Today we will tell you about weak gravitational lensing. The reason for this was Professor Matthias Bartelmann from the University of Theoretical Physics Heidelberg, which he wrote specifically for the educational project Scholarpedia.

First, a bit of history: the idea that massive bodies can deflect light goes back to Isaac Newton. In 1704, he wrote in his book “Optics”: “... do bodies influence light at a distance and deflect its rays by this influence; and is not this influence the stronger, the smaller the distance [between the body and the ray of light]? For a long time, the very formulation of such a question was controversial, because Newtonian physics works only with bodies that have mass, and the debate about the nature of light, the properties and the presence of mass in its particles went on for another two good centuries.

Nevertheless, in 1804, the German astronomer Johann von Soldner, assuming the presence of mass in photons that had not yet been discovered by that time, was able to calculate the angle by which light from a distant source would deviate if it "strikes" on the surface of the Sun and reaches the Earth - the beam had to deviate by 0.83 arc seconds (this is approximately the size of a penny coin from a distance of 4 kilometers).

The next big step in the study of the interaction of light and gravity was made by Albert Einstein. His work on the theory of relativity replaced Newton's classical theory of gravity, where forces are present, with a geometric one. In this case, the mass of photons is no longer important - the light will be deflected simply because the space itself near the massive object is curved. Before finishing work on general relativity, Einstein calculated the angle of deflection of a beam of light passing near the Sun and got ... exactly the same 0.83 arc seconds as von Soldner a hundred years before him. Only five years later, having completed work on general relativity, Einstein realized that it was necessary to take into account not only spatial but also temporal at th component of the curvature of our four-dimensional space-time. This doubled the calculated deflection angle.

Let's try to get the same angle. Passing by a massive body, a ray of light is deflected because it moves straight, but in a curved space. From Einstein's point of view, space and time are equal, which means that the time it takes for light to reach us also changes. Therefore, the speed of light changes.

The speed of light passing through the gravitational field of the lens will depend on the gravitational potential of the lens and will be less than the speed of light in a vacuum

This does not violate any laws - the speed of light can indeed change if the light travels through some substance. That is, according to Einstein, the deflection of light by a massive object is equivalent to its passage through a certain transparent medium. Wait, this is reminiscent of the refractive index of a lens that we all learned in school!

The ratio of the two speeds of light is the refractive index familiar to us from school

Now, knowing the speed of light in the lens, you can get something that can be measured in practice - for example, the angle of deflection. To do this, you need to apply one of the fundamental postulates of nature - Fermat's principle, according to which a beam of light moves in such a way as to minimize the optical path length. Writing it in the language of mathematics, we get the integral:

The deflection angle will be equal to the integral of the gradient of the gravitational potential

It is not necessary to solve it (and it is very difficult), the main thing here is to see the deuce in front of the integral sign. This is the same deuce that Einstein appeared when taking into account the spatial and temporal about th component and which doubled the deflection angle.

To take the integral, an approximation is used (that is, a simplified and approximate calculation). For this particular case, it is more convenient to use the Born approximation, which came from quantum mechanics and was well known to Einstein:

The same Born approximation for a simplified calculation of the deflection angle

Substituting the values ​​known for the Sun into the formula above and converting radians to arcseconds, we get the desired answer

The famous expedition led by Eddington observed the solar eclipse of 1919 in Africa, and the stars that were near the solar disk during the eclipse deviated by an angle of 0.9 to 1.8 arcseconds. This was the first experimental confirmation of the general theory of relativity.

Nevertheless, neither Einstein nor his colleagues thought about the practical use of this fact. Indeed, the Sun is too bright, and deviations are noticeable only in stars near its disk. This means that the effect can be observed only during eclipses, and it does not give astronomers any new data either about the Sun or about other stars. In 1936, Czech engineer Rudi Mandl visited a scientist at Princeton and asked him to calculate the angle of deflection of a star whose light would pass next to another star (that is, any star other than the Sun). Einstein made the necessary calculations and even published an article, but in it he noted that he considered these effects to be negligible and unobservable. However, the idea was seized on by the astronomer Fritz Zwicky, who by this time was closely involved in the study of galaxies (the fact that there are other galaxies in addition to the Milky Way became known eight years before). He was the first to understand that not only a star, but also a whole galaxy and even their cluster can act as a lens. Such a gigantic mass (billions and trillions of solar masses) deflects light strongly enough to be registered, and in 1979, unfortunately, five years after Zwicky's death, the first gravitational lens was discovered - a massive galaxy that deflected the light of a distant quasar passing through it. Now, contrary to Einstein's predictions, lenses are used not at all for testing general relativity, but for a huge number of studies of the largest objects in the universe.

