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, ημν(+) denotes the flat Minkowski metric with associated signature (−1,+1,+1,+1). Usually, one says that such a signature represents a world of (1+3)-dimensions.

If one now defines the linear momentum

with m0(+)≠0 a constant, one sees that (3) implies

Of course, m0(+) plays the role of the rest mass of the object. This is because setting 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 = cλ, y ² = w, y ³ = ρ and y ⁴ = ζ. Moreover, ημν(-) denotes the flat Minkowski metric with associated signature (+1,−1,−1,−1). One says that this signature represents a world of (3+1)-dimensions.

If one now defines the linear momentum

with m0(-)≠0 a constant, one see that (9) implies

Since, u > c one observes that in this case the constraint (10) corresponds to a tachyon system with mass m0(-).

Now, for the case of ordinary matter, if one wants to quantize, one starts promoting p ^µ as an operator identifying p^μ=-i∂μ.. Thus, at the quantum level (5) becomes

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 ◻(+)2=ημν+∂μ∂ν one notes that last equation reads

Analogously, in the constraint (10) one promotes the momentum P ^µ as an operator P^μ=-i∂μ and using ημν(+)=-ημν(-), the expression (10) yields

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 ηij(+)=diag(-1,1,...,1) is the Minkowski metric and the i index goes from 1 to d. and r02 is a positive constant. The line element is given by

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 xmxnηmn(+)=-x1x1+r2, where r ² = x ^a x ^b δ _ab , with a, b running from 2 to d. Moreover, dΩ ^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 xixjηij(+)-xd+12=-r02. This constraint will play an important role in Sec. 5.

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 gμν(0). At the end we shall associate gμν(0) with the de Sitter or anti-de Sitter space. So, the analogue of (21) becomes

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 Γμν(0)λ are the Christoffel symbols associated with g ⁽⁰⁾ _µν and Σμνλ is given by

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 gμν(0) as a de Sitter (or anti-de Sitter) background one obtains

As it is commonly done, in linearized gravity in four dimensions, one shall define h¯μν=hμν-1/2gμν0. Therefore, substituting this expression for h¯μν in (35), fixing the Lorentz gauge Dνh¯μν=0 and assuming the trace h¯=0, one finally gets

where d is the dimension of the space-time. It is important to observe that in (36), ◻2=ημν(+)∂μ∂ν is now generalized to the form ◻2=g      μν(0)DμDν.

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 ημν(+) and ημν(-) mentioned in section 3. In fact, for the metric ημν(+) one has following possibilities:

and

While for the metric ημν(-) one finds

and

Now, since one has the relation ημν(+)=-ημν(-), one sees that two sets of constraints (38)-(41) and (42)-(45) are equivalents. Hence, we shall focus only on the first set of constraints (38)-(41). Let us now rewrite (39) and (40) as

and

Observe that one may assume that xμxνημν+≶0, with ημν+=(-1,1,...,1). So, since right hand side of (46) is strictly positive, by consistency this constraint must be omitted. Similarly since the right hand side of (47) is strictly negative then this constraint must also be omitted. Thus, one only needs to consider the two constraints, (38) and (41) which corresponds to polynomial equations over the reals whose set of solutions can be associated with classically algebraic varieties. When this constraints are used at the level of line element, one discovers that they can be associated with manifolds for the de Sitter and anti-de Sitter space-time. It turns out that the constraints (38) and (41) can be rewritten as

and

where

and

Here, one is assuming that (49) allows for a different radius ρ ₀. This is useful for emphasizing that r02 refers to the de Sitter world and ρ02 to the anti-de Sitter world. Note that using ρ02 the expression (49) can also be rewritten as

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 Rμναβ(+) and Rμναβ- become

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 gμν(+). Multiplying this expression by 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 gμν(-)

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 gμν(+) and gμν(-). Such equations lead us to find a relations for Λ⁽⁺⁾ and Λ ^(-) . Now, if one substitutes the Eqs. (68) and (70) into (36) one obtains

and

Here, ⁽⁺⁾◻2=g       μν±(0)DμDν.

Let us now to consider, in the context of linearized gravity, the vielbein formalism for gμν(+) and gμν(-). One introduces the vielbein field eμa and writes

where

If one replaced (74) into (73), the metric gμν± becomes

Thus, one obtains

Since one is assuming that hμa≪1 then hμahνbηab±∼0 and therefore (76) is reduced to

If one establishes the identifications gμν±(0)=bμabνbηab± and hμν±=hμabνbηab±+bμahνbηab± one obtains

which is the expression (22) but with the signatures + or − in gμν±(0) and hμν± identified.

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 x(+)μ and x-μ if one makes the convention that the index µ in x(+)μ runs from 1 to 4, while the index µ in x-μ is changed to an index a running from 5 to 8. Thus, in tensorial notation one may write (83) and (84) as

and

Here, ημν+ and ηab- denotes flat Minkowski metrics with the signatures (−1,+1,+1,+1) and (+1,−1,−1,−1), respectively. It seems evident that one can reach a unified framework by adding (85) and (86) in the form

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 ◻(+)2=gμν(+)(0)DμDν and ◻(-)2=gab(0)(-)DaDb. One shall use now the separation of variables method. For this purpose let us assume that the perturbation h¯AB=h¯AB(xμ,xa) can be splitted in the form

Thus, one discovers that (95) becomes

Multiplying the last equation by h¯(+)AEh¯        D(-)B yields

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 Λ^ such that

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=-Λ^, while the mass in the de Sitter-world by m(+)2=(2/21)Λ+Λ^. Moreover, one can also introduce an effective 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 h¯    ab(+)=0 and h¯    μν(-)=0. Thus, we have shown that from linearized gravity in (4+4)-dimensions one can derive linearized gravities in (1 + 3)-dimensions and in (3+1)-dimensions. Moreover, the modes h¯    μν(+) can be associated with a massive graviton in the de Sitter space, while the modes h¯    ab(-) can be associated with the tachyonic graviton in the anti-de Sitter space.