Force field (physics)
In physics a force field is a type of physical field which describes a non-contact force acting on a particle at various positions in space through the use of a vector field. Specifically, a force field is a vector field , where is the force that a particle would feel if it were at the point .
- The subject of electric oscillation announced in a remarkable paper of Henry in 1842 and threshed out in its main features by Kelvin in 1856, followed by Kirchhoff's treatment of the transmission of oscillations along a wire (1857), has become of discriminating importance between Maxwell's theory of the electric field and the other equally profound theories of an earlier date. These crucial experiments contributed by Hertz (1887, et seq.) showed that electromagnetic waves move with the velocity of light, and like it are capable of being reflected, refracted, brought to interference and polarized. A year later Hertz (1888) worked out the distribution of the vectors in the space surrounding the oscillatory source. ...Some doubt was thrown on the details of Hertz's results by Sarasin and de la Rive's phenomenon of multiple resonance (1890), but this was soon explained away as the necessary result of the occurrence of damped oscillations by Poincaré (1891), by Bjerknes (1891) and others.
- Based on Faraday's earlier work, Maxwell stressed the notion of fields, in contrast to Newton's emphasis on the direct action of bodies on each other across empty space (action at a distance). Faraday and Maxwell regarded the effect of an electrically charged body as giving rise to stresses in its immediate surroundings... [and] in ever widening circles, gradually diminishing... These stresses... [i.e.,] the fields are intermediaries between the material particles and assume the burden of Newton's action at a distance. ...[O]ne set of Maxwell's equations is to the effect that, in the presence of a magnetic field which changes in the course of time, an electric field arises which is not caused by the presence of any electric charge. This [is] the law of electromagnetic induction... From his theory, Maxwell... predicted that magnetic fields propogate at... the speed of light. ...The laws of mechanics involve only accelerations, not velocities: the laws of electromagnetism involve a universal velocity [c]...
- Peter G. Bergmann, The Riddle of Gravitation: From Newton to Einstein to Today's Exciting Theories (1968)
- William Gilbert published a famous book on the magnet in 1600 and laid himself open to the gibes of Sir Francis Bacon for being one of those people so taken by their pet subject of research that they could only see the whole universe transposed into terms of it. Having made a spherical magnet called a terrella, and having found that it revolved when placed in a magnetic field, he decided that the whole earth was a magnet, that gravity was a form of magnetic attraction, and that the principles of the magnet accounted for the workings of the Copernican system as a whole. Kepler and Galileo were both influenced by this view, and with Kepler, it became an integral part of his system, a basis for the doctrine of almost universal gravitation.
- Herbert Butterfield, The Origins of Modern Science (1949)
- One possibility in this direction is to regard, classically, an electron as the end of a single Faraday line of force. The electric field in this picture from discrete Faraday lines of force, which are to be treated as physical things, like strings. One has then to develop a dynamics for such a string like structure, and quantize it.... In such a theory a bare electron would be inconceivable, since one cannot imagine the end of a piece of string without having the string.
- Paul Dirac, Bombay Lectures (1955)
- Classical mechanics has been developed continuously from the time of Newton and applied to an ever-widening range of dynamical systems, including the electromagnetic field in interaction with matter. The underlying ideas and the laws governing their application form a simple and elegant scheme, which one would be inclined to think could not be seriously modified without having all its attractive features spout. Nevertheless it has been found possible to set up a new scheme, called quantum mechanics, which is more suitable for the description of phenomena on the atomic scale and which is in some respects more elegant and satisfying than the classical scheme. This possibility is due to the changes which the new scheme involves being of a very profound character and not clashing with the features of the classical theory that make it so attractive, as a result of which all these features can be incorporated in the new scheme.
- Paul Dirac, The Principles of Quantum Mechanics (4th ed. 1958) I. The Principle of Superposition - 1. The Need for a Quantum Theory.
- We shall therefore assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.
- If de Sitter's solution were valid everywhere, then it would be thereby shown that the purpose which I pursued with the introduction of the λ-term has not been reached. In my opinion the general theory of relativity only forms a satisfactory system if according to it the physical qualities of space are completely determined by matter alone. Hence no gμv-field must be possible, i.e., no space-time-continuum, without matter that generates it.