There are strong, weak and microlensing. The difference between them lies in the location of the source, observer and lens, as well as in the mass and shape of the lens.

Strong gravitational lensing is characteristic of systems where the light source is close to a massive and compact lens. As a result, the light, diverging from the source on different sides of the lens, bends around it, bends and reaches us in the form of several images of the same object. If the source, lens and observer (that is, we) are on the same optical axis, then several images can be seen at the same time. The Einstein cross is a classic example of strong gravitational lensing. In a more general case, the lens greatly distorts the shape of the object, making it look like an arch.

An example of strong lensing of a distant galaxy (white object) by a massive galaxy closer to us (turquoise object)

Wikimedia Commons

Weak gravitational lensing, which will be the main story in our material, is not capable of forming either a clear image or even a bright beautiful arch - the lens is too weak for this. However, the image is still deformed, and this gives scientists a very powerful tool in their hands: there are few examples of strong lensing known to us, but a weak one, for which it is enough for two large galaxies or two clusters to be at an angular distance of about one second of an arc, is quite enough for statistical study of galaxies, clusters, dark matter, relic radiation and the entire history of the universe from the Big Bang.

And finally, gravitational microlensing is a temporary increase in the brightness of a source by a lens that is on the optical axis between it and us. Usually this lens is not massive enough to form a sharp image or even an arch. However, it still focuses some of the light that would otherwise not have reached us, and this makes the distant object brighter. This method is used to search (or rather to say - random detection) of exoplanets.

Recall that in this review, following Professor Bartelmann's article, we restrict ourselves to the discussion of nominal weak lensing. It is very important that weak lensing, in contrast to strong lensing, cannot create either arches or multiple images of the same source. It cannot even increase the brightness significantly. All it can do is slightly change the shape of a distant galaxy. At first glance, this seems like a trifle - are there many effects in space that distort objects? Dust absorbs light, the expansion of the Universe shifts all wavelengths, light, reaching the Earth, is scattered in the atmosphere, and then still passes through the imperfect optics of telescopes - where can we notice that the galaxy has become a little more elongated (considering that we did not know what was it originally? However, here statistics come to the rescue - if galaxies have a preferred direction of elongation in a small area of ​​the sky, then perhaps we see them through a weak lens. Despite the fact that modern telescopes can see about 40 galaxies in a square with sides of one arcminute (this is the size of the ISS as we see it from Earth), the distortion introduced by lensing into the shape of the galaxy is so insignificant (does not exceed a few percent), that we need very large and very powerful telescopes. Such as, for example, the four eight-meter telescopes of the VLT complex in Chile, or the 3.6-meter CFHT telescope located in Hawaii. These are not just very large telescopes - they can also image a large area of ​​the sky in one shot, up to one square degree (unlike, for example, the very powerful Hubble telescope, one frame of which covers a square with a side of only 2.5 arc minutes ). To date, several surveys with an area of ​​just over 10 percent of the sky have already been published, which have provided enough data to search for weakly lensed galaxies.

Matter distribution map, reconstructed after calculations of the effects of weak gravilizing; white dots represent galaxies or clusters of galaxies

It must be said that the method of searching for gravitational lenses by the orientation of galaxies has several assumptions. For example, that galaxies in the universe are arbitrarily oriented, which is not necessarily the case - since the 1970s, astrophysicists have been arguing about whether clusters should have some sort of ordered orientation or not. Recent studies show that most likely not - even in the nearest and most massive clusters of galaxies are randomly oriented, but this question has not been finally closed. However, sometimes physics is on the side of scientists - gravitational lenses are achromatic, that is, unlike ordinary lenses, they deflect light of all colors in exactly the same way and we do not have to guess: the galaxy looks red because it is actually red, or just because all the other colors flew past our planet?

Illustration of the effects of weak gravitational lensing. On the left, the most noticeable effects are shown - the appearance of elongation. In the center and on the right - the influence of parameters of the second and third orders - displacement of the source center and triangular deformation

Matthias Bartelmann et al. 2016

Is there any practical application for this complex method? There is, and more than one thing - weak gravitational lensing helps us in studying the distribution of dark matter, as well as the large-scale structure of the Universe. The elongation of galaxies along some axis can quite accurately predict the mass of the lens and its concentration in space. Comparing the resulting theoretical mass with the mass of visible galaxies, which we can reliably determine from the data of optical and infrared telescopes, it is possible to measure the mass of dark matter and its distribution in the galaxy or cluster of galaxies that acts as a lens. For example, we already know that the halo (that is, a cloud) of dark matter around individual galaxies is somehow flatter than we thought before. Another application of lensing could be the discovery of new clusters of galaxies - there is still a debate whether several galaxies can have one dark matter halo at all, but it seems that in some cases this is indeed the case. And then this halo will serve as a lens and indicate that these galaxies are not just next to each other, but are part of a cluster, that is, a gravitationally bound system in which the movement of each of them is determined by the influence of all members of the cluster.

