More than 100 years ago, Albert Einstein came up with his general theory of relativity (also known as general relativity), which unified the traditional concept of three-dimensional space and one-dimensional time into four-dimensional spacetime.The concept of extra dimensions was born almost simultaneously with or even earlier than the general theory of relativity. The correctness and accuracy of the general theory of relativity have been verified numerous times with high accuracy since the starlight deflection experiment was confirmed in 1919. Over the past century, humans have never stopped exploring extra dimensions both theoretically and experimentally, but the existence of extra dimensions has not been confirmed experimentally. Is there any dimension beyond the four-dimensional spacetime that humans cannot perceive yet and that has yet to be confirmed? If yes, what are its shape and size?

Recently, Postdoctoral Fellow Tang Ziyu from the School of Fundamental Physics and Mathematical Sciences of Hangzhou Institute for Advanced Study (HIAS), UCAS, and his collaborators have made the latest progress in exploring extra dimensions with the black hole shadow, and the findings were published in theScience Bulletin (Volume 67, 2022, Pages 2272-2275).Based on the black hole shadow of the five-dimensional rotating black string model, the Event Horizon Telescope (EHT) observations of supermassive black holes in the center of M87 and SgrA galaxies not only rule out the possibility of infinite extra dimensions in the model, but also demonstrate the compact property of the extra dimensions and constrain the length of the compact extra dimension to2.03125-2.6 mm.

This studysupports the idea that the extra dimensions are compact from the point of view of black hole shadow observations and avoids Gregory-Laflamme (GL) instability by giving a much smaller limiting range than the results given by GL instability.

Short Communication

The length of a compact extra dimension from black hole shadow

1Extra dimension

The conception of extra dimension was first introduced by Nordström in 1914, in order to unify electromagnetism and gravity.Nowadays, it has become a common idea that besides our four-dimensional spacetime, the extra spatial dimension is a relevant notion in the framework of particle physics, gravity and cosmology. Moreover, it appears in phenomenological approaches and quantum gravity models such as string theories. Among others, the five-dimensional Kaluza–Klein (KK) theory is shown to recover both the electrodynamics and general relativity in the four-dimensional spacetime, where the extra dimension is considered to be a compact circle, and the four-dimensional electromagnetic field can be regarded as a purely geometric effect. However, the mass of the non-zero KK model is far beyond the capability of the particle collision experiment and cannot be tested in the experiment. Then, the domain wall theory was constructed with an infinite extra dimension and a bulk scalar field, where the effective potential well along the extra dimension could localize the energy density of the scalar field on a three-dimensional hypersurface, i.e., the domain wall embedded in the five-dimensional spacetime. Unlike KK theory, the particles in the Standard Model can be generated from the degradation of scalar field perturbations, and the KK model of the fermionic field can be constructed by changing the form of the scalar field potential energy without requiring extra dimensions to be compact. Nevertheless, since the fifth dimension is infinite and flat, it is difficult to fix the zero model of gravity on the domain wall, and the hierarchy problem (the large discrepancy between the Planck scale and the electroweak scale) is difficult to solve. Later, the well-known RS-I and RS-II models were proposed by Randall and Sundrum as a well-defined extra-dimensional theory, in which a warped structure was introduced to the compact/infinite extra dimension, thus successfully solving problems such as hierarchy.

2GL instability of black strings

Higher dimensional black holes have always been attractive in the context of string theories and braneworld models.Our observable world is effectively four-dimensional, while we can treat higher-dimensional black hole solutions with extra dimensions as candidates for realistic models. Gregory and Laflamme made the pioneering attempt to generalize a four-dimensional Schwarzschild black hole to a five-dimensional black string by the extension to an extra dimension. A five-dimensional black string of the same mass has a higher entropy than a five-dimensional Schwarzschild black hole with hyperspherical symmetry, provided that the length of the black string is small enough, which is thought to be a mechanism that may trigger dimensional degradation. Soon, they found that the black strings/branes were unstable under long wavelength perturbations, and then they put forward the well-known GL instability.Nevertheless, the instability can be evaded by the compactification of the extra dimensions, which is consistent with both the numerical results and the discussion of entropy.The ultimate fate of long black string/brane instability has been studied and discussed, and it remains an open question whether the extra dimensions in the black string/brane model are compact or infinite.

3Black hole shadow

Physicists used to focus the detection of extra dimensions on high-energy experiments, but now the successful detection of gravitational waves (GWs) and the black hole photos taken by the EHT give new directions for extra-dimensional detection.The features of GWs in extra-dimensional theories that can distinguish the effects of extra dimensions from those in other modified gravity theories are mainly the discrete high-frequency spectrum and shortcuts. However, the discrete high-frequency spectrum (about ≥300 GHz) is far beyond the scope of GW detectors at present, and not all GWs in extra-dimensional theories can take shortcuts, so the information of extra dimensions from current GW detection is still difficult. We want to explore the extra dimensions in the strong field regime of the black hole through the black hole photos taken by EHT in 2019 and 2022 respectively, and find a suitable way to constrain its size.

The deflection of light near a massive object due to curved spacetime has been verified since 1919, as the first observational evidence of general relativity.While in spacetime near a black hole, the gravity will be so strong that the photons can orbit the black hole. Because such photon orbits are usually unstable and light cannot escape the black hole, it appears to the observer that there is a disc-shaped shadow around the black hole, that is, the shadow of the black hole. Black hole photos provide us with information on the geometry near the event horizon of the black hole, and the research on the black hole shadow has become a new research focus in recent years.