- Albert Einstein, (1918) as quoted by Jürgen Renn, "The Emergence of the Riddle of Gravitation," The Genesis of General Relativity: Sources and Interpretations (2007)
- If the idea of physical reality had ceased to be purely atomic, it still remained for the time being purely mechanistic; people still tried to explain all events as the motion of inert masses; indeed no other way of looking at things seemed conceivable. Then came the great change, which will be associated for all time with the names of Faraday, Clerk Maxwell, and Hertz. The lion's share in this revolution fell to Clerk Maxwell. He showed that the whole of what was then known about light and electro-magnetic phenomena was expressed in his well known double system of differential equations, in which the electric and magnetic fields appear as the dependent variables. Maxwell did, indeed try to explain, or justify, these equations by intellectual constructions. But... the equations alone appeared as the essential thing and the strength of the fields as the ultimate entities, not to be reduced to anything else.
- Albert Einstein, "Clerk Maxwell's Influence on the Evolution of the Idea of Physical Reality" in Essays in Science (1934)
- I just want to explain what I mean when I say that we should try to hold on to physical reality.
We are … all aware of the situation regarding what will turn out to be the basic foundational concepts in physics: the point-mass or the particle is surely not among them; the field, in the Faraday-Maxwell sense, might be, but not with certainty. But that which we conceive as existing ("real") should somehow be localized in time and space.
- Albert Einstein, "What must be an essential feature of any future fundamental physics?" Letter to Max Born (March 1948); published in Albert Einstein-Hedwig und Max Born (1969) "Briefwechsel 1916-55", and in Potentiality, Entanglement and Passion-at-a-Distance: Quantum Mechanical Studies for Abner Shimony, Volume Two edited by Robert Cohen, Michael Horn, and John Stachel (1997), p. 121
- What led me more or less directly to the special theory of relativity was the conviction that the electromotive force acting on a body in motion in a magnetic field was nothing else but an electric field.
- During that year in Aarau the question came to me: If one runs after a light wave with light velocity, then one would encounter a time-independent wavefield. However, something like that does not seem to exist! This was the first juvenile thought experiment which has to do with the special theory of relativity. Invention is not the product of logical thought, even though the final product is tied to a logical structure.
- Albert Einstein, from his "Autobiographische Skizze" (18 April 1955), original German version here. Translation from Subtle is the Lord: The Science and the Life of Albert Einstein by Abraham Pais (1982), p. 131. Pais notes that when he said "during that year", he was referring to some time between October 1895 and early fall 1896.
- The special theory of relativity owes its origins to Maxwell's equations of the electromagnetic field.
- Albert Einstein, as quoted in New Scientist, Vol. 130 (1991), p. 49.
- We postulate: It shall be impossible, by any experiment whatsoever performed inside such a box, to detect a difference between an acceleration relative to the nebulae and gravity. That is, an accelerating box in some gravitational field is indistinguishable from a stationary box in some different gravitational field. How much like Einstein this sounds, how reminiscent of his postulate of special relativity! We know the principle of equivalence works for springs, (as we knew special relativity worked for electrodynamics), and we extend it by fiat to all experiments whatsoever. We are used to such procedures by now, but how originally brilliant it was in 1911—what a brilliant, marvelous man Einstein was!
- Richard Feynman (1962-63) Feynman Lectures on Gravitation (1995), Lecture 7.
- From 1916 Einstein and de Sitter corresponded extensively on exactly what kind of universe best fit the relativity equations. De Sitter initially developed a model of a spherical universe, in contrast to the cylindrical one Einstein had envisioned. De Sitter also tried to map out the shape of the spherical universe in absence of all matter. Einstein's reaction to de Sitter's model was strong and negative...de Sitter's sphere described a universe that changed in size instead of remaining nicely constant. ... Einstein saw matter—and its corresponding gravitational field—as what inherently created the shape of the universe. He cited what he dubbed "Mach's principle,"...the movements of any object ...were determined by all other bodies in the universe. ...how a body moves through space is tantamount to what shape space is, the concept of "shape" without matter, Einstein insisted, was meaningless.
- Karen C. Fox, Aries Keck, Einstein A to Z (2004)
- It turns out that the energy of a gravitational field—any gravitational field—is negative. During inflation, as the universe gets bigger and bigger and more and more matter is created, the total energy of matter goes upward by an enormous amount. Meanwhile, however, the energy of gravity becomes more and more negative. The negative gravitational energy cancels the energy in matter, so the total energy of the system remains whatever it was when inflation started—presumably something very small. ...This capability for producing matter in the universe is one crucial difference between the inflationary model and the previous model.
