Gravitational measurements in higher dimensions

Proposing consistent theories of quantum gravity, such as string theory, has recently attracted lots of attention to theories with extra dimensions. In this letter we study the three famous experimental tests of Einstein's general theory of relativity in higher dimensions both in commutative and non-commutative spaces. In the context of non-commutative gravity, we consider exact solution of the Einstein's equation in higher dimensions whose source is given by a static spherically symmetric, Gaussian distribution of mass. The resulting metric describes a regular, i.e. curvature singularity free, black hole in higher dimensions. The metric smoothly interpolates between Schwarzschild geometry at large distance, and de-Sitter spacetime at short distance. We consider gravitational redshift, lensing, and time delay in each sector. We show that, compared to the four dimensional spacetime, in some cases there can be significant modifications due the presence of extra dimensions.


Introduction
After Einstein proposed his general theory of relativity in 1915, a large amount of research has been devoted to unify General Relativity (GR) and Electromagnetism as two fundamental forces of nature.
However, first proposals date back to the 1920s, through KaluzaKlein theory to unify these forces [1,2].
It was a classical unified field theory built in five dimensional spacetime. Recently, motivated by string theory as a requirement for describing a consistent theory of quantum gravity, extra dimensions have been the subject of much interest. Beside string theory, there are some other theories proposing the necessity of extra dimensions: • Large extra dimensions, mostly motivated by the ADD model, by Arkani-Hamed, Dimopoulos, and Dvali together with Antoniadis in Refs. [3][4][5] to solve the hierarchy problem in which the difference between the Standard Model interactions and GR manifests itself impressively in their dissimilar coupling strengths. While the electromagnetic, weak and strong forces differ by just 6 orders of magnitude, the gravitational interaction falls apart by further 33 orders.
• Warped extra dimensions, such as those proposed by the RandallSundrum model [6], in which our observable universe is modeled as a four dimensional hyper surface, known as the 3-brane, embedded in a five dimensional space, usually called the bulk. The novel idea of the Brane world is that all the gauge interactions, described by the Standard Model, are confined to live in the 3-brane while the gravitational interaction can spread into the fifth dimension of the space.
• Universal extra dimensions, proposed and first studied in Ref. [7], assume, at variance with the ADD and RS approaches, that all fields propagate universally in the extra dimensions.
The size and shape of extra dimensions should be related to fundamental energy scales of particle physics: the cosmological scale, the density of dark energy, the TeV electroweak scale, or the scale of ultimate unification. More likely, the extra dimensions are microscopic, in this case high-energy particle accelerators [8,9] and cosmic-ray experiments [10,11] are the only ways to detect their physical effects. The LHC experiments will have direct sensitivity to probe extra dimensions, through the production of new particles that move in the extra space. There is also a chance that, due to the existence of extra dimensions, microscopic black holes may be detected at the LHC [12,13] or in the highest energy cosmic rays [14][15][16].
On the other hand, the Einstein's work which derived gravitation from the underlying spacetime concept, was not provoked by observational facts, but was motivated on a purely theoretical basis. This theory changed fundamentally our understanding of space-time, mass, energy, and gravity. GR had some features and effects beyond Newton's theory of gravitation, such as light bending, time dilation, and gravitational redshift [17]. These effects have been verified experimentally and to this date are being tested to higher and higher accuracies. The observation of gravitational waves which recently was detected by LIGO and Virgo collaborations [18][19][20] is also another profound implication of GR. The detected signals perfectly agree with predictions based on black holes in GR up to 5σ [21].
Gravitational redshift is a very useful tool in astrophysics. It helps us to test our knowledge of the structure of those stars whose internal structures are different from the sun and other normal stars and has an important effect in satellite-based navigation systems such as global positioning system GPS [22]. Gravitational lensing occurs when light rays passing close to a massive body and confirmed by Eddington for the first time [23]. About one century after the first measurement it is still one of the major tools of cosmology [24,25], astrophysics [26,27] and astronomy [28][29][30]. Time dilation measures the amount of time elapsed between two events by observers at different distances from a gravitational mass. This phenomenon has been confirmed by PoundRebka experiment [31] and its corrections are also very important in GPS. The clocks on GPS satellites tick faster than the clocks on Earths surface, so we have to put a correction into the satellite measurements.
The other important implication motivated by string theory, in addition to the idea of extra dimensions, was the non-commutativity of space [32][33][34][35]. It has drawn a lot of interest in a wide range of areas from condensed matter physics to cosmology, high energy physics, and astrophysics [36]- [38]. The simplest non-commutativity that one can postulate is the commutation relation [x i , x j ] = iθ ij , where θij is an antisymmetric (constant) tensor of dimension (length) 2 . The parameter θ measures the amount of coordinate non-commutativity in the coordinate coherent states (CCS) approach [39,40] in which the concept of point-like particle becomes physically meaningless and must be replaced with its best approximation, i.e., a minimal width Gaussian distribution of mass. This effective approach may be considered as an improvement to semiclassical gravity and a way to understand the non-commutative effects.
Motivated by this idea, models of noncommutative geometry inspired Schwarzschild black holes were obtained in [41], which was extended to the Reisnner-Nordstrom model in [42,43], and generalized to higher dimensions in [44], and to charged black holes in higher dimensions [45][46][47]. Further, recent years witnessed a significant interest in this non-commutative approach from cosmology [48,49], holography [50][51][52], and the black hole physics [53]- [63]. However, in this letter we want to study the three mentioned tests of GR in higher dimensions both in commutative and non-commutative spaces. It would be of interest to obtain explicit expressions for the gravitational redshift, deflection of light, and time delay in spaces with extra dimensions that we think it has not been studied so far. This issue deserves further research along the lines that have already been proposed in [64].