Galaxies are very good, but is it possible to look even further with the help of gravitational lensing - into the past, when there were no galaxies and stars yet? It turns out you can. CMB radiation - electromagnetic radiation that appeared in the universe just 400,000 years after the Big Bang - has been present in every cubic centimeter of space for the past 13.6 billion years. All this time it is spreading in different directions and carries the "imprint" of the early Universe. One of the key areas of astrophysics in recent decades has been the study of cosmic microwave background radiation in order to find inhomogeneities in it that could explain how such an inhomogeneous and disordered structure could appear from such a symmetric and anisotropic (in theory) primordial Universe, where in one place a cluster of thousands of galaxies , and in the other - emptiness for many cubic megaparsecs.

The satellites RELIKT-1, COBE, WMAP, Planck measured the homogeneity of the CMB with increasing accuracy. Now we see it in such detail that it becomes important to “cleanse” it from various noises introduced by sources that are not related to the initial distribution of matter in the Universe - for example, due to the Sunyaev-Zeldovich effect or that very weak gravitational lensing. This is the case when it is recorded in order to then be removed as accurately as possible from the cosmic microwave background radiation and continue to consider whether its distribution in the sky fits into the standard cosmological model. In addition, even the most accurate pictures of the CMB cannot tell us everything about the Universe - it's like a problem where we have only one equation in which there are several unknowns (for example, the density of baryonic matter and the spectral density of dark matter). Weak gravitational lensing, even if it does not give such accurate results now (and sometimes it does not agree well with the data of other studies - see the picture below), but this is the second independent equation that will help determine the contribution of each unknown to the general formula of the Universe.

refractive index

Experience in 1919 on the observation of the deflection of light rays in the gravitational field of the Sun. Gravity lenses

All material particles, by virtue of Newton's theory of gravitation, must be attracted to the Sun. On the other hand, from the standpoint of classical physics, light is wave, and not a particle - therefore, the equations for the propagation of a light wave in a gravitational field do not differ from the equations in its absence. As a result, light rays in classical physics do not bend in the gravitational field of the Sun. When observing stars near the solar disk, diffraction effects can be neglected, since the radius of the first Fresnel zone (see Arago–Poisson diffraction experiment) is

where is the wavelength of light, is the distance from the earth to the sun, is the radius of the sun.

Note that the equations for the propagation of a light wave are relativistic, so that the absence of deflection of rays in the Newtonian gravitational field is not the result of applying a nonrelativistic apparatus to motion at the speed of light. Indeed, if we consider relativistic particle with mass in the same gravitational field, then, according to the special theory of relativity, we have the equations of motion:

those. gravity, generally speaking, bends the trajectory of motion. The mass of the test particle is reduced, and then in the ultrarelativistic limit we get:

where is the unit vector in the velocity direction. For light , and we get the absence of curvature of the trajectory!

Such an interesting result leads to a consistent consideration of the problem of the deflection of light rays within the framework special theory of relativity. If we want to put forward a generalization of Newton's theory of gravity that does not violate the equivalence principle, we must choose one of two alternatives:

1. Neither light waves nor ultrarelativistic particles bend their path in a gravitational field (an example is special relativity);
2. Ultrarelativistic particles are deflected by the gravitational field - but the latter also deflects waves. The presence of wave deflection should mean that the gravitational field creates an effective refractive index in vacuum, due to the inhomogeneity of which the beams are bent.

In particular, if we simply add a factor to the Newtonian force of gravity, ultrarelativistic particles will begin to deviate as they fly near the Sun - however, the light described by Maxwell's equations will continue to travel in a straight line. On the one hand, this violates de Broglie's hypothesis - light, considered as a particle and as a wave, must propagate along different trajectories. On the other hand, the difference in the trajectories of a light beam and an electron accelerated almost to the speed of light can be used to distinguish the action of gravity from the action of inertial forces - in other words, the principle of equivalence is violated.