- Alan Guth, "A Universe in Your Backyard," in Third Culture: Beyond the Scientific Revolution (1996) ed. John Brockman, p. 279.
- What it takes to produce a gravitational repulsion is a negative pressure. According to general relativity, it turns out... both pressures and energy densities can produce gravitational fields, unlike Newtonian physics, where it's only mass densities that produce gravitational fields.
- Alan Guth, The Early Universe (2012) Lecture 1 ibid.
- A positive pressure produces an attractive gravitational field... Positive pressures are just sort of normal pressures and attractive gravity is normal gravity, so normal pressures produce normal gravity, but it is possible to have negative pressures, and negative pressures produce repulsive gravity, and that's the secret of what makes inflation possible.
- Alan Guth, The Early Universe (2012) Lecture 1 ibid.
- The universe would have expanded in a smooth way from a single point. As it expanded, it would have borrowed energy from the gravitational field, to create matter. As any economist could have predicted, the result of all that borrowing, was inflation. The universe expanded and borrowed at an ever-increasing rate. Fortunately, the debt of gravitational energy will not have to be repaid until the end of the universe.
- Stephen Hawking, The Beginning of Time (1996).
- Newton's system was for a long time considered as final and the task... seemed simply to be an expansion.... The first difficulty arose in the discussion of the electromagnetic field in... Faraday and Maxwell. In Newtonian mechanics the gravitational force had been considered as given... In the work of Faraday and Maxwell... the field of force... became the object of the investigation... they tried to set up equations of motion for the fields, not primarily for the bodies... This change led back to a point of view...held... before Newton. An action could... be transferred... only when the two bodies touched... Newton had introduce a very new and strange hypothesis by assuming a force that acted over a long distance. Now in the theory of fields... action is transferred from one point to a neighboring point... in terms of differential equations. ...the description of the electromagnetic fields... by Maxwell's equations seemed a satisfactory solution of the problem of force. ...The axioms and definitions of Newton had referred to bodies and their motion; but with Maxwell the fields... seemed to have acquired the same degree of reality as the bodies in Newton's theory. This view... was not easily accepted.; and to avoid such a change in the concept of reality... many physicists believed that Maxwell's equations actually referred to the deformations of an elastic medium... the ether... the medium was so light and thin that it could penetrate into other matter and could not be seen or felt. ...[H]owever ...it could not explain the complete absence of any longitudinal light waves.
- Werner Heisenberg, Physics and Philosophy: The Revolution in Modern Science (1958)
- The theory of relativity... showed in a conclusive way that the concept of the ether as a substance, to which Maxwell's equations refer, had to be abandoned. ...the result was that the fields had to be considered as an independent reality.
- Werner Heisenberg, Physics and Philosophy: The Revolution in Modern Science (1958)
- He was little interested mathematics or theory; for example, when his ideas on magnetic fields were extensively developed later by [James Clerk Maxwell] (1831-1879), Faraday was little concerned with the results. His own scientific career was characterized by simple ideas and simple experiments.
- Aaron J. Ihde, The Development of Modern Chemistry (1970) Ch. 5, p. 136.
- The invention of the wheel was perhaps rather obvious; but the invention of an invisible wheel, made of nothing but a magnetic field, was far from obvious, and that is what we owe to Nikola Tesla.
- Reginald Kapp (1956) as quoted by Christopher Lavers, Edmund G.R. Kraal, Stanley Buyers, Reeds Marine Engineering and Technology, Vol 6: Basic Electrotechnology for Marine Engineers (1965)
- In view of the facility with which Lorentz's theory explains the dispersion and observation phenomena, a direct proof of its truth was hardly required. But that was also forthcoming. In 1896 a pupil of Lorentz, P. Zeeman, discovered a phenomenon whose existence Faraday had vainly sought for in 1862. If a luminous vapour, say a sodium flame, is brought into a strong magnetic field, the spectrum lines of the vapour show peculiar changes, consisting of a doubling or trebling, according to the line of vision. These changes are predicted by Lorentz's theory. The Zeeman phenomenon further permitted a determination of the inert mass connected with the vibrating charges, and then a striking result was obtained: the vibrating electron is always negatively charged, while the positive charge is stationary. ...The original and almost tacit assumption that the whole ion—i.e., the chemical atom plus its valency charge—was in oscillation must, therefore, be abandoned. We must suppose that the charge, just as is the case in electrolysis, has also an independent mobility in the light-emitting molecule, and that the mass concerned in the Zeeman phenomenon is that of the electron itself.
