The great Einstein returns and jokingly tries to give a relativistic solution, in his distinctive way, to problems such as the rotation of galaxies, the measured redshift of stars and the apparent need for dark matter. He does so, as far is possible, using a humble and cheerful language, since understanding the Universe is, in Einstein’s opinion, simple and wonderful.
Some time ago a small group of theoretical physicists asked the great Albert Einstein what his considerations on the apparent need for dark matter could be, since in his time that problem was not known. The following is a summary of his answer.
“Before answering, colleagues, since in death I had a lot of time to think a bit about everything, I’ll propose you a mind game. Follow me and, at the end, let’s see if you’re going to repeat that interesting question in the same way.”
“Consider a star near the galactic edge that is currently participating in the Milky Way rotation at about 250 km/s. We know that, taking into account the total mass of ordinary galactic matter, it should orbit at no more than 50 km/s. Now suppose, just to play together our mind game, that the difference, 200 km/s, might really be due to a strange kind of space-time rotation at that speed in that region.
On the other hand, we know that space-time can be expanded, can be bent, can be twisted.. why not be spun around? Obviously I’m not talking about highly relativistic and quantum phenomena such as ergosphere or others, but just a new type of classical rotation, concerning space-time. I do know where your mind is going right now: “what about the doppler effect by which we measure the relative speed of stars, if space itself runs away?” Be patient.. and confident, because our game will offer some hidden surprises!
Only remind that until Copernicus’ time, in order to explain the motion in the sky of the five known planets, and above all the retrograde component of their motion, seen during significant periods, Ptolemy’s epicycles were used, i.e. spheres spinning around points, which were rotating around other spheres, up to different orders of complexity. Then, Copernicus hypothesized a single movement, the revolution of the Earth around the Sun, and the need for epicycles disappeared: it was discovered that there was only an apparent need for them, which lasted until we were finally willing to admit the possibility of that single unforeseen component (and until we were finally willing to renounce a hitherto undisputed certainty, namely that the Earth was the center of the Universe).”
“Now, how could we better represent a rotation involving space? Let’s take the example of a ferris wheel, a simple horizontal type, with passenger seats swinging strictly in a radial way. All of us know how passenger seats (and people sitting on) behave while the wheel spins. But now space itself is spinning, not a wheel; how could we imagine an equivalent situation?
Maybe it’s a bit as I had to run around and after that rotating wheel, attaching each passenger seat over and over again! So, let’s try to attach at least one seat to the great wheel, whilst it is spinning: since any possible swing is merely radial, we see immediately that the inertia of the seat makes it swing towards the inside of the wheel for some time, so that those people feel their own inertia behaving initially in a centripetal way. But if they were hung up again and again, this impression would be constant. Is this maybe the reason why, if space itself is spinning, our outer star does not flee away from our galaxy? Should we assume that a rotating space acquires an inertial potential whose lowest point is the centre of rotation? We will come back to that. In the meantime, let’s see what feature a supposed inertial potential should have, to join a possible and arguable space rotation.
Imagine now a horizontal spinning wheel the size of a big car wheel, but thin and made of lead; on the upper face, along one of its radii, there are a series of handles at a regular distance. If I want to stop this spinning wheel what handle will I choose to avoid dislocating my wrist? Obviously the outermost one, because, even if I know that inner ones are turning more and more slowly, I realize that the inertia of the whole area of the wheel that is outside of each handle plays against me: so I decide without a doubt. Therefore, let’s try to balance those two components, the inertia of the passenger seat or of my wrist, and the inertia of the wheel.
If we came back to the big rotating wheel, the one with the seats, with a new variant i.e. our possibility of attaching a seat to every point of any spoke of the wheel, and we wanted to achieve the objective of making each of the seats (therefore each of those people) feel the same traction towards the centre, once they are attached to any point of the radius, well, we would find out that this shared effect is impossible to reach by using a real wheel: the linear decrease of rotation speed does not balance the quadratic growth of wheel’s inertia outside the attachment point, while we move towards the centre. If we assume instead that the rotation of space-time around the centre of our galaxy is doing just that, i.e. imposing the same “traction” to each point of the galactic disk, in proportion to mass density of a neighbourhood of that point, we should conclude that space-time rotation decreases more than linearly moving towards the centre (i.e. it grows more than linearly moving towards the edge), and then it does not behave like the rotation of a material wheel!