Schwarzschild black hole in higher dimensions
The space-time around an uncharged, stationary, spherically symmetric black hole, known as Schwarzschild black hole in (d + 1) dimensions [65][66][67][68], is described by (2. 2) The mass parameter µ 0 is related to the mass of black hole with (2.4) For later convenience we use G 4 = 1 and define dimensionless variables  1) this occurs in smaller positions in higher dimensions, reflecting the fact that, the gravity is more stronger in four dimensions than higher dimensions. This fact can also be checked by noting that, in higher dimensions the g 00 curves tend more rapidly to the g 00 of flat space-time.

Gravitational effects in commutative higher dimensions
In this section we are going to obtain expressions for the three aforementioned effects of GR in the case of an extra dimensional Schwarzschild black hole as the gravitational system. In order to compare the behavior in extra dimensions with GR we perform a numerical analysis by plotting the quantities.
Redshift: When the light passes in opposite direction of gravitational field some of its energy has been wasted and it is transmitted to red-shift wavelength. In fact, around a Schwarzschild metric (2.1), there is a shift in the spectral lines of light given by [17] where ω 2 and ω 1 are the frequencies received by the observer and emitted by the source, respectively.
Let the light was emitted from radius r 1 and received at r 2 → ∞, then the redshift measured by an asymptotic observer turns out to be We have plotted the redshift factor (3.2) for different spatial dimensions in Fig. (2). Comparing the graphs confirms this statement that in higher dimensions the space-time foam has lower curvature than GR.
where r • is the closest distance to the massive object. The integration yields the following expression for bending of light caused by a Schwarzschild metric (2.1), We have plotted (3.4) for different dimensions in Fig. (3). It shows that the deflection of light in higher dimensions is weaker than GR.
where the time required for light to go from r 0 to r is t(r 0 , r) = Finding an exact expression for the time delay with general d is a tedious work, so we only derive some approximate relations in (3.5) for d = 3, 4, 5, and 6 spatial dimensions respectively, (∆t) max r 0 8αη πx 2 arccos[ in which we have used the previous dimensionless variables α, η, x and r 0 is the closest distance to the black hole that here is approximately equal to the event horizon radius. The parameters δ and σ are the orbital radius of the Earth and of the reflecting planet around the center of black hole (r 1 and r 2 in Fig. (4b)). We have plotted the excess time delays (∆t)max r0 , given by (3.7)-(3.10) in Fig. (4). tends more rapidly to zero in higher dimensions which again approves that, gravity is weaker in higher dimensions than GR.

