Weak ﬁeld limit for embedding gravity S. S. Kuptsova M. V. Ioffeb S. N. Manidab S. A. Pastonb aSt. Petersburg Department of Steklov Mathematical Institute of RAS

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Weak ﬁeld limit for embedding gravity

S. S. Kuptsova,∗

, M. V. Ioffeb,†

, S. N. Manidab,‡

, S. A. Pastonb,§

aSt. Petersburg Department of Steklov Mathematical Institute of RAS,

St. Petersburg, Russia

bSaint Petersburg State University, St. Petersburg, Russia

Abstract

We study a perturbation theory for embedding gravity equations in a background for

which corrections to the embedding function are linear with respect to corrections to the

ﬂat metric. The arbitrariness remaining after solving the linearized ﬁeld equations is ﬁxed

by an assumption that the solution is static in the second order. A nonlinear diﬀerential

equation is obtained, which makes it possible to ﬁnd the gravitational potential for a

spherically symmetric case if a background embedding is given. An explicit form of a

spherically symmetric background parameterized by one function of radius is proposed. It

is shown that this function can be chosen in such a way that the gravitational potential is

in a good agreement with the observed distribution of dark matter in a galactic halo.

∗E-mail: s1t2a3s4@yandex.ru

†E-mail: ioﬀe000@gmail.com

‡E-mail: s.manida@spbu.ru

§E-mail: pastonsergey@gmail.com

arXiv:2210.13272v2 [gr-qc] 30 Nov 2022

1 Introduction

Embedding gravity (also called embedding theory) [1] is a modiﬁed theory of gravity based on

the idea of our spacetime as a 4D surface in a 10D ﬂat space. From a geometric point of view,

this approach makes theory of gravity similar to the string theory. In this case, space-time

metric is considered as induced and has the form

gµν = (∂µya)(∂νyb)ηab,(1)

where ηab is a ﬂat metric of the ambient space, and ya(xµ)is an embedding function deﬁning

the shape of the surface. Here and further µ, ν, . . . = 0,...,3and a, b, . . . = 0,...,9. After the

pioneering work [1], the ideas of the embedding approach were repeatedly used in the works of

various authors to describe gravity, including the problem of its quantization, see, for example,

works [2–11].

The embedding gravity equations of motion, called Regge-Teitelboim equations, turn out

to be more general in comparison with Einstein’s equations Gµν =κTµν and have the form:

(Gµν −κTµν )ba

µν = 0,(2)

where ba

µν is a second fundamental form of a 4D surface deﬁned by an embedding function

ya(xµ). It can be seen that all "Einsteinian" (i. e. satisfying Einstein’s equations) solutions

satisfy the Regge-Teitelboim equations, but the full set of solutions is not exhausted by them.

There may be extra solutions for which Gµν 6=κTµν , and nevertheless (2) is satisﬁed. Initially,

this was seen as a disadvantage of the approach, since embedding gravity was understood as

some reformulation of GR, potentially more convenient for quantization due to the presence of

a ﬂat ambient space. Therefore, it was proposed to additionally introduce Einstein constraints

into the theory, making it truly equivalent to GR [1, 12]. However, nowadays the presence of

extra solutions in the theory can be considered as an advantage.

If we assume that just the Regge-Teitelboim equations describe the gravitational interaction,

then from the point of view of describing observations in terms of the usual GR, extra solutions

will manifest themselves as an additional contribution to the right side of Einstein’s equations.

It can be interpreted as a presence of some additional ﬁctitious embedding matter, which has

nothing to do with ordinary matter. This contribution depends solely on the gravitational

variables (which in embedding theory are ya(xµ)), and its appearance is connected only with

an attempt to reformulate a new theory in the old language. Since direct detection of dark

matter does not currently yield results [13, 14], it is of interest to try to treat the embedding

matter of the embedding theory as a dark matter (and possibly as a dark energy). Then the

eﬀects associated with dark matter and energy turn out to be purely gravitational. In this way,

observational problems that have no explanation within the framework of GR can be solved.

It is known the Regge-Teitelboim equations (2) can be rewritten as a pair of equations

Gµν =κ(Tµν +τµν ),(3)

τµν ba

µν = 0,(4)

i. e. in the form of a set of Einstein’s equations with the contribution of an energy-momentum

tensor τµν of ﬁctitious embedding matter and its equations of motion. It was ﬁrst pointed

out in [15]. It should be noted that there are also modiﬁed theories of gravity alternative to

embedding gravity, the equations of motion of which can be written in the form of Einstein’s

equations supplemented by other equations. The most famous is mimetic gravity [16, 17].