Now, each spoke of the big wheel is behaving like a lever, showing less and less positive effect as we move towards the edge; even if it is difficult to imagine something like a leverage effect about stars, what certainly is important for us (and for the continuation of our game) is this: if we compose that strange inertial rotation curve with the one we derive (from gravitational masses in play) as the expected rotation curve, as a result this addition shows the motion of stars in the Milky Way is described, according to observational data, better than with any possible density distribution of dark matter we are able to imagine.
But what that constant “traction” we introduced might be opposed to? Let’s move on to another step of our mind game.”
“Now suppose there are eight of us, and we want to hang on to a large rotating wheel that is located in deep space, while we all are close and just over it, in zero gravity. The face of this big wheel, the one we see, has eight pivoting handles on its outer edge. Before we hang on, the wheel turns following its own inertia. Since we don’t have to unbalance it, everyone chooses and grabs his/her handle simultaneously. After a little settling, if each handle can rotate frictionless, we will see that we have subtracted spin from the big wheel by gaining reverse spin ourselves. If we are not sorted as in an inflatable dinghy, fat ones facing fat ones and skinny ones facing skinny ones, the wheel is going to get unbalanced anyway, but we’re smart enough to have thought about this before.
Now each of us rotates around the handle that he/she keeps in his/her hand. Before we were in zero gravity, in free fall, and we felt weightless, and without experiencing our own inertia; now what weight do we experience? The opposite of a weight, indeed! In fact, we feel the strength we need to use to remain attached to our handle.
If you let me, we choose to define “proper inertia” this negative quantity (this sub-zero weight) that each of us experiences. So, we are getting the new step of our mind game, wherein we suppose that the rotation of space in the Milky Way just applies that “traction” that balances its opposite, i.e. what we have just named proper inertia. (And you can see that this is exactly the contrary of what the large wheel in deep space is doing with each astronaut – to whom, by contrast, it gives, adds, proper inertia, changing his previous condition of free fall). Consequently, we will name, if you agree, “remote inertia” the positive quantity that we have termed “traction” so far. So, by revolving, the whole rest of galactic mass is adding remote inertia to each point of the galaxy, in proportion to the mass density of a neighbourhood of that point: it all is just balancing the proper inertia of that neighbourhood. By balancing and taking off proper inertia, remote inertia lets any massive body continue its gravitational free fall (without “feeling” neither its own weight nor its proper inertia while doing it).”
“Imagine now that, in lieu of our outer star that is rotating around the galactic centre, there is a candidate black hole composed as follows: its mass a few times the solar mass, and made of objects evenly spaced in a spherical volume; each object is nothing but a small primordial black hole, with the mass of Mount Everest and the size of a few nanometres. Since everyone is so small that until the end of the common gravitational collapse any collision is very improbable, we can follow the deviation of the geodesics (crossed by those bodies) during a continuous free fall [see Footnote 1]. From the remote point of view, the non-rotating centre of rotation of the galaxy, we observe the proper inertia of the candidate black hole balanced by its centripetal remote inertia due to space rotation. To obtain instead the fast-ticking point of view [see Fn 2], given that we have to imagine it neither accelerated nor rotating, we assume the equivalent of the condition in which a single astronaut (the candidate b.h.) has absorbed and self-centered all the spin of the wheel (the galaxy space between the candidate and the f.t. point itself) in the opposite direction (as we see the astronaut is doing). So, from that fast-ticking point we observe our mini black holes gravitationally converging, but on the condition of applying to the deviation of their geodesics, in the same time, a component of reverse spin (rotation whose axis is orthogonal to the galactic plane). As a result deviations in an inversely rotating frame will have the characteristic of making the single accelerations of each object or mini black hole converge towards a spheroid surface whose polar minimum radius is the Schwarzschild radius [see Fn 3] of the candidate black hole. In fact the radial centrifugal component due to inertia is always the same (per our definition of space rotation, supposed before, a feature now extended to astronaut’s body) along the candidate’s radius, and it will therefore have proportionally greater influence on the minor accelerations, those of the objects placed more centrally: so every geodesic will converge towards the shortest relative path (the one that maximizes proper time) just by lying on that spheroidal surface, where every mini b.h. approaches but never reaches the speed of light.