Einstein's general theory of relativity takes the second of two paths: light actually bends in a gravitational field, regardless of whether the wave or particle description is used. This result is achieved automatically, since Einstein's theory - metric theory of gravity. In other words, gravity is perceived as the curvature of space-time, and the curvature itself is determined by setting the distances between its infinitely close points:

Material points (including massless photons) in curved space-time move along trajectories of the smallest length - geodesics. It can also be shown that wave packets also move along them - thus, the wave-particle duality is not destroyed. The curvature itself is proportional to the difference between the sum of the angles of a small triangle, built from geodesic segments, from 180 degrees. Below are slices of two-dimensional spaces with constant curvature: Lobachevsky space (hyperboloid, negative curvature) and Riemann space (sphere, positive curvature).

Examples of Lobachevsky space are a saddle on a horse, as well as chips Pringles(see below).

Even the first astronomers could check the presence of deflection of rays in the gravitational field of the Sun, if the need arose. Since the competition between different theories of gravity (Newtonian, Einsteinian, Nordström's theory, etc.) only intensified at the beginning of the 20th century, the first observations of this effect date back only to 1919. This date is also due to experimental and historical circumstances. First, it is realistic to observe stars near the solar disk (i.e. during the day!) only during a total solar eclipse. Secondly, the outbreak of the First World War suspended all research.

It is interesting to note that even Henry Cavendish, based on contemporary physics, predicted the deflection of rays near the Sun. In 1801, the magnitude of this effect was calculated by Johann von Soldner (1776–1833). This is not surprising - after all, in non-relativistic mechanics, rays must deviate, like any other bodies. However, Albert Einstein, already after the creation of the special theory of relativity, carried out the same calculation, obtaining a non-zero result (1907). Only in 1915, after a deep analysis of the consequences of the equivalence principle, which led him to the formulation of the general theory of relativity, did Einstein recalculate the deflection of the rays - and it turned out to be twice about big. So, we have the following predictions of the deflection angle of various theories:

Thus, in Einstein's general theory of relativity, the deflection angle of the rays is twice the non-relativistic value. This effect leads to a shift in the apparent positions of stars near the solar disk during an eclipse. The picture below is a star light B observer A seems to come from a point B` , separated from B per angular distance on the celestial sphere.

It was this effect that Arthur Stanley Eddington (1882–1944) studied during the eclipse of 1919: photographs of the sky during a solar eclipse were compared with photographs taken at night six months earlier (then the Earth was facing the celestial sphere in exactly the same way). The observations were carried out independently at different points of the globe where a total solar eclipse was observed. The results of the experiments coincided with Einstein's predictions within 25%. Further experiments also confirmed this result.

Now the effect of deflection of rays in a gravitational field has become quite familiar in astronomy: massive clusters of galaxies create a gravitational field around them, which acts as a collecting gravitational lens. At the same time, this lens is by no means thin, so the images of galaxies behind the cluster are distorted. One light source can form after lensing einstein circle(Fig. 1), as well as several copies of the same image, for example, einstein cross(Fig. 2). Finally, fig. 3 shows in animation the structure of Einstein's circles near a black hole.

Any theory is valid if its consequences are confirmed by experience. This was the case with many well-known theories, including Einstein's GR theory. It was a timely and necessary stage in physics and was confirmed by numerous experiments. Its essential element was the representation of gravity as a curvature of space, which can be described by various metrics (the geometry of space). According to the curvature of space by stars, galaxies deflect light rays by gravity. Astronomical observations have brilliantly confirmed this geometric concept. The artificiality of general relativity is still in doubt and dissatisfaction among some physicists. It is necessary to find a physical justification for the observed phenomena and the nature of gravity in general. The author put forward a hypothesis about the nature of gravity. It is based on the study of the electrical component of the vacuum structure and further supplemented by the magnetic continuum component. In this form, the physical vacuum is a medium for the propagation of electromagnetic waves (EMW); the birth of a substance when the necessary energy is introduced into it; the medium for the formation of "allowed orbits" of electrons in atoms, the wave properties of particles, etc.

The speed of light is not constant in outer space. This is the main difference between the vacuum theory of A. Einstein's theories. Based on astronomical observations and the theory of the structure of vacuum, the following formula is proposed for the dependence of the speed of light on the acceleration of gravity:

(1)

α –1 = 137.0359895 is the reciprocal of the radiation fine structure constant;

r= 1.39876 10 –15 m is the dipole distance of the electric component of the vacuum structure;

g[m/s 2 ] – local acceleration of gravity;

Eσ = 0.77440463 [ a –1 m 3 c–3] is the specific electric polarization of the vacuum;

S= 6.25450914 10 43 [ a· s· m–4] is the deformation polarization of the vacuum.

