1. Introduction

It is known that there are a number of works relating tachyons with *M*-theory [^{1}] (see also ^{Ref. 2} and references therein), including the brane and anti-brane systems [^{3}], closed-string tachyon condensation [^{4}], tachyonic instability and tachyon condensation in the *E*(8) heterotic string [^{5}], among many others. Part of the motivation of these developments emerges because it was discovered that the ground state of the bosonic string is tachyonic [^{6}] and that the spectrum in *AdS=CF T* [^{7}] can contain a tachyonic structure.

On the other hand, it is also known that the (5 + 5)-signature and the (1 + 9)-signature are common to both type IIA strings and type IIB strings. In fact, versions of *M*-theory lead to type IIA string theories in space-time of signatures (0 + 10), (1 + 9), (2 + 8), (4 + 6) and (5 + 5), and to type IIB string theories of signatures (1+9), (3+7) and (5+5) [^{8}]. It is worth mentioning that some of these theories are linked by duality transformations. So, one wonders whether tachyons may also be related to the various signatures. In particular, here we are interested to see the possible relation of tachyons with a space-time of (4 + 6)-dimensions. Part of the motivation in the (4 + 6)-signature arises from the observation that (4 + 6) = (1 + 4) + (3 + 2). This means that the world of (4 + 6)-dimensions can be splitted into a de Sitter world of (1 + 4)-dimensions and an anti-de Sitter world of (3 + 2)-dimensions. Moreover, looking the (4 + 6)-world from the perspective of (3+7)-dimensions obtained by compactifying-uncompactifying prescription such that 4 *→* 3 and 6 *→* 7, one can associate with the 3 and 7 dimensions of the (3 + 7)-world with a *S*
^{3} and *S*
^{7}, respectively, which are two of the parallelizable spheres; the other it is *S*
^{1}. As it is known these spheres are related to both Hopf maps and division algebras (see ^{Ref. 9} and references therein).

In this work, we develop a formalism that allows us to address the (4 + 6)-dimensional world *via* linearized gravity. In this case, one starts assuming the Einstein field equations with cosmological constant Λ in (4 + 6)-dimensions and develops the formalism considering a linearized metric in such equations. We note that the result is deeply related to the cosmological constant Λ ≶ 0 sign. In fact, one should remember that in (1 + 4)-dimensions, Λ is positive, while in (3 + 2)-dimensions, Λ is negative. At the level of linearized gravity, one searches for the possibility of associating these two different signs of Λ with tachyons. This leads us to propose a unified tachyonic framework in (4 + 6)-dimensions which includes these two separate cases of Λ. Moreover, we argue that our formalism may admit a possible connection with the increasing interesting proposal of duality in linearized gravity (see Refs. [^{10}-^{12}] and references therein).

In order to achieve our goal, we first introduce, in a simple context, the tachyon theory. Secondly, in a novel form we develop the de Sitter and anti-de Sitter space-times formalism, clarifying the meaning of the main constraints. Moreover, much work allows us to describe a new formalism for higher dimensional linearized gravity. Our approach is focused on the space-time signature in any dimension and in particular in (4 + 6)-dimensions.

A further motivation of our approach may emerge from the recent direct detections of gravitational waves [^{13}-^{15}]. According to this detection the upper bound of the graviton mass is *m*
_{
g
}
*≤* 1*:*3 *×* 10^{
-58
} kg [^{15}]. Since in our computations the mass and the cosmological constant are proportional, such an upper bound must also be reflected in the cosmological constant value.

Technically, this work is structured as follows. In Sec. 2, we make a simple introduction of tachyon theory. In Sec. 3, we discuss a possible formalism for the de Sitter and anti-de Sitter space-times. In Sec. 4, we develop the most general formalism of higher dimensional linearized gravity with cosmological constant. In Sec. 5, we establish a novel approach for considering the constraints that determine the de Sitter and anti-de Sitter space. In Sec. 6, we associate the concept of tachyons with higher dimensional linearized gravity. In Sec. 7, we develop linearized gravity with cosmological constant in (4 + 6)-dimensions. We add an Appedix A in attempt to further clarify the negative mass squared term-tachyon association. Finally, in Sec. 8, we make some final remarks.

