Philosophia Naturalis - Questions on the notion of a “warped space”
by Paolo Pasqualucci
What is “warped” by the Sun’s gravitational field, is it space itself or “the physical objects which define space”?
1. Eddington’s denial of the distinction as “metaphysical”. I know such a question is considered abstruse and therefore uninteresting by the majority of Physicists, dismissed as merely philosophical, in the sense of the old, discredited “metaphysics”. Eddington himself, in replying to his critics, simply discarded it.
“These experimental proofs, that space in the gravitational field of the sun is non-Euclidean or curved, have appeared puzzling to those unfamiliar with the theory. It is pointed out that the experiments show that physical objects or loci are “warped” in the sun’s fields; but it is suggested that there is nothing to show that the space in which they exist is warped. The answer is that it does not seem possible to draw any distinction between the warping of physical space and the warping of physical objects which define space. If our purpose were merely to call attention to these phenomena of the gravitational field as curiosities, it would, no doubt, be preferable to avoid using words which are liable to be misconstrued. But if we wish to arrive at an understanding of the conditions of the gravitational field, we cannot throw over the vocabulary appropriate for that purpose, merely because there may be some who insist on investing the words with a metaphysical meaning which is clearly inappropriate to the discussion”.
In his reply, Eddington assumes, as if it were obvious, that it is impossible to draw any distinction between space and the “physical objects” that occupy it or “that define it”. In other words: void does not exist, therefore the warping of spacially existing physical objects is the warping of the space they “define” (a term perhaps not overwhelmingly clear, though traditional).
Eddington seems to share the opinion of all those who have denied any distinction between space and what occupies it, whether moving in it or not (from Aristotle to Einstein, via Descartes, Leibniz and others). But Newton, if I am not mistaken, shared the opposite view and his opinion we surely cannot qualify nor dismiss as “metaphysical”. I mean, when he thought of ether as a medium existing other than in bodies also “in the void celestial space” extended among the sun, the stars, the planets, the comets: “Qu. 21. Is not this Medium much rarer within the dense Bodies of the Sun, Stars, Planets and Comets, than in the empty celestial Space between them?”
The effective “shape of space” (to use the terminology of philosopher Graham Nerlich) seems to have its importance for an exact knowledge of the physical world.
2. Death of the “closed universe” hypothesis? Briefly resuming for the larger public the theory of “dark matter”, the astrophysicist, prof. Russell Stannard, wrote: “Most of the matter we see around us today did not, in fact, originate at the instant of the Big Bang; it was created a fraction of a second later during inflation. Moreover, the amount produced is such that the overall density should end up with exactly the critical value. The agreement between this requirement and the experimentally measured value for the mean density of the universe, provides powerful evidence in favour of the inflation theory. So what does this mean in terms of the size of the universe? Because the density is critical - and does not exceed that value – three-dimensional space does not curve back on itself. So the closed universe hypothesis – attractive though it might be – is dead […] This means that we are left with the answer that the universe is infinite. But what kind of answer is that? What do we actually mean by saying it is infinite?”
If the “closed universe hypothesis” is to be considered “dead”, what would be the consequences for the prevailing Image of the World? None, it seems: the “closed universe hypothesis”, that shapes the universe as the continuous, unlimited “curved space” of a spheroid, is still taught in the Universities and seems to be still predominant with the larger public . Yet, we know that the tridimensional, infinite nature of the Euclidean “flat” space was never effectively eliminated from physical theory: it is still alive as the only “shape of space” suitable for quantum mechanics. In his recent best-selling booklet, prof. Carlo Rovelli has reminded us of the ambivalent vision of space still haunting contemporary Physics. “In the morning, when lectured on general relativity, the students are taught that the world is a curved space where everything is continuous; in the afternoon, when lectured on quantum mechanics, they are confronted with a flat space in which quanta of energy jump all over”. The overcoming of this “schizofrenic” dualism, he points out, may succeed only if a new vision of the physical world appears, capable of unifying quanta and gravity in a new concept of gravity (quantum gravity), so far not yet attained.
Given this unsatisfactory situation, I dare say that reasonable questions on the effective “shape of space” are legitimate, even on the part of the non-specialist.