Knowing the speed of light, measured under Earth conditions as 2.99792458(000000) 10 8 m/s, we determine the speed by formula (1) in open space With 0 = 2.997924580114694 10 8 m/s. It differs little from the earth's speed of light and is determined with an accuracy of 9 decimal places. With further refinement of the earth's speed of light, the indicated value for open space will change. From the wave theory of light by Fresnel and Huygens, it is known that the refractive index upon transition from a medium with a speed With 0 to Wednesday with speed c e equals

In our case, the angle of incidence of the beam to the normal of the surface of the Sun is equal to i 0 =90°. To estimate the amount of light deflection by the Sun, two models of light propagation can be given.

1. Model of light refraction during the transition from the "empty" half-space to the half-space with solar acceleration of gravity 273.4 m/s 2 . Naturally, this simplest model will give a deliberately incorrect result, namely: according to the reduced refractive index, the angle is determined as

13.53" (arcseconds).

2. A more accurate model must be calculated by the differential-integral method, based on the beam propagation function, in the field of increasing and decreasing according to the law 1/ R 2 gravitational potential of the Sun. Help came from a completely unexpected quarter – from seismology. In seismology, the problem of determining the course of a beam of elastic waves in the Earth from a source (earthquake, underground atomic explosion) on the surface and its exit angle up to the opposite side of the Earth has been solved. The exit angle will be the desired analogy of the Sun's deflection of the beam from the source either on a sphere that includes the Earth's orbit, or at a great distance from the Sun. In seismology, there is a simple formula for determining the angle of exit of a seismic wave through a constant beam parameter

p = [R 0 / V(R)] cos( i) = const, where:

R 0 is the radius of the Earth; V(R) is a function of elastic wave velocity as a function of distance (radius from the center of the Earth); i- exit angle.

Let's transform the seismological formula for cosmic distances and the speed of light:

Ms is the mass of the sun. R is the variable radius of the sphere in the center of which the Sun is located, determined by along a beam to a light source passing in close proximity to the Sun; 2.062648 10 5 is the conversion of angle radians to seconds.

The question arises about the constant in this formula. It can be resolved on the basis of world fundamental constants well known to science. The experimental value of the deflection angle is 1.75".

Based on this value, we determine that

const = Δ t const (MxR 2 sun / M sun R x 2) / (π 137.0359) 2 .

The number π and the reciprocal of the fine structure constant are the fundamental constants of our modern world. Number Δ t const = 1[s] is required to enter the dimension. Ratio ( MxR 2 sun / M sun R x 2) - introduced for all possible masses in the Universe and their sizes, as is customary in astronomy: to bring all masses and sizes to solar parameters.

On fig. 1 shows the dependence of the angle of deflection of a beam of light by the Sun depending on the distance to its source.

Rice. one. Dependence of the angle of deflection of a beam of light by the Sun on the distance to the source along the path passing near the Sun

We got full agreement with the exact experimental data. It is curious that when the source moves inside the sphere corresponding to the Earth's trajectory, the angle of deflection of the beam by the Sun decreases according to the graph of the figure. The prediction of this theory can be attributed to the fact that a beam of light from a source on the surface of the Sun or near it will deviate by only 1.25 ".

Schwarzschild solution:

Here Rg = 2MG / c 2 - Schwarzschild radius or gravitational radius.

Beam deflection i = 4MG / c 2 R= 1.746085", where R is the impact distance, which in our case is equal to the radius of the Sun.

Formula (1) gives: i= 1.746054". The difference is only in the 5th digit.

1. The results obtained indicate at least the consistency of the proposed concept. The formation of so-called "gravitational lenses" in space is also explained by the dependence of the speed of light on gravity.
2. In general relativity and in vacuum theory there are identical experimental confirmations.
3. General relativity is rather a geometric theory supplemented by Newton's law of gravity.
4. The theory of vacuum is based only on physical relations, which made it possible to discover gravity in the form of vacuum polarization in the presence of masses that are attracted by the vacuum structure according to Faraday's laws of induction.
5. General relativity has exhausted itself in the development of physics, the theory of vacuum has opened up the possibility of studying vacuum as a natural environment and opens the way for the progress of physics and technology related to the properties of vacuum.

In conclusion, I express my deep gratitude to the astrophysicist P.A. Tarakanov for a very useful remark regarding the variable mass in the formula for the deflection ray, where the mass of the Sun can be replaced by any other mass known to science.

Literature

1. Rykov A.V. The beginnings of full-scale physics // OIPH RAS, 2001, p. 54.
2. Savarinsky E.F., Kirnos D.P. Elements of seismology and seismometry // Gos. tech.-theor. Published, M.: 1955, p. 543.
3. Clifford M. Will. The Confrontation between General Relativity and Experiment // Preprint of Physical Reviewer (arXiv: gr-qc/ 0103036 v1 12 Mar 2001).