Japan
Department of System Design, Osaka Institute of Technology, Osaka City, Osaka 530-8568, JapanAbstract. We numerically investigated how the nonlinear dynamics depends on the dimensionality and on the higher-order curvature corrections in the form of Gauss-Bonnet(GB) terms, with a model of colliding scalar pulses in plane-symmetric space-time. Weobserved that a collision of large scalar pulses will produce a large-curvature region, ofwhich the magnitude depends on αGB . The normal corrections (αGB > 0) work for avoiding the appearance of singularity, although it is inevitable.

1 Introduction
The Einstein-Gauss-Bonnet (GB) gravity theory is one of the plausible candidates which describes
the early Universe, but so far little is known of its nonlinear dynamical behaviors. We numericallyinvestigated the dynamics in higher-dimensional space-time with and without the GB terms. (Pleaserefer the detail report [1]).
The Einstein-GB gravity is derived from string theory, with additional higher-order-curvature correction terms to GR in the form of the Lagrangian,
LGB = R2 − 4Rµν Rµν + Rµνρσ Rµνρσ ,(1)where R, Rµν , and Rµνρσ are the n-dimensional scalar curvature, the Ricci tensor, and the Riemanncurvature, respectively. This particular combination gives us several reasonable properties, as suchghost-free combinations, and a set of equations up to the second derivative in spite of the highercurvature combinations. The theory is expected to have singularity-avoidance features in the context
of gravitational collapses and/or cosmology. However, only a few studies so far have reported on
the investigation of nonlinear dynamical features in Einstein-GB gravity (e.g., numerical studies on
critical phenomena [2, 3], black hole formation in AdS [4]).
Our investigative model deals with colliding wave packets. Due to the nonlinear features of thetheory, in GR, gravitational waves interact with themselves when they pass through each other. Considering a collision of plane gravitational waves is the simplest scenario of this nonlinear interaction
problem (see Ref. [5] and references therein). In fact, Penrose [6] pointed out that the future light coneof a plane wave is distorted as it passes through another plane wave. Since these solutions assume aplane-symmetric space-time, this singularity does not have a horizon; it is a “naked" one.
Our attention to this problem focuses on the diﬀerences in the growth of curvature, especially thedependences on the dimension and the GB terms.
⋆ e-mail: hisaaki.shinkai@oit.ac.jp⋆⋆ e-mail: takashi.torii@oit.ac.jp

2 Field Equations and numerical technique
2.1 Action

The Einstein-GB action in n-dimensional space-time (M, gµν ) is described as∫]√ [ 1 ()
S =dn x −g 2 R + αGB LGB + Lmatter ,2κM

(2)where LGB is the GB term [Eq. (1)], κ2 is the n-dimensional gravitational constant, and Lmatter isthe matter Lagrangian. This action reproduces the standard n-dimensional Einstein gravity, if we set
αGB = 0. In the actual simulations, we set κ2 = 1 and change αGB as a parameter.
The action (2) gives the gravitational equation asGµν + αGB Hµν = κ2 T µν ,(3)1where Gµν = Rµν − gµν R, and2() 1Hµν = 2 RRµν − 2Rµα Rαν − 2Rαβ Rµανβ + Rµαβγ Rναβγ − gµν LGB .
2We assume massless scalar field, ψ, of which energy-momentum tensor is given by1∂µ ψ∂ν ψ − gµν (∇ψ)2 ,2which obeys the Klein-Gordon equations, □ψ = 0.
T µν

=(4)

(5)

2.2 Dual-null formulation

We use dual-null formulation for expressing space-time which has planar symmetry. The use of dualnull coordinates simplifies the treatment of horizon dynamics, enables us to approach close to largecurvature regions, and also clarifies radiation propagation in far regions. We implement our dual-nullevolution code which was used for four-dimensional GR [7] so as to follow higher-dimensional spacetime with the GB terms.
We adopt the line element+−ds2 = −2e f (x , x ) dx+ dx− + r2 (x+, x− )γi j dzi dz j ,(6)where the coordinates (x+ , x− ) are along the null propagation directions, and γi j dxi dx j is the metric of
the flat (n − 2)-dimensional space.
For writing down the field equations, we introduce the variables1,
ϑ± ≡ (n − 2)∂± r,
ν± ≡ ∂± f,π± ≡ ∂± ψ(7)rwhere ∂± ≡ ∂/∂x± , and these are the conformal factor, expansions, and inaﬃnities, respectively.
Please refer [1] for a set of equations.
Ω≡