- The impressions received by the two observers A0 and A would be alike in all respects. It would be impossible to decide which of them moves or stands still with respect to the ether, and there would be no reason for preferring the times and lengths measured by the one to those determined by the other, nor for saying that either of them is in possession of the "true" times or the "true" lengths. This is a point which Einstein has laid particular stress on, in a theory in which he starts from what he calls the principle of relativity, i.e., the principle that the equations by means of which physical phenomena may be described are not altered in form when we change the axes of coordinates for others having a uniform motion of translation relatively to the original system.
I cannot speak here of the many highly interesting applications which Einstein has made of this principle. His results concerning electromagnetic and optical phenomena ...agree in the main with those which we have obtained... the chief difference being that Einstein simply postulates what we have deduced, with some difficulty and not altogether satisfactorily, from the fundamental equations of the electromagnetic field. By doing so, he may certainly take credit for making us see in the negative result of experiments like those of Michelson, Rayleigh and Brace, not a fortuitous compensation of opposing effects, but the manifestation of a general and fundamental principle.
Yet, I think, something may also be claimed in favour of the form in which I have presented the theory. I cannot but regard the ether, which can be the seat of an electromagnetic field with its energy and vibrations, as endowed with a certain degree of substantiality, however different it may be from all ordinary matter. ...it seems natural not to assume at starting that it can never make any difference whether a body moves through the ether or not, and to measure distances and lengths of time by means of rods and clocks having a fixed position relatively to the ether.
It would be unjust not to add that, besides the fascinating boldness of its starting point, Einstein's theory has another marked advantage over mine. Whereas I have not been able to obtain for the equations referred to moving axes exactly the same form as for those which apply to a stationary system, Einstein has accomplished this by means of a system of new variables slightly different from those which I have introduced.
- Hendrik Lorentz, The Theory of Electrons and Its Applications to the Phenomena of Light and Radiant Heat (1916) Ch. V Optical Phenomena in Moving Bodies.
- The general equations are next applied to the case of a magnetic disturbance propagated through a non-conductive field, and it is shown that the only disturbances which can be so propagated are those which are transverse to the direction of propagation, and that the velocity of propagation is the velocity v, found from experiments such as those of Weber, which expresses the number of electrostatic units of electricity which are contained in one electromagnetic unit. This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.
- For decades, new-energy researchers talked about the possibility of treating a magnet so that its magnetic field would continuously shake or vibrate. On rare occasions, [Floyd] Sweet saw this effect, called self-oscillation, occur in electric transformers. He felt it could be coaxed into doing something useful, such as producing energy. Sweet thought that if he could find the precise way to shake or disturb a magnet's force field, the field would continue to shake by itself. It would be similar to striking a bell and having the bell keep on ringing. Sweet - who said his ideas came to him in dreams - turned for inspiration to his expertise in magnets. He knew magnets could be used to produce electricity, and wanted to see if he could get power out of a magnet by something other than the standard induction process. What Sweet wanted to do was to keep the magnet still and just shake its magnetic field. This shaking, in turn, would create an electric current. One new-energy researcher compares self-oscillation to a leaf on a tree waving in a gentle breeze. While the breeze itself isn't moving back and forth, it sets the leaf into that kind of motion. Sweet thought that if cosmic energy could be captured to serve as the breeze, then the magnetic field would serve as the leaf. Sweet would just have to supply a small amount of energy to set the magnetic field in motion, and space energy would keep it moving.
- Jeane Manning, The Coming Energy Revolution: The Search for Free Energy (1996) p.72.