2. Special relativity and the signature of the space-time

Let us start considering the well known time dilatation formula

Here, τ is the proper time, *v*
^{2} ≡ (*dx*/*dt*)^{2} + (*dy*/*dt*)^{2} + (*dz*/*dt*)^{2} is the velocity of the object and *c* denotes the speed of light. Of course, the expression (1) makes sense over the real numbers only if one assumes *v* < *c*. It is straightforward to see that (1) leads to the line element

In tensorial notation one may write (2) as

where the indices *µ, ν* take values in the set {1,2,3,4}, *x*
^{1} = *ct*, *x*
^{2} = *x*, *x*
^{3} = *y* and *x*
^{4} = *z*. Moreover,

If one now deﬁnes the linear momentum

with

Of course, *p*
^{
i
} = 0, with i ∈ {2,3,4}, in the rest frame and deﬁning E = *cp*
^{1}, the constraint (5) leads to the famous formula *E* =±*m*
_{0}
*c*
^{2}.

Let us follow similar steps, but instead of starting with the expression (1), one now assumes the formula

where *u*
^{2} = (*dw*/*dξ*)^{2} +(*dρ*/*dξ*)^{2} +(*dζ*/*dξ*)^{2}. Note that in this case one has the condition *u* > *c*. Here, in order to emphasize the differences between (1) and (6), we are using a different notation. Indeed, the notation used in (1) and (6) is introduced in order to establish an eventual connection with (4+6)-dimensions. From (6) one obtains

In tensorial notation, one may write (7) as

where *y*
^{1} = *c*λ, *y*
^{2} = *w*, *y*
^{3} = *ρ* and *y*
^{4} = *ζ*. Moreover,

If one now deﬁnes the linear momentum

with

Since, *u* > *c* one observes that in this case the constraint (10) corresponds to a tachyon system with mass

Now, for the case of ordinary matter, if one wants to quantize, one starts promoting *p*
^{
µ
} as an operator identifying

It is important to mention that here we are using a coordinate representation for *ϕ* in the sense that *ϕ*(*x*
^{
µ
} ) =< *x*
^{
µ
} |*ϕ* >.

By deﬁning the d’Alembert operator

Analogously, in the constraint (10) one promotes the momentum *P*
^{
µ
} as an operator

The last two expressions are Klein-Gordon type equations for ordinary matter and tachyonic systems, respectively. In fact, these two equations will play an important role in the analysis in Sec. 6, concerning linearized gravity with positive and negative cosmological constant.

3. Clarifying de Sitter and anti-de Sitter space-time

Let us start with the constraint

where *i* index goes from 1 to *d*. and

It is not difficult to see that the corresponding Christoffel symbols and the Riemann tensor are given by

and

respectively.

Here, the metric *g*
_{
ij
} is given by

It is worth mentioning that one can even consider a flat metric *η*
_{
ij
} = *diag*(*-*1*, -*1*, ….,* 1*,* 1), with *t*-times and *s*-space coordinates and analogue developments leads to the formulas (14)-(18).

Of course, the line element associated with the metric (18) is

which in spherical coordinates becomes

Here, one is assuming that *r*
^{2} = *x*
^{
a
}
*x*
^{
b
}
*δ*
_{
ab
} , with *a, b* running from 2 to *d*. Moreover, *d*Ω^{
d-2
} is a volume element in *d -* 2 dimensions. The expression (20) is, of course, very useful when one considers black-holes or cosmological models in the de Sitter space (or anti-de Sitter space).

In the anti-de Sitter case, instead of starting with the formula (14) one considers the constraint is

4. Linearized gravity with cosmological constant in any dimension

Although in the literature there are similar computations [^{16}], the discussion of this section seems to be new, in sense that it is extended to any background metric in higher dimensions. Usually, one starts linearized gravity by writing the metric of the space-time *g*
_{
µν
} = *g*
_{
µν
} (*x*
^{
α
} ) as

where *η*
_{
µν
} = *diag*(−1,−1,....1,1) is the Minkowski metric, with *t*-times and s-space coordinates, and *h*
_{
µν
} is a small perturbation. Therefore, the general idea is to keep only with the ﬁrst order terms in *h*
_{
µν
} , in the Einstein ﬁeld equations.

Here, we shall replace the Minkowski metric *η*
_{
μv
} by a general background metric denoted by

The inverse of *g*
_{
µν
} is

Here, (23) is the inverse metric of (22) at ﬁrst order in *h*
_{
µν
} . Also, the metric *g*
^{(0)}
_{
µν
} is used to raise and lower indices. Therefore, neglecting the terms of second order in *h*
_{
µν
} one ﬁnds that the Christoffel symbols can be written as

where *g*
^{(0)}
_{
µν
} and

Here, the symbol *D*
_{
µ
} denotes covariant derivative with respect the metric *g*
^{(0)}
_{
µν
} .

