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,
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,
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
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
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
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
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
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
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
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
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].