- Weyl considered an aspect about general relativity... the nonpreservation of direction in a curved space. ...[He] decided to consider the possibility that length was also not preserved. ...To effect this change mathematically, Weyl had to make a slight modification in the structure of general relativity. He assumed that in addition to the usual metric (set of numbers or variables) that described the gravitational field, there was another one related to length. ...amazingly when the result was analyzed Maxwell's equations mysteriously appeared. It almost seemed as if a bit of magic had occurred and scientists quickly became interested in the miracle. ...but with detailed analysis the theory was shown to be flawed. Einstein was the first to put his finger on the flaw. ...Weyl soon acknowledged the flaw and laid his theory to rest. It may have been a failure (actually it was not an entire failure; a similar idea is used today in modern field theory), but it did accomplish something important: it got people interested in the possibility that the electromagnetic and gravitational field could be unified. Einstein soon began working on an alternative theory, as did others.
- Barry Parker, Einstein's Dream: The Search for a Unified Theory of the Universe (1986).
- I want to talk about thought experiments and how they can work, and I want to do that by talking about my favorite example which is Maxwell's equations, the laws of electromagnetism. Again, these are more equations, but it's ok because they're on a T-shirt. So these laws govern the behavior of electric and magnetic fields, but actually, when Maxwell was a boy... there was a missing term. ...When Maxwell got into the field these were the equations, and they had been discovered experimentally, and I want to say a little bit about them. So this bit here is Gauss's law
- J. J. Thomson was about to make the most significant find of the late nineteenth century... Thompson had been investigating the nature of cathode rays. He was convinced that they were some kind of electrified particles and, to prove his theory, began testing their behavior in electric or magnetic fields. By measuring both the extent to which such fields deflected them and their electric charge, he discovered that cathode rays consisted of very small negatively charged particles whose mass was about eighteen hundred times smaller than the lightest known substance—the hydrogen atom. ...He initially named these tiny carriers of electricity "corpuscles." Later they would become known as "electrons." The corpuscles were, in fact, the first subatomic particles to be found...
- Dianna Preston, Before the Fallout from Marie Curie to Hiroshima (2005).
- There is no direct observational evidence for the curvature [of space], the only directly observed data being the mean density and the expansion, which latter proves that the actual universe corresponds to the non-statical case. It is therefore clear that from the direct data of observation we can derive neither the sign nor that value of the curvature, and the question arises whether it is possible to represent the observed facts without introducing the curvature at all. Historically the term containing the 'cosmological constant λ' was introduced into the field equations in order to enable us to account theoretically for the existence of a finite mean density in a static universe. It now appears that in the dynamical case this end can be reached without the introduction of λ.
- Tesla, with his almost preternatural insight into alternating current phenomenon... enabled him... to revolutionize the art of electric power transmission through the invention of the rotary field motor...
- John Stone Stone, "John Stone Stone on Nikola Tesla's Priority in Radio and Continuous-Wave Radiofrequency Apparatus" (1915)
- If, in the very intense electric field in the neighbourhood of the cathode, the molecules of the gas are dissociated and are split up, not into the ordinary chemical atoms, but into these primordial atoms, which we shall for brevity call corpuscles; and if these corpuscles are charged with electricity and projected from the cathode by the electric field, they would behave exactly like the cathode rays.
- The physicist's definition of a field is the following. It's something that ... is spread everywhere throughout the universe. It's something that takes a particular value at every point in space. And what's more, that value can change in time.
- David Tong, Quantum Fields: The Real Building Blocks of the Universe - with David Tong YouTube, 15 February 2017, (quote from 10:45 of 1:00:17).
- The conception of lines of force was introduced by Faraday to form a mental picture of the processes going on in the electric field. To him these lines were not mere mathematical abstractions. He ascribed to them properties that gave them a real physical significance. They terminate on opposite charges, are always in a state of tension, tending to shorten themselves, and are mutually repellent. The direction of a line of force at any point gives the direction of the field at that point. With the help of these properties of lines of force it is possible to obtain an idea of the distribution of the intensity of the field surrounding electrically charged bodies.