Similarly, one obtains that at ﬁrst order in *h*
_{
µν
} , the Riemann tensor becomes

which can be rewritten as

where

Then, using the definition (28), the Riemann tensor becomes

Note that in this case the covariant derivatives *D*
_{
µ
} do not commute as is the case of the ordinate partial derivatives *∂*
_{
µ
} in a Minkowski space-time background.

One can show that the term *D*
_{
α
}
*D*
_{
β
}
*h*
_{
µν
} −*D*
_{
β
}
*D*
_{
α
}
*h*
_{
µν
} leads to

Then using (29), (30) and properties of the Riemann tensor, one can rewrite *R*
_{
µναβ
} as

Multiplying (31) by *g*
^{
µν
} , as given in (23), leads to the Ricci tensor

Thus, the scalar curvature *R* = *g*
^{
µν
}
*R*
_{
µν
} becomes

Now one can use (32) and (33) in the Einstein gravitational field equations with a cosmological constant

When one sets

As it is commonly done, in linearized gravity in four dimensions, one shall deﬁne

where *d* is the dimension of the space-time. It is important to observe that in (36),

At this point, considering the(4+6)-signature (which can be splitted into a de Sitter and an anti-de Sitter space according to (4+6) = (1+4)+(3+2)) one has to set *d* = 8 since there are two constraints, one given by the de Sitter world and another from the anti-de Sitter world. Consequently, the Eq. (36) becomes

One recognizes this expression as the equation of a gravitational wave in *d* = 8.

5. Constraints in de Sitter and anti-de Sitter space

When one considers the de Sitter space, one assumes the constraint (14). However, one may notice that actually there are eight possible constraints corresponding to the two metrics

and

While for the metric

and

Now, since one has the relation

and

Observe that one may assume that

and

where

and

Here, one is assuming that (49) allows for a different radius *ρ*
_{0}. This is useful for emphasizing that

Now, using (50) and (51) one can write the line elements in

and

From (38) one obtains

So, differentiating (55) one obtains

Similarly, from (52) one gets

Hence, with the help of (56) and (57), one can rewrite (53) and (54) as

and

respectively.

Thus, one learns that the metrics associated with (58) and (59) are

and

respectively.

Using (60) and (61) one sees that according to (17) the Riemann tensors

and

The corresponding curvature scalars associated with (62) and (63) are

and

Now, let us consider the Einstein gravitational ﬁeld equation (see Eq. (34))

for *g*
^{
µν(+)
} one sees that (66) leads to

Solving (67) for Λ^{(+)} leads to

where the Eq. (64) was used. In analogous way, by considering the Einstein gravitational ﬁeld equations for

one obtains

Note that, since Λ^{
(-)
} refers to the anti-de Sitter space, (70) agrees with the fact that Λ^{
(-)
}
*<* 0.

6. The signature of the space-time in linearized gravity

In the previous section, the Einstein gravitational field equations were considered for ^{(+)} and Λ^{
(-)
} . Now, if one substitutes the Eqs. (68) and (70) into (36) one obtains

and

Here, ^{(+)}

Let us now to consider, in the context of linearized gravity, the vielbein formalism for

where

If one replaced (74) into (73), the metric

Thus, one obtains

Since one is assuming that

If one establishes the identiﬁcations

which is the expression (22) but with the signatures + or − in

Now, we shall compare the Eqs. (71) and (72) with (12) and (13), respectively. Since Λ(+) > 0 and Λ(−) < 0 one can introduces the two mass terms

and

For *d* = 4, corresponding to the observable universe, and for ordinary matter one has

This mass expression must be associated with a systems traveling lower than the light velocity (*c > v*). In the case of particles traveling faster than light velocity (*v > c*), corresponding to tachyons, one obtains

Note that since Λ^{(+)}
*>* 0 and Λ^{
(-)
}
*<* 0, both rest masses *m*
^{(+)2} and *m*
^{
(-)2
} are positives.