Gravitational effects in non-commutative higher dimensions
In CCS formalism, originated in [39], the usual definition of mass density in terms of Dirac delta function in commutative space does not hold good in non-commutative space because of the position-position uncertainty relation. So, instead of the point mass, M , described by a δ-function distribution, a static, spherically symmetric, Gaussian-smeared matter source whose non-commutative scale is determined by the parameter √ θ, is defined in the following way [44,46] ρ M (r) = i.e. the particle mass M is diffused throughout a region of linear size √ θ. It is generally assumed that √ θ is closed to the Planck length, and as such it would be unaccessible both to present and future experimental observations. However, there is no problem in defining the line element and Einsteins equations with de-localized matter sources giving regular, i.e. curvature singularity free, metrics. [41,44]. This is exactly what is expected from the existence of a minimal length.
The particle-like d + 1-dimensional solution of Einstein's equation with this source is described by the metric (2.1) such that where N C refers to the non-commutative space and γ(a/b, z) is the Euler lower Gamma function For an observer at large distances, r √ θ → ∞ or √ θ r → 0, this smeared density looks like a small sphere of matter with radius about √ θ, so it assures that the metric to be Schwarzschild. In contradiction to the usual Schwarzschild black hole in GR which has a single horizon, in 3+1-dimensional non-commutative space we have different possibilities: • For η = M √ θ < 1.9 there is no horizon for (4.2) shown by red solid curve in Fig. (5a) • For η = M √ θ = 1.9 there is a degenerate horizon in x = r √ θ = 3 shown by blue curve in Fig. (5a).
where M = 1.9 √ θ is the mass of an extremal black hole and represents its final state at the end of Hawking evaporation process. As also indicated in Fig. (5b) by increasing spatial dimension d, more and more mass is needed to create an extremal black hole of a given radius. They are plotted for η = 1.9, which according to the above discussion represents an extremal black hole depicted by solid blue curve. Redshift: In the context of non-commutative geometry in CCS approach the red-shift function is obtained by evaluating (3.1) with the function (4.2), so the red-shift measured by an asymptotic observer, r 2 → ∞, is given by where in the limit √ θ r1 → 0, it leads to (3.2) for higher dimensional Schwarzschild solution. We have plotted the red-shift function calculated by (4.5) for different spatial dimensions in Fig. (6) in terms of dimensionless radial coordinate x = r1 √ θ and α = 3. As expected for far regions from the gravitational systems, all of them get to zero and there is no shift in the light wavelength. Also in contrast to GR predictions, there is a finite extremum for the red shift value which occurs at x = 3 for η = 1.9 in four dimensions [64] and may be expected from the existence of a minimal length. Comparing the plots in Fig. (6), we see this value has decreased for higher dimensional cases relative to GR red-shift. In Fig. (7a) we have plotted again the red-shift function, but the size of the extra dimensions is variable. As seen the shape of the curves will change by growing α. To better understand this point we have plotted the red-shift for α = 6.5 in Fig. (7b) and the behavior of curves clearly differ with the one for α = 3 depicted in Fig. (6).
and after integration the result is as follows where in the limit √ θ r → 0, it gives the predicted deflection as denoted by (3.4). According to the Figs. (8a,8b), the curves again have different behaviors for α = 3 and α = 6.5.

Conclusions
In this letter, we investigated well-known predictions of Einsteins general relativity in the case of higher dimensional Schwarzschild black holes. We obtained expressions for the gravitational red-shift, lensing, and time delay of light passing close these black holes in commutative space but we were unable to provide an exact expression for the time delay in non-commutative geometry. As shown in Figs. (2)- (4) for commutative sector, by increasing the dimension of spacetime the effects of gravity becomes weaker than GR which is consistent with the fact that gravitational effects propagate into the extra dimensions.
That is gravity gets diluted in the large volume of the extra dimensions [3]. So the amounts of red-shift, deflection and time delay will decrease when we study higher dimensional black holes.
On the other hand, in non-commutative geometry which was based on the CCS formalism, we observed that existence of Schwarzschild black hole with a degenerate horizon (extremal black hole) is tightly depended on the ratio of M √ θ , where √ θ is a minimal length of order the Planck scale. It has been shown in Fig. (5b) that by increasing the dimension we need more mass to generate extremal Schwarzschild black holes in non-commutative space. We have also obtained exact expressions for the gravitational red-shift and deflection of light in this geometry. In spite of GR in which the red-shift factor does not have a finite value, here we get a finite extremum value in which light might shift to the red wavelength. It could even be seen from Fig. (6) that this value will suppressed by increasing the dimension of spacetime. However, there is a different behavior when we increase the size of extra dimensions. This anomalous behavior was observed in both red-shift and deflection of light in non-commutative higher dimensional solutions.
Similar to the red-shift effect, there is a finite extremum value for the amount of the deflection of light in each dimension, essentially different from the commutative case and this value will decrease by increasing the dimension of spacetime.