- In deriving... the energy of the moving electron, it was assumed that the field of the moving electron is the same as that of the stationary electron. This is, however, only the case if the electron moves slowly, because when a Faraday tube is moved it tends to set itself at right angles to the direction of motion. The tubes constituting the electron therefore tend to crowd together in a plane perpendicular to the direction of motion of the electron. The result is an increase in the inertia or mass of the electron, because more work must be done to move a Faraday tube parallel to itself than along its own direction, just as it is harder to move a log of wood in the water parallel to itself than to move it endwise. This increase in the mass of the electron only becomes appreciable when it moves with a speed greater than about one-tenth that of light... The mass of the electron is measured by the ratio of the force to the acceleration to which it gives rise. According to the theory of Abraham and Lorentz the electron has two masses: the longitudinal mass, when it is accelerated in the direction of motion, and the transverse mass, when it is accelerated perpendicular to the direction of motion of the electron. If m represents the mass of the slow moving electron, then the longitudinal and transverse masses m1 and m2 are given by
- where c is the speed of light. It is seen that as the speed of the electron approaches that of light its electromagnetic mass tends to become infinitely large. The transverse mass of the highspeed electron for various speeds has been determined by Kaufmann and Bucherer. Their experiments verify the above expression for the transverse mass. From this it would seem that the mass of the electron is entirely electromagnetic. Later developments of the Theory of Relativity have rendered this conclusion somewhat questionable, so that there does not seem to be definite experimental evidence to indicate that the electronic mass is entirely electromagnetic.
- With Mie's view of matter there is contrasted another, according to which matter is a limiting singularity of the field, but charges and masses are force-fluxes in the field. This entails a new and more cautious attitude towards the whole problem of matter.
- Hermann Weyl, Space—Time—Matter From the Author's Preface to 4th Edition (1920)
- First, the physicists in the persons of Faraday and Maxwell, proposed the "electromagnetic field" in contradistinction to matter, as a reality of a different category. Then, during the last century, the mathematicians, … secretly undermined belief in the evidence of Euclidean Geometry. And now, in our time, there has been unloosed a cataclysm which has swept away space, time, and matter hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope and entailing a deeper vision. This revolution was promoted essentially by the thought of one man, Albert Einstein.
- Hermann Weyl, Space—Time—Matter (1952) Introduction
- From quantum theory there follows the existence of so called zero-point oscillations; for example each oscillator in its lowest is not completely at rest but always is moving about its equilibrium position. Therefore electromagnetic oscillations also can never cease completely. Thus the quantum nature of the electromagnetic field has as its consequence zero point oscillations of the field strength in the lowest energy state, in which there are no light quanta in space... The zero point oscillations act on an electron in the same way as ordinary electrical oscillations do. They can change the eigenstate of the electron, but only in a transition to a state with the lowest energy, since empty space can only take away energy, and not give it up. In this way spontaneous radiation arises as a consequence of the existence of these unique field strengths corresponding to zero point oscillations. Thus spontaneous radiation is induced radiation of light quanta produced by zero point oscillations of empty space.
- Victor Weisskopf, "Probleme der neueren Quantentheorie des Elektrons," Naturwissenschaften' (1935) Vol.23, pp. 631–637.
- Although the Special Theory of Relativity does not account for electromagnetic phenomena, it explains many of their properties. General Relativity, however, tells us nothing about electromagnetism. In Einstein's space-time continuum gravitational forces are absorbed in the geometry, but the electromagnetic forces are quite unaffected. Various attempts have been made to generate the geometry of space-time so as to produce a unified field theory incorporating both gravitational and electromagnetic forces.
- Gerald James Whitrow, The Structure of the Universe: An Introduction to Cosmology (1949)
- Changing electric fields produce magnetic fields, and changing magnetic fields produce electric fields. Thus the fields can animate one another in turn, giving birth to self-reproducing disturbances that travel at the speed of light. Ever since Maxwell, we understand that these disturbances are what light is.
- Frank Wilczek, The Lightness of Being (2008)
- String theory is extremely attractive because gravity is forced upon us. All known consistent string theories include gravity, so while gravity is impossible in quantum field theory as we have known it, it is obligatory in string theory.
- It was found [in the 1970s], unexpectedly and without anyone really having a concept for it, that the rules of perturbation theory can be changed in a way that makes relativistic quantum gravity inevitable rather than impossible. The change is made by replacing point particles by strings. Then Feynman graphs are replaced by Riemann surfaces, which are smooth - unlike the graphs, which have singularities at interaction vertices. The Riemann surfaces can degenerate to graphs in many different ways. In field theory, the interactions occur at the vertices of a Feynman graph. By contrast, in string theory, the interaction is encoded globally, in the topology of a Riemann surface, any small piece of which is like any other. This is reminiscent of how non-linearities are encoded globally in twistor theory.
