“…Atom contains nucleus and electrons and mainly a void between them. So the matter in this sense contains almost nothing except the space. What geometry type of space is there? …”

Matter’s spacetime is [5]4D Euclidian empty container [by definition and in accordance with few rational suggestions that explain what is “[5]4D”], i.e., the “voidness” everywhere. And this fact doesn’t depend on something inside the container, including on the GR and merging of the GR with QM. It is [5]4D Euclidian inside atoms and outside them…

Cheers

Dear Sirs,

All physics is the experiment. So please propose at least the experiment confirming that the general relativity space-time continuum is contained in the above voids or the empty space between nucleus and electrons inside the matter. I am pleased to see the references too!

What is about singularities in the location of elementary particle having non zero mass? Could here be some GR effects like in case of black holes?

Spatial structure is flat within atomic length scales (10-8 cm). Note however that electron moves with relativistic velocities. Relativistic effects such as the length contraction and time dilation will apply in this case. In usual calculations of quantum field theory structure of space-time is assumed to be that of  a Minkowski space-time, where inertial frames are related by Poincare transformations. Minkowski space time is a flat 4 dimensional geometry having endowed with a specific form of inner product, which is governed by the signature (-- + + +)

An exception will occur if the atom is near the event horizon of a black hole where space-time is not flat enough due to gravitational pull exerted by the Black Hole. In such a position one sees that virtual pairs may to teared apart causing a reduction of mass-energy of a Black hole and emission of an associated Hawking radiation.

https://www.britannica.com/science/space-time#ref206

“…Spatial structure is flat within atomic length scales (10 -8 cm)…”

there cannot be a “spatial structure”, Matter’s spacetime as the possibility for material objects to exist and to change

[when every material object uninterruptedly changes its states – as positions in the 3D space and/or internal states, the last is changes of its position [i.e., the motion] in 4-th “coordinate time” 1D ctcoor dimension. Besides every/any change of every object’s state is accompanied obligatorily  by “true time” interval, what is changes of object’s position [i.e., the motion] in 5-th “true time” 1D cttrue dimension; thus every material object simultaneously moves in 4D sub-spacetime and along the true time axis with 4D and 1D speeds of light]

is an [5]4D empty container, emptiness cannot have a structure.

 But it seems as very probable that this container is filled by the dense regular 4D lattice of 4D fundamental logical elements, when particles are some disturbances in the lattice. In this case indeed motion of every material object is motion step by step, where the steps’ sizes are equal to the sizes [when all 4 FLE's "sizes" are identical] of FLE.

“…In usual calculations of quantum field theory structure of space-time is assumed to be that of  a Minkowski space-time…”

real Matter’s spacetime is a simple Euclidian, or even “Cartesian”, since all axis are mutually orthogonal, real manifold;

 when the Minkowski spacetime, which has imaginary  either temporal or spatial coordinates and where essential part of distances between points are imaginary also, isn’t, of course, the real Matter’s spacetime. It  is only convenient mathematical tool that is applicable only at solving physical tasks in strongly enough interacting systems of objects; and which isn’t applicable for systems of free objects. Though yeah, in atoms electrons and nuclei aren’t free and so this tool is applicable.

More see https://www.researchgate.net/publication/273777630_The_Informational_Conception_and_Basic_Physics

Cheers

Article The Informational Conception and Basic Physics

Yes, we suppose the space is flat. So it is Minkowski space-time. But electron is elementary particle so it is a point object having mass so we have singularity.  If the straight proton beam (light ray analogue) will be near the electron then any General relativity effects could be? How singularity problem is solved?

Anatoly, to form a Black Hole using the mass of electron, naive calculations will show that Schwarzschild radius will be 1.253 x 10-57 m. This is too small a scale to apply Einstein's gravitational equations. Quantum Gravity will take over at these scales.

Biswajoy Brahmachari, why the electron schwarzschild radius is small for application of Einstein equations? Please specify or give some reference preferentially famous book that seems I could find in Russian.

Best wishes

Well known Schwarzschild solution of Einstein's Equations for the line element near a massive object of mass M is:

ds2=( 1- 2GM / rc2 ) c2 dt2 - (1- 2GM / rc2)-1 dr2 - r2 (dθ2 + sin2 θ dφ2)

You can just read off that the line element diverges for the value of radial coordinate  rs= 2GM/c2. This quantity rs , which is known as the Schwarzschild radius, depends solely only on the mass of the object. Now put M as the mass of the elementary particle under consideration.

This is a naive solution because it ignores effects of electric charge and angular momentum.

