Gravitational lensing is a very useful tool for astrophysics and cosmology. At first the gravitational lensing mainly was investigated on a theoretical basis in the weak gravitational field. Using the gravitational lensing, we determine the cosmological constant, the distribution of dark matter and the Hubble constant, the existence of extrasolar planets and so on.
For the last decade, gravitational lensing in the strong gravitational field has been studied eagerly. It is well known that, for the lensing by a black hole, an infinite number of Einstein rings are formed by the light rays which wind around the black hole nearly on the photon sphere, which are called relativistic Einstein rings. This is also the case for the lensing by a wormhole. In this paper, Tsukamoto and colleagues studied the Einstein ring and relativistic Einstein rings for the Schwarzschild black hole and the Ellis wormhole, the latter of which is an example of traversable wormholes of the Morris-Thorne class.
“Given the configuration of the gravitational lensing and the radii of the Einstein ring and relativistic Einstein rings, we can distinguish between a black hole and a wormhole in principle.”— Told Tsukamoto, lead author of the study
They considered the experimental situation where they know, the separation “Ds”, between the observer and the source and the separation “Dl”, between the observer and the lens. They assumed that they do not know whether the lens object is a black bole or a wormhole and do not know its parameter, i.e., the mass ‘M’ or the radius ‘a’ of the throat.
They need at least two observable quantities to determine whether the lens object is a black hole or wormhole since the lens system has one parameter in this situation. First, they observed an Einstein ring and determine the parameter for both possibilities. Second, they observed relativistic Einstein rings and tell the wormhole from the black hole. If the predicted relativistic ring angles by the black hole and by the wormhole were of similar size, they could not discern the difference. However, Eqs. (20), (25) given in paper and Fig. 1 showed that they do not confuse them.
They concluded that they can detect the relativistic Einstein rings by wormholes which have radius, a ≃ 0.5pc at a galactic center with the distance Dl = Dls = 10Mpc and which have a ≃ 10AU in our galaxy with the distance Dl = Dls = 10kpc using the most powerful modern instruments which have the resolution of 10¯2 arcsecond such as a 10-meter optical infrared telescope. They note that the corresponding black holes which have the Einstein rings of the same size are galactic supermassive black holes with 1010M and 107 M, respectively, and that the relativistic Einstein rings by these black holes are too small to measure with the current technology.
In fact, their results imply that they can distinguish between slowly rotating Kerr-Newmann black holes and the Ellis wormholes with their Einstein-ring systems. This is because the leading term of the deflection angle for the lensing by the Kerr-Newmann black holes in the weak-field regime is equal to the one for the lensing by the Schwarzschild black holes, while the black hole charge and small spin only slightly change the radii of the relativistic Einstein rings. Moreover, this also suggests that it is much more challenging to determine the charge and/or small spin of black holes than to distinguish between black holes and the Ellis wormholes.
They assumed that the observer, the lensing object and the source object are directly aligned, though such a configuration is fairly rare. In general the strong gravitational lensing effect is observed as broken-ring images which are called relativistic images. Therefore, more realistic problem is to size the relativistic images. Their result suggests that we can distinguish black holes and wormholes by using the relativistic images. To observe the relativistic images is one of the challenging works with many difficulties. Bozza et al. pointed out that relativistic images are always very faint with respect to the weak field images. The Very Large Telescope Interferometer (VLTI) has high resolution but it will not work because of this demagnifying effect.
They also assumed point-like sources, although astrophysical sources have their own size. If the source object is a galaxy, it may conceal the relativistic Einstein rings, especially, in the case that the lens object is a black hole. Testing some hypotheses of astrophysical wormholes by using the relativistic Einstein rings and the Einstein ring is left as future work.
“Tejeiro and Larranaga investigated the gravitational lensing effect of the wormhole solution obtained by connecting the Ellis solution as an interior region and the Schwarzschild solution as an exterior region. They concluded that we cannot distinguish the Schwarzschild black hole and the wormhole unless the discontinuity of the magnification curve at the boundary is observed. This does not contradict our results because their wormhole solution behaves as the Schwarzschild solution in the weak-field regime.“— told Tsukamoto, lead author of the study
Featured image: Einstein ring. Artwork of an Einstein ring, formed when two massive objects are perfectly aligned with each other as seen from Earth. Here, a black hole (centre) is between Earth and a galaxy. Light from the distant galaxy is bent around the black hole by the latter’s immense gravitational field, forming a ring of light. The phenomenon is known as gravitational lensing. The idea that light could be bent by gravity was put forward by Albert Einstein in his general theory of relativity (1915). Several examples of gravitational lenses have been discovered in recent years. Credit: Jon Lomberg / Science Photo Library
Reference: Naoki Tsukamoto, Tomohiro Harada, Kohji Yajima, “Can we distinguish between black holes and wormholes by their Einstein-ring systems?”, Phys. Rev. D 86, 104062 – Published 27 November 2012. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.86.104062
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