From the fast-ticking point pov, above that surface both combined remote inertia and the mutual proper inertias of objects are not only balanced, but minimized too, up to a level of combined proper inertia equal to the one of the whole candidate (the pivot of the handle), as measured from the remote point (the galactic centre of rotation). Interesting, isn’t it?”
“On the other hand we notice a very significant fact in relation to what we’re talking about. In the galactic bulge the maximum density of gases and the maximum forge line of new stars are reached in an area that is just outside, but very close to the galactic Schwarzschild radius, the way we are able to calculate it according to the current estimation of galactic mass. More inside, from that area to the central supermassive body, the particle density just drops off completely and in a truly inexplicable way, if not taking into account that gravity and the minimizing of the proper inertia of matter makes it converge with its maximum density towards that spheroidal surface (which is not other than the relativistic-inertial dilatation of the galactic centre of mass as was meant in the newtonian gravity theory).
Continuous oscillations, above and below (or outside and inside) that ideal surface make however some matter (gases or stellar bodies) lie further, that is more inside. While in the previous case, the stellar black hole, matter will collimate towards the outside as it regains proper inertia (it undergoes a deficit of remote inertia), beyond the galactic centre of mass inner matter must instead collimate towards the inside in order to do the same thing, and every geodesic can only end into the central supermassive b.h.: in fact in that zone space rotation is taking another sense, which we could name time-verse [see Fn 4].
On second thought I should say that this is a fortune, because it’s just the inevitable existence of this buffer zone at the centre of galaxies that did not make the Universe homogeneous at any scale, but only at scales much higher than the galactic ones. So since a very remote era wherein space has not yet had enough time to fully unfold its spins. In fact, if the spin of space had been transmitted instantly, we would not have had the central supermassive b.h.s, produced by a momentary excess of local spin (That is, we are in the situation wherein the astronaut has just grasped the handle and his/her wrist sustains excessive twisting and excessive acceleration compared to the final ones, so when the excess of torsion is passed on, that momentary surplus of acceleration – in that time before the astronaut is finally straight and stretched – doomed a part of matter to be, sooner or later, confined in the central compact body and in a time-verse rotating space [still see Fn 4]: without this phenomenon no buffer zones and therefore no relevant local lack of homogeneity).
With regard to the supermassive central b.h. itself, well we notice that therein both remote point and proper p.o.v. coincide, so: the imaginary fast-ticking point of view, to keep itself non-rotating and non-accelerated, continues to see the rotation of space in reverse, i.e. like the spin of the astronaut rather than the spin of the wheel, and our considerations are the same as in the case of the stellar mass black hole (matter is spinning at a speed close to that of light, emitting energy in the form of evenly spaced radio waves); instead, from its proper point of view, the central b.h. sees that its proper inertia is balanced by nothing, because nothing having any significant mass is space-revolving around it, i.e. along its time-verse direction of rotation [still Fn 4] (as well as its own space is not rotating around any remote point – like the big wheel in deep space, whose rotation has no pivot). As a result it can’t help but drag us into its inertial free fall, the one that makes us move away from almost all the other galaxies of the Universe.”
“At this point, let me say that – if our mind game wants to make any sense! – well we should envisage that every mass, small or large (the edge star, the mini black holes, the star-sized black hole, the super massive black hole, a planet, an asteroid, a proton etc.) possesses so much proper inertia, how much its components (come to make it a gravitational bonded object) have inversely imparted to all the rest of the matter of the Universe in the past, through the spin with which they have locally imprinted space, thus co-generating the global counter-component, or remote inertia. Therefore, the universal composition of those two quantities is zero, in every era.
If all this correctly describes the inertial aspect of mass, the homogeneous and isotropic large-scale distribution of matter in the Universe satisfies the probabilistic logic of retaining universal remote inertia as minimal, once viewed by any observer in any time: our one is simply the most probable universe we could observe. If galaxies, or galaxy clusters, fail to compose respective spins, they move away in inertial free fall, that is, each from all the others towards the point with the highest gravitational potential (that is also the one with the lowest inertial potential), because while they gather this acceleration they compensate their proper inertia (they do not feel their own inertial weight).
The perturbations of respective spins form galaxy filaments, super clusters, etc. The composition of all universal spins is zero, that is, the Universe does not rotate and has no inertia; in this sense, the relativistic description of the Universe is independent and remains independent of any need to have an absolute framework of space and time, even according to the conclusions of our mind game.”
“If so, you’ll tell me, what about the equivalence principle?