- Edward Witten, "The Past and Future of String Theory" in The Future of Theoretical Physics and Cosmology: Celebrating Stephen Hawking's Contributions to Physics (2003) ed. G. W. Gibbons, E. P. S. Shellard & S. J. Rankin.
The Evolution of Scientific Thought from Newton to Einstein (1927)Edit
- A. D'Abro, source at archive.org
- With the new views advocated by Riemann... the texture, structure or geometry of space is defined by the metrical field, itself produced by the distribution of matter. Any non-homogeneous distribution of matter would then entail a variable structure of geometry for space from place to place. ...
Riemann's exceedingly speculative ideas on the subject of the metrical field were practically ignored in his day, save by the English mathematician Clifford, who translated Riemann's works, prefacing them to his own discovery of the non-Euclidean Clifford space. Clifford realised the potential importance of the new ideas and suggested that matter itself might be accounted for in terms of these local variations of the non-Euclidean space, thus inverting in a certain sense Riemann's ideas. But in Clifford's day this belief was mathematically untenable. Furthermore, the physical exploration of space seemed to yield unvarying Euclideanism. ...it was reserved for the theoretical investigator Einstein, by a stupendous effort of rational thought, based on a few flimsy empirical clues, to unravel the mystery and to lead Riemann's ideas to victory. (In all fairness to Einstein... he does not appear to have been influenced directly by Riemann.) Nor were Clifford's hopes disappointed, for the varying non-Euclideanism of the continuum was to reveal the mysterious secret of gravitation, and perhaps also of matter, motion, and electricity. ...
Einstein had been led to recognize that space of itself was not fundamental. The fundamental continuum whose non-Euclideanism was fundamental was... one of Space-Time... possessing a four-dimensional metrical field governed by the matter distribution. Einstein accordingly applied Riemann's ideas to space-time instead of to space... He discovered that the moment we substitute space-time for space (and not otherwise), and assume that free bodies and rays of light follow geodesics no longer in space but in space-time, the long-sought-for local variations in geometry become apparent. They are all around us, in our immediate vicinity... We had called their effects gravitational effects... never suspecting that they were the result of those very local variations in the geometry for which our search had been in vain... the theory of relativity is the theory of the space-time metrical field.
- Faraday was the first scientist to realise the enormous importance of the electromagnetic field. He saw in it a reality of a new category differing from matter. It was capable of transmitting effects from place to place, and was not to be likened to a mere mathematical fiction such as the gravitational field was then assumed to be. In his opinion, the phenomena of electricity and magnetism should be approached via the field rather than via the charged bodies and currents. In other words, according to Faraday, when a current was flowing along a wire, the most important aspect of the phenomenon lay not in the current itself but in the fields of electric and magnetic force distributed throughout space in the current's vicinity. It is this elevation of the field to a position of preeminence that is often called the pure physics of the field. Faraday was not a mathematician and was unable to co-ordinate the phenomena he foresaw in a mathematical way, and derive the full benefit from his ideas. Before dying, however, he entrusted this task to his colleague Maxwell; and one of the most astonishing theories of science, eclipsed only in recent years by Einstein's theory of relativity, was the outcome.
- In order to appreciate the nature of Maxwell's contributions, let us recall how matters stood in his day. ...Faraday's law of induction ...states that a variable magnetic field generates an electric field. Maxwell, however, considered that this law, standing alone, lacked symmetry; so he formulated the hypothesis that conversely a variable electric field should generate a magnetic one, and proceeded to construct his theory... no experimental results could be claimed to have justified any such assumption... His celebrated equations of electromagnetics represented, therefore, the results of experiment, supplemented by this additional hypothetical assumption. The advisability of making this hypothesis was accentuated when it was found to ensure the law of conservation of electricity. ...In the particular case of free space in which only fields but no charges or currents are present, Maxwell's equations of electromagnetics, termed field-equations (since they describe the state of the electromagnetic field), can be written:
- where E represents electric field intensity, H magnetic field density, t denotes time, and c a most important constant... (2) represents Faraday's law of induction, while (4) constitutes the hypothetical law postulated by Maxwell... Prior to Maxwell's investigations the fourth equation would have been written "curl H = 0"...From his field equations, Maxwell... was able to deduce the two additional ones:
- ...these last two equations connote that varying electric and magnetic intensities will be propagated through the ether in wave form with a velocity c... This discovery removed all possibility of action at a distance, since the field perturbations now appeared to be propagated from place to place with a finite velocity. It was... of interest to determine the precise value of c. ...Physicists ...were unable, in Maxwell's day, to devise a means of performing such delicate experiments. ...Maxwell remarked that it would be given by the ratio of the magnitude of any electric charge, measured in terms of electrostatic units (based on electricity), and then of electromagnetic units (based on magnetism).