In the Oxford Dictionary of Astronomy we read that Eddington “obtained observational proof that gravity bends light, as predicted by the general theory of relativity, when he measured slight apparent changes in the position of stars seen near the sun during the total solar eclipse of 1919; the accuracy of his results has since been questioned, but their announcement influenced the acceptance of general relativity.” Eddington’s accuracy should perhaps be questioned again but we know that subsequent and repeated “astrometric” measurements of cosmic radiowaves using very sophisticated instruments (Hubble Space Telescope, ESA Hipparcus Satellite), have confirmed the apparent changes. “Even for stars in line with the Sun, the shift in apparent position is less than two seconds of arc, or a few ten-thousands of a degree.”
In this short paper, I will try to pose some questions on the “shape of space” problem, with special regard to the revolutionary notion of a “warped space”, key-notion to Einstein’s theory of gravitation.
A. What might be the consequences, as far as the notion of space is concerned, of the fact that light coming from the Hyades cluster to skim the Sun and reach our retinas here on the Earth, seems to travel always on a straight line, as if it were always traveling in a vacuum?
B. The light beams deflected by the warping of space around the Sun originate from stars situated in a cluster calculated to be at 153 light years from the Earth, quite a remarkable distance. If this whole distance is covered by light always traveling on a straight line (a straight line that is also maintained in its course from the Sun to the Earth, after the slight deflection caused by the Suns’s gravitational field), can we assume that space is in itself flat (“euclidean”) and subject to a “warping” only in the proximity of celestial bodies, under the action of their gravitational field? That is, that it is “warped” at the level of the small scale only?
3. Light always traveling on a straight line, as if it were always rushing ahead in a smooth and transparent medium. Explaining in a drawing the meaning of the famous Isle of Prince observations of 29 May 1919, Eddington wrote: “The main part of the bending of the ray [of light] occurs as it passes the sun S; and the initial course PQ and the final course FE are practically straight.” The “initial course” PQ is the course of light from its source, located in the Hyades cluster, to the Sun; the “final course” FE is the distance from the Sun to the Earth. Given the “deflection” suffered in the proximity of the Sun by the beam of light coming from the Hyades, the position of some of these stars, as seen from the Earth, must appear slightly different than it normally does, as confronted with background stars of the same cluster. This apparent change in the position of the stars is caused by a deviation or deflection from a course of light that must necessarily be straight, given the fact that we become aware of it only when a total solar eclipse reveals that the light beams are running on a tangent aligning the Earth, the Moon and the Sun; and straight must also be the course of the slightly deflected light from the perifery of the Sun to us, otherwise we would not be able to see it as a luminous point on that same tangent.
As said above, the Hyades cluster is calculated to be at 153 light years from the solar system. In one year, as we know, light covers 9,463 billion of km.  9,463 times 153 = 1, 447,839. This means : one million 447,839 thousand billions of km, if I am not mistaken. This enormous distance is the “initial course” mentioned by Eddington, while the last jump of “only” 150 million km, from the Sun to us, is its “final course”.
Eddington’s last statement is supported by the visual evidence offered to us by the geometry of lines and solids belonging to reality outside us. Such an evidence seems to be a multiple one. Let’s consider some other examples.
i. Personal experience. During Spring 2007, on the 29th of March, I witnessed an extraordinary astronomical event. A lunar eclipse took place exactly when, by a rare coincidence, Saturn, the Moon and the Earth were all aligned on the same straight line: the Earth and Saturn were simultaneously on the tangent of the Moon. The night was clear, my wife Sandra Anne and I, alerted by the Media, looking from the tarmac in front of our bungalow, located in the South-East of Ireland, were able to see with our naked eye a minuscule but very bright point that appeared to be attached to the small crescent of sun light left by the umbra projected by the Earth on the Moon. That implied, if I am correct, that the sunbeams exposing Saturn had travelled in a straight line for about 1,434 billion km, which is Saturn’s mean distance from the Sun. This experience confirms, in my opinion, that light travels in a straight line and practically so in the whole solar system, not only within the limited space between the Sun and the Earth.
ii. Einstein’s Rings. We could say, I think, that the same occurs in the case of the so called Einstein’s rings. In this case, the gravitational lensing effect is produced, as we know, when a galaxy “bends the light emanating from a galaxy that is directly behind it”, creating a “ring of light warped by the gravitational pull of the galaxy on the forefront.” We can observe this phenomenon only when there is “ exact alignment of the source [the galaxy behind], lens [the galaxy upfront] & observer [the Earth or the Hubble Space Telescope].
Exact alignment means a display of objects all placed exactly on the same straight line. But how far are these “rings” from one another and from the Earth? “The thin blue bull’s-eye pattern in these eight Hubble Space Telescope images appears like neon signs floating over reddish-white blobs. The blobs are giant elliptical galaxies roughly 2 to 4 billion light-years away. The bull’s-eye patterns are created as the light from galaxies twice as far away is distorted into circular shapes by the gravity of the giant elliptical galaxies.”