A corrected Anatoly's last question:

Why the electron schwarzschild radius is too small for application of Einstein equations?

Perhaps the radius  is even smaller after taking into account the EM-field. What then would be  wrong with the too small radius?

Regards

“…Why the electron schwarzschild radius is too small for application of Einstein equations?…”

there is nothing wrong in the GR when a schwarzschild radius of something is infinitesimal – in the GR the Matter’s spacetime is a continuum. Just therefore in physical mainstream official journals, Net sources and in “near scientific” media there are a huge number of quite legitimated in the mainstream physical society “theories”, with “spacetime babbles” and corresponding “Multiverses”, “spacetime foams”, etc. Which are rather evident fantasies – as the GR/SR “relativistic effects”, though,  when “the Matter’s spacetime  is Minkowski space” that can be “contracted”, “dilated”, “curved” etc.

Again – all above are nothing more then a fantasy that appeared in the theories since authors didn’t understand – what are the notions/phenomena “Space”, “Time” and “Spacetime”. But on the another hand the “relativistic effects” have some “experimental confirmations” – when indeed real length’s contractions of concrete material bodies, slowing down of internal processes rates in particles and in concrete clocks in the gravity fields are observed.  That is another thing that [in contrast to  bubbles’/foams’ etc. “theories”, which are non-provable and non-testable]

authors of the SR/GR by the reason above erroneously interpreted these real changing of concrete material objects’ properties at concrete material impacts on the objects as real changes/transformations of Matter’s spacetime – seems such suggestion was “more fundamental”.

In the reality, again,  Matter’s spacetime is absolute [5]4D Euclidian empty container that has no other properties besides that Matter can be – and is -  placed and evolve inside; emptiness cannot be transformed, and, besides, the emptiness cannot really impact on concrete material bodies by some mystic ways “forming” the bodies' “geodesics in the spacetime”;   when the gravity is only fourth “usual” fundamental Nature force that is rather similar to the EM force and has the same problems with singularities  as the  EM force has.

The singularities  exist only in the theories, though, in the reality there cannot be the singularities   either at gravity or at EM interactions/fields, since objects Matter’s cannot interact on space distances lesser then the FLE’s size [see above, seems the Planck length], moreover the “sizes” of real particles, which really interact, are in many orders of magnitude are larger then the FLE size.

Cheers

Planck Length is defined as, lp=Sqrt( ħ G / c3) = 1.616 x 10-35m. At this length scale Einstein's gravity breaks down. Schwarzschild radius of an electron rs= 2Gme/c2 ≈ 10-57m is much smaller than Planck length, which is clearly the domain of quantum gravity.

“…At this [Planck length] length scale Einstein's gravity breaks down…”

Einstein’s gravity breaks down in much larger distances – because of in  the reality there is no postulated in the GR “spacetime curvature” that by some mystic way forces, as that the GR claims, for example, Earth to move around Sun.

Besides in this theory it is rather evidently incorrectly postulated that photons, again by some mystic way, when moving “along geodesics in the curved spacetime” don’t change their energies, when photons are rather banal particles, when all/every particles have inertial and gravitational masses and so obtain/loose their energies at motion in gravity fields.

Moreover, if photons don’t change their energies, then from the theory and the Pound-Rebka-Snider experiments immediately follow that the GR predicts the gravitational mass defect of a system of masses value that is twice larger then the evident really possible value.

Correspondingly to show that the GR is rather questionable theory it is enough to make rather simple and cheap experiment, see https://www.researchgate.net/publication/277710038_The_informational_model_-_gravity_a_next_experiment

All for such experiment exists yet now; and it can be made in a week – if instead of lifting a balloon in atmosphere, as  that is suggested in the link above, to make experiment in some skyscraper, spending on the experiment seems no more then a couple of $millions; when that would be indeed the experimental testing of the GR – in contrast to, say, the seeking of  evidently fantastic “ripples of the spacetime” with spending $Billions.

The experiment was suggested yet in Jun 2015, but no one physical group till now didn’t attempt to make it…

“…Planck length, which is clearly the domain of quantum gravity…”

As well as the domain of quantum gravity isn’t only in Planck length scale. However, because of the G-force is extremely weak force, the quantum gravity reveals itself only when extremely small masses are impacted by the gravity. Thus quantum effects can be detected, rather probably, for example, in experiment with photons see   - https://www.researchgate.net/publication/215526868_The_informational_model_-_possible_tests   ;  at least the section 2.1.2. “Monochromatic photon beam distortion”.

The experiment was proposed yet in 2007…

Cheers

Research The informational model – gravity; a next experiment

Article The informational model - possible tests