2.3 Numerical integration scheme

The basic idea of numerical integration is as follows. We prepare our numerical integration range
as drawn in Fig. 1. We give initial data on a surface Σ0 , where x+ = x− = 0, and the two null
hypersurfaces Σ± , generated from it, where x∓ = 0 and x± > 0. Due to the dual-null decomposition,
the causal region of a grid is clear, and there are in-built accuracy checks: the integrability conditions
or consistency conditions ∂− ∂+ u = ∂+ ∂− u.

Evolution

x+x−Figure 1. Numerical grid structure. Initial data are given on null
hypersurfaces Σ± (x∓ = 0, x± > 0) and their intersection Σ0 .
Generally, the initial data have to be given as (Ω, f, ϑ± , ψ) on Σ0 , and(ν± , π± ) on Σ± . We then evolve the data u = (Ω, ϑ± , f, ν± , ψ, π± ) on aconstant-x− slice to the next.

Σ03 Numerical evolutions of the collision of scalar pulses
There are several exact solutions of the colliding plane waves, which produce curvature singularity
after their collisions. We prepare a similar situation in our code and examine such a strong curvatureeﬀect in higher-dimensional GR and in Einstein-GB gravity. We first note that in the construction ofexact solutions, the wave fronts are assumed to be a step function, while in our simulations the wave
fronts are a continuous function. We have found that we can compare the behaviors more easily when
we place colliding matter rather than colliding gravitational waves.
We put a perturbed scalar field (ψ) in the flat background on the initial surfaces and evolve it. Theinitial scalar field is set as ψ = 0 and has momentum{{√
π+ = 0π+ = a exp(−b(x+ / 2 − c)2 )√on Σ+ ,on Σ− .
(8)
π− = 0π− = a exp(−b(x− / 2 − c)2 )3.1 Evolutions in GR

The two typical evolutions are shown in Fig. 2. We plot the behaviors of the scalar field and theKretschmann scalar, I(5) , for five-dimensional GR. We set a = 0.2 and 0.4, and b = 10, c = 2 forthese plots. For small pulses [Fig. 2(a)], we see that two pulses just pass through each other and thecurvature I(5) turns back to the flat again. On the contrary, for large pulses [Fig. 2(b)], the nonlinear
curvature evolution appears after the collision of pulses. The latter behavior is similar to the exact
solutions of the plane-wave collision (see, e.g., figures in Ref. [8]). We actually find that in all blowup regions, both expansions are ϑ± < 0. In four-dimensional plane-symmetric space-time, if thecurvature blows up, then it means the appearance of a naked singularity, since there is no chance toform a horizon. However, in higher dimensions, we expect such a blow-up will be hidden in a horizon
as the expansions suggest.
3.2 Evolutions in Einstein-GB

We also evolved the same initial data by the set of evolution equations with nonzero αGB .
Figure 3(a) displays the Kretschmann scalar, I(5) (= Ri jkl Ri jkl ), for both αGB = +1 and αGB = −1
cases for the same initial data with the large-amplitude case (a√= 0.4) in Fig. 2(b). We see that thelocal peak of I(5) at the collision of two pulses (at x+ = x− = 2 2) is smaller (larger) when αGB > 0(αGB < 0) than that in GR.
In Fig. 3(b), we plot the “evolution" behavior of the Kretschmann scalar, I(5) , at the origin (x+ =−x ) where two pulses collide. At later times, we see that the curvature will diverge for all the cases(GR and Einstein-GB) due to the large amplitude of the initial pulses, but these growing behaviors are(a) small amplitude case