Well, we could say that we have converted it into an emergent property: any observer will always locally observe the equivalence principle, because he/she cannot avoid staying in a rotating space or in a condition of acceleration (and subjected to inertial tide); so the astronaut on the Moon who drops the hammer and the feather, will always observe the principle as reproduced, because all four bodies, the Moon, the astronaut, the hammer and the feather, undergo an interaction in the meantime, proportional to their own mass, by the remote inertia of all other galactic bodies, due to the common rotation: inertia that is exactly inverse to the universal centrifugal feature of four objects’ proper inertia. If we bring the Moon, the astronaut, the hammer and the feather between two galaxies in an equipotential condition (i.e. supposed absence of space-related spin), our satellite cannot help but accelerate in an inertial free fall, meanwhile it will gather gradually an increasing spin for a sort of inertial tide effect; so that these two components will make our experiment succeed, even if in the opposite direction: in time, astronaut hammer and feather (and also a lot of dust and so many boulders) will begin to move away from the Moon because our satellite is not structured like a galaxy (and that’s precisely because it was, inside a galaxy, born).
So, at this point, we have to conclude that inertial mass always exceeds, and in a non-microscopic way, gravitational mass, but there is no possible observer that can experience this within his/her laboratory.
All this also provides a response to the initial objection about the doppler effect: if I consider the outer star, in relation to its differential speed (200 km/s), as being carried away by space, I have to assume a point of view in which I remain behind, I make myself overcome by space; in other words, I need to keep myself accelerated. So, attributing the differential redshift to my own speed, or to the speed of the star, or to its relativistic remote-inertial mass, or to my reverse acceleration (i.e. my relative proper inertia), is, according to the equivalence principle, telling four equivalent things.”
“Thanks to our mind game, i.e. through passing from the treatment of uniformly accelerated systems to the treatment of space-rotating systems, my Relativity becomes more comprehensive (and even more funny!). In the time to come, the following four cornerstones, i.e. the consideration of space-rotating systems, the emergence of the equivalence principle, the different conception of the cosmological horizon, and the emergent property of proper time (going to be treated as an exact measure of a special metric), will support the foundations for transforming my theory in a theory of Universal Relativity.”
“Furthermore Relativity, in its essence, is looming likely to be a Machian theory [see Fn 5]; and, it had taken centuries, this is the real reason of my coming here today from beyond the grave, and of my speaking with you: I see the girl who is feeling her arms get up while she is looking at all the sky and at all its stars turning together. Do what you wish, it goes without saying, even if I suggest what the emperor Septimius Severus said on his deathbed: Laboremus! [Let’s work!]”
“Now if you want to ask me again what I think about dark matter.. well! Otherwise please offer me a drink..”
Edited by Giorgio Mistenda.
Dedicated to Albert Einstein and Roger Penrose.
First upload 2019/01/19. Releases: Patamu register #98605 2019/01/28; Patamu register #98381 2019/01/24; Patamu register #97905 2019/01/14. Italian releases: certified mail 2019/01/07; certified mail 2019/01/04.
1) In General Relativity, geodesic deviation is the approaching or receding from one another of objects while moving along a gravitational field, due to the gradient or difference in strength of the same field in different points of space. So if two objects are moving along two parallel trajectories, the presence of a tidal gravitational gradient will cause the trajectories to bend towards or away from each other, producing a relative acceleration between the two objects. See https://en.wikipedia.org/wiki/Geodesic_deviation and https://en.wikipedia.org/wiki/Tidal_force.
2) The fast-ticking observer is, in General Relativity, the inertial observer, the one not subject to any acceleration, located very far from the gravitational field, and, in our case, placed in polar position, that is, not rotating; his/her clock, as noticing the relative slowness of all other clocks, is fast-ticking, i.e. with the faster ticking; consequently, his/hers is the clock that records the longest duration relatively to an event.
3) The Schwarzschild radius is the theoretical minimum radius of the sphere within which all the mass of a body should be compressed in order to have an escape velocity on body’s surface equal to or greater than the speed of light; if the body proceeds in its collapse, the Schwarzschild radius remains the radius of the sphere that delimits the event horizon, since no information can come out from within that sphere.
4) In this case space rotation takes a “time-verse” feature. For a more detailed and formal exposition refer to the dedicated article of mine: The centre of our galaxy: a holostar.
5) For Mach’s principle of inertia see Mach’s principle.