7. Linearized gravity in (4+6)-dimensions

The key idea in this section is to split the (4 + 6)-dimension as (4 + 6) = (1 + 4) + (3 + 2). This means that the (4 + 6)-dimensional space is splitted in two parts the de Sitter world of (1 + 4)-dimensions and anti de Sitter world of (3 + 2)- dimensions. In this direction, let us write the line elements in (3) and (7) in the form

and

respectively. One can drop the parenthesis notation in the coordinates *µ* in *µ* in

and

Here,

Let us assume that in a world of (4 + 6)-signature one has the two constraints

and

where Λ_{(+)}
*>* 0 and Λ_{
(-)
}
*<* 0 again play the role of two cosmological constants. Following similar procedure as in Sec. 5, considering the constraints (88) and (89) one can generalize the the line element (87) in the form

where

and

Using (91) and (92) one can define a background matrix *γ*
_{
AB
} , with indexes *A* and *B* running from 1 to 8 in the form

Thus, one can write the linearized metric associated with (93) as

Hence, following a analogous procedure as the presented in section 4, one obtains the equation for *h*
_{
AB
} in *d* = 4+4 = 8- dimensions,

Here, one has

One can split □^{2} in the form

where

Thus, one discovers that (95) becomes

Multiplying the last equation by

Thus, one observes that while the left hand side of (100) depends only of *x*
^{
μ
} and the right hand side depends only of *x*
^{
a
} one may introduce a constant

and

One may rewrite (101) and (102) in the for

and

According to the formalism presented in Sec. 5, one can identify the tachyonic mass in the anti-de Sitter-world by *M*
^{2} = (2/21) Λ in the (4+4)-world. Note that the effective mass *M*
^{2} can be written as *M*
^{2} = *m*
^{(+)2} + *m*
^{(−)2}. Thus, (103) and (104) can be rewritten as

and

Here, we ﬁxed the gauges

8. Final remarks

In this work we have developed a higher dimensional formalism for linearized gravity in the de Sitter or anti-de Sitter space-background which are characterized by the cosmological constants Λ^{(+)}
*>* 0 and Λ^{
(-)
}
*<* 0, respectively. Our starting point are the higher dimensional Einstein gravitational field equations and the perturbed metric ^{(+)}, Λ^{
(-)
} and the Minkowski flat metric *η*
^{(±)}
_{
μv
} . After straightforward computations and after imposing a gauge conditions for *m*
^{(+)2} can be associated with ordinary graviton which “lives” in the de Sitter space, while the massive graviton with mass *m*
^{
(-)2
} is a tachyonic graviton which “lives” in the anti-de Sitter space. We should mention that these results agree up to sign from those described by Novello and Neves [^{17}]. The origin of this difference in the signs is that although they consider a version of linearized gravity their approach refers only to four dimensions and rely in a field strength *F*
_{
μvαβ
} which is not used in our case. Here, we get a four dimensional graviton mass ^{18}] we can set *m*
_{
g
}
*~* 3*.*0 *×* 10^{
-69
} kg, while the current upper bound obtained by the detection of gravitational waves is *m*
_{
g
}
*≤* 1*.*3 *×* 10^{
-58
} kg [^{15}].

Furthermore, in the previous section, we discuss the case of the (4 + 6) signature where we identify *m*
^{(+)2} and *m*
^{
(-)2
} as a contribution to an effective mass *M*
^{2} in the unified framework of (4 + 4)-dimensions. It would be interesting for a future work to have a better understanding of the meaning of the mass *M*
^{2}. Also, it may be interesting to extent this work to a higher dimensional cosmological model with a massive graviton.

On the other hand, it is worth mentioning that our proposed formalism in (4 + 4)-dimensions may be related to the so called double field theory [^{19}]. This is a theory formulated with *x*
^{
A
} = (*x*
^{
μ
}
*, x*
^{
a
} ) coordinates corresponding to the double space *R*
^{4}
*× T*
^{4}, with *A* = 1*,* 2*, …,* 8 and *D* = 8 = 4 + 4. In this case the constant metric is given by

Moreover, the relevant group in this case is *O*(4*;* 4) which is associated with the manifold *M*
^{8}. It turns out that *M*
^{8} can be compactified in such a way that becomes the product *R*
^{4}
*× T*
^{4} of flat space and a torus. In turn the group *O*(8*,* 8) is broken into a group containing *O*(4*,* 4) *× O*(4*,* 4; *Z*). A detail formulation of this possible relation will be present elsewhere.

Finally, it is inevitable to mention that perhaps the formalism developed in this work may be eventually useful for improvements of the direct detection of gravitational waves. This is because recent observations [^{20}] established that the cosmological value has to be small and positive and that the observable universe resembles to a de Sitter universe rather than an anti de Sitter universe. Also, it will be interesting to explore a link between this work and the electromagnetic counterpart of the gravitational waves [^{21}].