π+x+x−I (5)ψ

π−x+x−x−x+x−x+(b) large amplitude caseπ+ψ

π−x+x−I (5)x+x+x−x−x−x+Figure 2. Evolutions of colliding two scalar pulses in five-dimensional GR: (a) the small-amplitude case [a = 0.2in Eqs. (8)], and (b) the large-amplitude case (a = 0.4). The scalar momentum π± , scalar field ψ, and theKretschmann scalar I(5) are plotted in the (x+ , x− ) coordinates. Initial data were set at both Σ− (x+ = 0, x− > 0)and Σ+ (x+ > 0, x− = 0) and evolved. For small pulses, we see that they just cross, and space-time turns backtoward flat again, while for large pulses, we see that nonlinear curvature evolution appears after the collision ofpulses. The latter behavior is similar to the exact solutions of the plane-wave collision.

again ordered by αGB . Supposing that the curvature singularity will be formed at the final phase ofthis evolution (analogues to the plane-wave collision), then we can say that introducing the GB terms
cannot stop the formation of the singularity, but it will shift its appearance later if αGB > 0.
Figure 3(right) shows the magnitude of the Kretschmann scalar, I(n) , at the moment of the collisionof scalar pulses (at the first peak of I(n) ). We plot the cases αGB = 0, ±0.1, ±0.5, ±1.0 and thedimensions n = 4, 5, 6, and 7. We see that for n = 4, all three cases have the same magnitude, whichis consistent with the fact that the GB correction does not appear at n = 4. For larger dimensions,the magnitude becomes lower. We also find that introducing positive αGB (i.e. the normal highercurvature correction) reduces its magnitude.

4 Summary and Discussions
We investigated fully nonlinear dynamics in n-dimensional Gauss-Bonnet gravity using a model ofcolliding scalar pulses. We observed that curvature (Kretschmann scalar) evolves more mildly in thepresence of the normal GB terms (αGB > 0) and in higher-dimensional space-time. The appearanceof the singularity is inevitable in our model, but the basic feature is matched with the expected eﬀectof the cosmologists; i.e., the avoidance (or lower possibility) of the appearance of the singularity.
As is shown in other models (e.g., Refs. [1, 9]), in higher-dimensional GR, the chance of appearances of naked singularities is suppressed compared to the four-dimensional GR cases. This issuggested by the existence of many freedoms in gravity which suppresses the growth of curvature andmakes the formation of horizons less eccentric. The introduction of the GB terms seems to work forthis direction. We hope that these results will be used as a guiding principle for understanding thefundamental dynamical features of the Einstein-GB gravity.
This work was supported in part by the Grant-in-Aid for the Scientific Research Fund of the JSPS
(C) No. 25400277, and also by MEXT KAKENHI Grants No. 17H06357 and No. 17H06358.

(a) large amplitude case, Gauss-Bonnet gravity

αGB = +1I (5)αGB = −1I (5)x+x+x−x−(b)Kretschmann scalar at origin

2.5GRGB alpha= +1
GB alpha= -1

2.0

1.5

1.0

0.5

0.0
0

123456xminus

Figure 3. (a) Kretschmann scalar, I(5) , of the evolutions of colliding two scalar pulses in five-dimensional
Einstein-GB gravity with αGB = ±1. The initial data are the same with the large-amplitude
case in Fig. 2(b).
√We see that the local peak of I(5) at the collision of two pulses (at x+ = x− = 2 2) is smaller (larger) when
αGB > 1 (αGB < 1). (b) Kretschmann scalar, I(5) , at the origin (x+ = x− ), of these evolutions together with onewith αGB = 0√(i.e. GR). (Right) The Kretschmann scalar, I(n) , at the moment of the collision of scalar pulses (atx+ = x− = 2 2). We plot for the models with αGB = 0, ±0.1, ±0.5, ±1.0 and for the dimensions n = 4, 5, 6, 7.
For larger dimensions, the magnitude becomes lower in GR. We also find that introducing positive αGB (i.e. thenormal higher-curvature correction) reduces its magnitude.

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