Two to four billion light-years away from the Earth but twice as far away from one another, that is: 4 to 8 billion light-years away from one another. Therefore, the galaxy providing the “source” can be 10 to 12 billion light-years away from the Earth. Indeed, the first Einstein-ring to be discovered was “10 billion light-years away from Earth (or a redshift of z = 1.849).” This means that, in this case, the galaxy acting here as “source” might be 14 to 18 billion light-years away from the Earth. This implies, in my opinion, that we have solid evidence of the fact that light can travel on a straight line to our retinas all along these monstrous distances: two to four to ten or more billion light-years for the image of the “Einstein’s ring”!
iii. Light never scattering, space run across appearing always transparent and smooth. The man-on-the-street can’t but marvel at all this. How is it possible that light can travel such distances (in our case, around 1/3 of the alledged radius of the visible Universe) always maintaining the same velocity and a rectilinear path, always immediately resumed once it is (slightly) deflected by the gravitational field of a massive celestial body? How is it possible that light never seems to scatter, during such an immense intergalactic voyage? If space is “bent”, as a result of the density of matter everywhere superior to 0, a density furthermore “warped” by the gravitational fields of massive stellar bodies, how could the straightlinear course of light not be significantly affected by this all pervading, bent and locally warped density? And yet light does not seem to be affected at all, in its cosmic traveling: when deflected, it resumes immediately its straight thrust forward. This peculiarity is best understood from the standpoint of quantum physics. Prof. Steven Weinberg explains it this way:
“An ordinary light wave contains a huge number of photons traveling along together, but if we were to measure the energy carried by the train of waves very precisely, we would find that it is always some multiple of a definite quantity, which we identify as the energy of a single photon. As we shall see, photon energies are generally quite small, so that for most practical purposes it appears as if an electromagnetic wave could have any energy whatever. However, the interaction of radiation with atoms or atomic nuclei usually take place one photon at a time, and in studying such processes it is necessary to adopt a photon rather than a wave description. Photons have zero mass and zero electrical charge, but they are real nonetheless – each one carries a definite energy and momentum, and even has a definite spin around its direction of motion.
What happens to an individual photon as it travels along through the universe? Not much, as far as the present universe is concerned. The light from objects some 10,000 million light years away seems to reach us perfectly well. Thus whatever matter may be present in intergalactic space must be sufficiently transparent so that photons can travel for an appreciable fraction of the age of the universe without being scattered or absorbed.”
There is, then, a precise relationship between the energy of the single photon and the energy of the wave in which it is traveling. Indeed, Einstein, applying the quantum theory, has formulated the correct explanation, according to which, “the energy of any photon is inversely proportional to the wavelenght”. Therefore, light bolts forward in a continuous straight path and never gets lost along the road, so to say: “never scatters nor is absorbed.” There seems to be an experimental confirmation to this from a relatively recent research on the possibility of diffraction of light in space.
“In a crystal, the rows and columns of atoms create countless apertures. Sending waves of a comparable wavelength through these gaps makes a diffraction pattern that can be measured, so we can work out the structure of the crystal. Observing how light scatters can also reveal the structure of empty space. If space is perfectly smooth, it won’t scatter light. If it is constructed from minuscule building blocks, as many physicists suspect, it should scatter different wavelengths of light by different amounts – albeit tiny amounts, since the structure is much smaller than any observable wavelenghts, blurring out the effects of the scattering. Last year  scientists using NASA’s Fermi Gamma-ray Space Telescope observed three photons with different wavelenghts arriving at Earth simultaneously. The photons emanated from a gamma-ray burst seven billion years ago – enough time to accumulate a noticeable difference in arrival times if one wavelength takes a slightly more wiggly path than another due to scattering. The observed simultaneity puts a limit on the size of any fundamental unit of space, if they exist at all.” 
So, no “wiggly path” for the three photons which traveled in a parallel route for 7 billion years! The “observed simultaneity” of their arrival makes the size of any possible “minuscule building block” so minuscule as to disappear completely from space. A path that does not “wiggle”, is generally considered a straight one. The accepted postulate on light dictates that light travels in a straight line in a vacuum. Since it seems evident that, in the above quoted experiences and experiment, light is effectively traveling in a straight line, maybe we have here reliable evidence of the fact that light has effectively traveled in a tridimensional vacuum, i.e. in an Euclidean space?