4Constrain extra dimensions with the black hole shadowConstrain extra dimensions with the black hole shadow

In general, astrophysical black holes are rotating and uncharged, which can be described by Kerr metric in four dimensions. When an extra spatial dimension is introduced in the simplest uniform way, the constructed spacetime is still a solution to the vacuum Einstein equations of general relativity in five dimensions. We mainly studied the black hole shadow of the rotating five-dimensional uniform black string.

We first calculated the geodesic equations of massive particles.We found that in the geodesic equations of the first four dimensions, the momentum of the particle along the extra dimension always appears in the form of a sum of squares with the square of the particle mass, indicating that the momentum of the particle along the extra dimension participates as an effective mass for the particle. In particular, for massless particles such as photons, the absolute value of the momentum along the extra dimension is exactly the same as the effective mass of the first four dimensions. When the extra dimensions are not taken into account, or the particle only moves in the hypersurface of the first four dimensions, the geodesic equation retreats back to the case of four-dimensional Kerr spacetime. This means that if a particle cannot move along the extra dimension, then we cannot recognize the existence of extra dimensions via the motion of the particle or the shadow of the black hole.

Then we calculated the photon region, namely the spherical orbital region of a photon, and its stability.We found that the existence of photons moving along the extra dimension enlarges the photon region outside the event horizon of the black hole, and as the momentum of the photons moving along the extra dimension increases, the entire photon region moves outward to a larger radius of the orbit. In contrast, the photon region inside the Cauchy horizon narrows down to an orbit with a smaller radius. By studying the stability, we found that the photon region outside the event horizon is still unstable, while the photon region inside the Cauchy horizon is divided into a stable region and an unstable region, and the stable region gradually decreases until it disappears as the momentum of photon motion along the extra dimension increases.

Then we calculated the shadows of the black hole observed by observers at finite and infinite distances from the black hole respectively. The existence of extra dimensions has a similar effect on both shadows of the black hole, and as the momentum of photons moving along the extra dimensions increases, the shadow area of the black hole gradually expands while keeping its shape basically unchanged. In order to compare with observations and evaluate parameters, we calculated several observables such as radius, deformation, area, flatness, mean radius, circular deviation, axial ratio, angular diameter, etc. of the black hole shadow using the data of distance, mass and inclination angle of the supermassive black hole in the center of M87 and SgrA galaxies, and plotted density maps of various observables with the ratio of momentum to energyP_z/E₀ of photons moving along extra dimensions and the ratio of rotational acceleration to mass a/M of the black hole as variables. We found that in all parameter ranges, the circular deviation and axial ratio of the black hole shadow are within the observation range given by EHT, i.e., the circular deviation

and the axial ratio

, so we can only limit the parameters by the size of the angular diameter.

In the angular diameter density map plotted from black hole data in the center of the M87 galaxy, for fixedP_z/E₀, we can evaluate the parameter of

using the lower boundθ_d = 39μ as of the diameter of the observed bright region.However, this constraint could not be proper for an infinite extra dimension because the value ofP_z/E₀ can be arbitrary between 0 and 1.In addition, the result shows that the shadow boundary may go beyond the outer border, which indicates that the luminosity distribution produced by these photons is beyond the observed bright region and hence it contradicts the EHT observations. We propose that a compact extra dimension can be a suitable option for the particular choices ofP_z/E₀.For a compact extra dimension with lengthℓ, the momentum is limited to be box normalized

(n = 0, ±1，±2...), such that the length of the extra dimension can be related by the way

whereλ₀ is the wavelength of the photons.Then, recalling the EHT observing wavelength 1.3 mm,

given by upper boundθ_d＝45μ as and the velocityv_z/c ≤ 1 only adopt the cases withn = 0 andn = 0,1.The former case implies that the photons cannot move along the extra dimension, and requires the length of the compact extra dimension to be smaller than the wavelength of the photons. While the latter case in turn constrains the length of the extra dimension as 2.03125 mm≲ℓ≲2.6 mm, and the radius of the corresponding compact extra dimension is between 0.323283 mm and 0.41 mm. Parallel analysis is also performed on the angular diameter density map drawn from the data of the SgrA black hole. The length of the compact extra dimensional is constrained to 2.28070-2.6 mm, and the radius of the corresponding compact extra dimensional is between 0.362985 mm and 0.41 mm.

Postdoctoral Fellow Tang Ziyu from the HIAS School of Fundamental Physics and Mathematical Sciences, UCAS, and Professor Kuang Xiaomei from the Center for Gravitation and Cosmology, School of Physical Science and Technology at Yangzhou University, are co-corresponding authors of this paper.Professor Wang Bin of Yangzhou University/Shanghai Jiao Tong University and Associate Professor Qian Weiliang of the University of São Paulo also participated in the study and they are also the authors of this paper. This work was supported by theGeneral Program of the China Postdoctoral Science Foundation, the Special Program for Theoretical Physics of the National Natural Science Foundation of China, and the Natural Science Foundation of Jiangsu Province.

Source | School of Fundamental Physics and Mathematical Sciences

Editor | Jiang Xuchen