In order to obtain properties of the embedding matter generated by embedding gravity

(and, therefore, to understand whether they are similar to the properties of dark matter), it is

necessary to investigate solutions of the equations (2). In general, due to their nonlinearity, this

is too diﬃcult a mathematical problem. And so, we will limit ourselves to the most physically

interesting case of weak gravity, when the metric gµν is close to the ﬂat metric ηµν . In this case

the problem arises to choose a background value ¯ya(xµ)of the embedding function, which would

correspond to the ﬂat metric. The simplest choice in the form of ¯ya(xµ)which deﬁnes a 4D

plane in the ambient space turns out to be unsuitable, since in such a background the Regge-

Teitelboim equations (2) are not linearized [2]. To linearize the equations, it is necessary to

choose "unfolded" embedding [18] as the background, which means that the second fundamental

form of the surface is nondegenerate in some sense.

In this paper we will use unfolded embedding of the Minkowski metric, which is the product

of a timelike line yI=const (we use indexes I, K, . . . = 1,...,9;i, k, . . . = 1,2,3) on 9D

unfolded embedding ¯yI(xi)of the euclidean 3D metric, i. e.

¯ya=x0

¯yI(xi), ∂i¯yI∂k¯yI=δik.(5)

Note that with such a choice of background embedding, a nonrelativistic motion of embedding

matter is possible [19].

The purpose of this work is to study the Regge-Teitelboim equations (2) linearized in the

background of (5). We will look for a solution that corresponds to a galaxy which rotates so

slowly that eﬀect of the rotation can be neglected, i. e. a static and spherically symmetric on

average distribution of ordinary matter. At the same time, we will assume that the metric (1)

is also static (and spherically symmetric), which corresponds to the time independence of the

value τµν describing embedding matter. The resulting solution determines the dependence of

the gravitational potential on the distance to the center of the galaxy and it can be compared

with observations of the rotation curves of galaxies.

In sections 2 and 3, we obtain linearized equations and ﬁnd their solution. In section 4, the

inﬂuence of the assumption of the exact static nature of the solution on its behavior in a linear

approximation is investigated. In section 5, we study how the problem is simpliﬁed in the case

of spherical symmetry. We propose an explicit form of a spherically symmetric background

embedding in section 6. In section 7, we study the possibility of choosing this embedding in

such a way that the corresponding gravitational potential is in agreement with the observed

rotation curves of galaxies.

2 Linearization of the Regge-Teitelboim equations

Let us recall some formulas of the embedding theory. All convolutions by Latin indexes are

carried out using the ﬂat metric of the ambient space ηab. The induced metric is expressed

in terms of the embedding function by the formula (1). We will use the space-time signature

(−,+,+,+). Note that the signature is changed by changing the sign of ηab, and the induced

metric changes the sign as a consequence. The second fundamental form of a 4D surface is

expressed in terms of the covariant derivative Dµof the embedding function consistent with

the metric or through the projector Π⊥a

bon a subspace transverse to the surface (see, for

example, [12]):

µν =Dµ∂νya= Π⊥a

b∂µ∂νyb.(6)

We will mark with a line the values corresponding to the background embedding function

¯ya(5), for example ¯

µν is the second fundamental form of the background surface. We will raise

and lower the 4D indexes of values with a line using the background metric. Since it is ﬂat,

the background connection is zero, and the covariant derivative in (6) is reduced to the usual

one. The second derivatives of ¯y0are zero, as are the derivatives by x0of ¯yI, hence the nonzero

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WeakeldlimitforembeddinggravityS.S.Kuptsova;*,M.V.Ioffeb;,S.N.Manidab;,S.A.Pastonb;aSt.PetersburgDepartmentofSteklovMathematicalInstituteofRAS,St.Petersburg,RussiabSaintPetersburgStateUniversity,St.Petersburg,RussiaAbstractWestudyaperturbationtheoryforembeddinggravityequationsinabackgroundforwhi...

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Weak ﬁeld limit for embedding gravity S. S. Kuptsova M. V. Ioffeb S. N. Manidab S. A. Pastonb aSt. Petersburg Department of Steklov Mathematical Institute of RAS

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