The result of the experiment commented by dr. Evans makes it clear that space should be considered “smooth”, an adjective which implies, in my opinion, the notion of a “flat”, Euclidean space . On the other hand, as underlined by professor Weinberg, if space is full of matter, whatever its status, all this matter must be sufficiently “transparent” as to let light pass through undisturbed in its (rectilinear) motion. Therefore: whether full of matter or empty, space shouldn’t be always considered “smooth”, i.e. Euclidean, as demonstrated by the continuous, rectilinear proceeding of light from regions as far away as billions of years and light years?
Space that to us appears empty is in itself smooth and the matter it contains must be transparent to light traveling in it, no matter how great the distances run across. This image of space seemingly applies to the whole cosmic space. But how do we relate it to the notion of a space warped by the masses of the stars, the space that deflects light?
4. Interstellar space not shaped by masses. Alfred Einstein and Leopold Infeld have written, in their classic Development of Modern Physics: “Our world is not Euclidean. The geometrical nature of our world is shaped by masses and their velocity. The gravitational equations of the general relativity theory try to disclose the geometrical properties of our world.”
Which world is meant, here, by the two Authors? The solar system or the whole universe? We know that for Einstein the whole space (space-time) is a field of electromagnetic “geodetics”, bent in a spheroidal continuum of matter and energy, irregularly shaped by the cosmic gravitational fields of the celestial “masses” in perpetual motion. Therefore, the “warping of space” in the gravitational field or heliosphere of a star like the Sun, is the “warping” of a “fabric” that is already curved in itself.
If the heliosphere made of hot, ionized, gaseous fluid called plasma by Physicists identifies the space warped by the gravitational field of the Sun, it is perfectly logical to consider this space as embedded, so to say, in a continuum made of an infinite number of plasmata, emanating by the infinite number of stars that populate the universe. Prof. Ester Antonucci, a renown specialist in Sun studies, stated, as if it were not only a common opinion but also an established truth, that: “The universe is mainly made of plasmata, like our solar system, where the prevailing matter is plasma, at least because almost all the mass is concentrated in its center of gravity, in the Sun”. This is like saying that the universe is mainly made of the warped spaces surrounding the masses of the stellar bodies.
The heliosphere is so defined by the Oxford Dictionary of Astronomy: “ The region of space around the Sun which the solar wind flows. The heliosphere is thought to be about 100 AU in radius, and is bounded by the heliopause, beyond which interstellar gas exerts an equal pressure from outside. The shape of the heliosphere is unknown, but if there is a flow of interstellar material around it from a particular direction (an interstellar wind), the heliosphere may be like the Earth’s magnetosphere: spherical on one side, but drawn out into a long tail on the other.”
Following Einstein’s postulate quoted above, shouldn’t the gaseous pressure from outerspace be caused too by a “plasma” created in its own turn by the mass of a star? Now, the nearest mass to our solar system belongs to the star Proxima Centauri or Alpha Centauri C, situated at about 4.2465 light years from us, a distance deemed to be the average distance among the stars inhabiting the disc of our Galaxy. If the heliosphere radius is about 100 AU, this means that it flows until about 15 billion km from the Sun. Therefore its border, ignoring the heliopause, is situated at approximately 37,837 billion km from Proxima Centauri : 4 light years = 37,852 billion km minus 15 billion km = 37,837 billion km.
But we know that Proxima Centaury is a red dwarf which has a mass 12.5% of the Sun’s mass while its actual diameter is about one-seventh (14%) of the diameter of the Sun. To reach us, its light has to cover a distance 300,000 times longer that the actual distance between our Earth and the Sun. The trip requires four years. Given its smaller size, the radius of its plasmosphere (if I may so say) should be considerably less extended than the radius of the Sun’s plasmosphere (heliosphere). Indeed, Proxima Centauri “has two confirmed exoplanets: Proxima Centauri b & c. Proxima Centauri b orbits the star at a distance of roughly 0.05 AU (7.5 million km)” while Proxima Centauri c “orbits roughly 1.5 AU (220 million km) away”.
5. A massive, empty interstellar space between any two stars? Therefore, can we admit, at this point, that between the thin plasmosphere of the masses of these two stars there is a massive interstellar gap of about 38,000 billion km, a space totally deprived of any stellar mass?
Proxima Centauri is the nearest star to the Sun, but in the vast space between the two there is no mass whatsoever to “warp” it with its own gravitational fields. Clouds of gases, stellar dust, waves and rays of energy in its various forms, all sorts of cosmic débris seemingly flow and float in it, in a continuous “intergalactic tide”, but this “tide” seems to be a physical status quite different from the masses requested by Einstein’s postulate. One could therefore ask: can we still apply here the notion of the universe as a continuum of stellar plasmata? And if there is no mass to generate gravitational fields for billions and billions of km, how can all this empty space be considered “bent”?
At this point of our analysis, we can perhaps provisionally conclude: space is not a continuum of stellar plasmata or “masses”: it is an Euclidean vacuum interspersed with the ebullient plasmata surrounding stellar masses, run across by all sorts of débris and energies. Indeed, such a conclusion seems to maintain the ambivalent vision of space lamented by prof. Rovelli. But does it, really?
15 December 2021
 Sir Arthur Eddington, Space, Time & Gravitation. An Outline of the General Relativity Theory, Cambridge UP, 1920, repr. 1995, p. 126. Emphasis added.
 Isaac Newton, Opticks, Book Three, Part I, 1730 ed. , p. 339, Query 21, Dover Publications, New York - Internet Archives pdf, archive.org/details/Opticks. Emphasis added..
 Russell Stannard, The End of Discovery, Oxford UP, 2010, pp. 48-49. Emphasis by the Author. See also: Stuart Clark, The Universe, in: The Big Questions, series edited by Simon Blackburn, Quercus, 2010, p. 91. For more recent popular science sources: Shape of the Universe, en. wikipedia.org/wiki/Shape_of_the_universe; Leah Crane, Cosmological crisis: We don’t know if the universe is round or flat (www.newscientist.come/article/2222159-cosmological-crisis-we-dont-know-if-the-universe-is-round-or-flat/; Natalie Wolchover, What Shape Is the Universe? A New Study Suggests We’ve Got It All Wrong, Nov 4, 2019 (www.quantamgazine.org/print); Cody Cottier, What shape is the universe? As far as cosmologists can tell, space is almost perfectly flat. But what does this mean? (astronomy.com/news/2021/02/what-shape-is-the-universe.); Ethan Siegel, Ask Ethan: Why Is The Universe Flat?, Mar 5 2021 (www.forbes.com/sites/startswithabang.)
 Carlo Rovelli, Sette brevi lezioni di fisica [Seven short lectures on physics], Adelphi, Milano, 2014, pp. 46-47. My translation from the Italian original.
 Oxford Dictionary of Astronomy, OxfordUP, 2nd revised edition, 2012, entry: Eddington, Arthur Stanley.
 For all these data, see: Relativity and the 1919 eclipse, in the European’s Spacial Agency’s blog (www.esa.int).
 Eddington, Space, Time & Gravitation, p. 112.
 See: Hyades (star cluster), en.wikipedia.org.
 Quotations from: A Gallery of Einstein Rings, from the site: hubblesite.org/contents/media/images/2005/32/1788-Image.html; Chelsea Gohd, Astronomers turn back time to solve Einstein ring mistery, in: www.space.com/firts-einstein-ring-mystery-hubble-telescope.html, June 03, 2020; Einstein Ring, in: hyperphysics.phy-astr.gsu.edu/hbase/Astro’einring.htmil.
 A Gallery of Einstein’s Rings, p. 2 of 6. Emphasis added.
 Article by Chelsea Gohd, quoted above, p. 2 of 12.
 Steven Weinberg, The First Three Minutes. A Modern View of the Origin of the Universe, updated edition, 1993, pp. 53-54. Emphasis added.
 Weinberg, ibidem, p. 61.
 Dr. Mike Evans, What is Light?, article in: ‘Sky at Night’, July 2013 # 98, pp. 66-69; p. 68. Emphasis added.
 Albert Einstein, Leopold Infeld, The Evolution of Physics. From early concepts to relativity and quanta, 1938, with a new introduction by Walter Isaacson, Touchstone, New York, London, Toronto, Sidney, 2007, p. 235. Emphasis added.
 Ester Antonucci, Dentro il Sole [Inside the Sun], il Mulino, Bologna, 2014, p. 44. Emphasis added. My translation from the original. Most of my information on the Sun comes from this excellent book for the non-specialist.
 Oxford Dictionary of Astronomy, entry: heliosphere.
 Clark, The Universe, p. 14.
 Https;//en.wikipedia.org/wiki/Proxima_Centauri, p. 1 of 20.
 Antonucci, ibidem., p. 16.
 En. wikipedia. org., ibidem, p. 2 of 20.