If two magnetic poles of equal strength, situated in empty space... one centimetre apart, attract or repel each other with a force of one dyne, either pole is said to represent one unit of magnetic pole strength in the electromagnetic system of units. Owing to the interconnections between magnetism and electricity, we can deduce therefrom the unit of electric charge also in the electromagnetic system. Likewise, if two electric charges of equal strength... in empty space at a distance of one centimetre apart, attract or repel each other with a force of one dyne, either charge is said to represent one unit of electric charge in the electrostatic system of units. From this we can derive the unit of magnetic pole strength in the electrostatic system.
Precise measurements... then proved that the value of this ratio was about 186,000 miles per second; whence it became necessary to assume that periodic perturbations in the strains and stresses of the field would be propagated in the form of waves moving through the ether with this particular speed. But this velocity was precisely that of light waves propagated through the luminiferous ether.
- Although experimenters had attempted by various means to submit Maxwell's views to a test, the technical difficulties were so great that no success had been achieved. It appeared clearly from Maxwell's equations that no appreciable effects could be anticipated unless dE/dt was very great; and this meant that the electric intensity E would have to vary with extreme rapidity. The simplest means of obtaining a result of this kind would be to produce an oscillating field of electric intensity in which the oscillations were extremely rapid, say, several millions per second. But no mechanical contrivance could yield such rapid vibrations, and... no other methods suggested themselves. ...
In 1885 Helmholtz directed the attention of his pupil, Hertz, to the problem. Hertz was one of the most remarkable experimenters of the nineteenth century; he succeeded in at last vanquishing the technical difficulties and in generating by purely electrical means an oscillating electric field of extremely high frequency. Electromagnetic waves of sufficient intensity were thus produced; and after having been sidetracked for a time by a secondary phenomenon whose nature was elucidated by Poincaré, Hertz verified the fact that the waves advanced with the speed of light and indeed possessed all the essential properties of light waves other than those of visibility to the human eye. Thus, as a result of Hertz's experiments, the foundations were laid for the commercial use of wireless and radio; but, more important still, Maxwell's electromagnetic theory of light establishing the intimate connection between electricity and optics had been at last vindicated.
- The most precise experiments have proved the correctness of the Einsteinian laws of mechanics and...Bucherer's experiment proving the increase in mass of an electron in rapid motion is a case in point.
Very important differences distinguish the theory of Einstein from that of Lorentz. Lorentz also had deduced from his theory that the mass of the electron should increase and grow infinite when its speed neared that of light; but the speed in question was the speed of the electron through the stagnant ether; whereas in Einstein's theory it is merely the speed with respect to the observer. According to Lorentz, the increase in mass of the moving electron was due to its deformation or Fitzgerald contraction. The contraction modified the lay of the electromagnetic field round the electron; and it was from this modification that the increase in mass observed by Bucherer was assumed to arise. In Einstein's theory, however, the increase in mass is absolutely general and need not be ascribed to the electromagnetic field of the electron in motion. An ordinary unelectrified lump of matter like a grain of sand would have increased in mass in exactly the same proportion; and no knowledge of the microscopic constitution of matter is necessary in order to predict these effects, which result directly from the space and time transformations themselves.
Furthermore, the fact that this increase in mass of matter in motion is now due to relative motion and not to motion through the stagnant ether, as in Lorentz's theory, changes the entire outlook considerably. According to Lorentz, the electron really increased in mass, since its motion through the ether remained a reality. According to Einstein, the electron increases in mass only in so far as it is in relative motion with respect to the observer. Were the observer to be attached to the flying electron no increase in mass would exist; it would be the electron left behind which would now appear to have suffered the increase. Thus mass follows distance, duration and electromagnetic field in being a relative and having no definite magnitude of itself and being essentially dependent on the conditions of observation.
Owing to the general validity of the Lorentz-Einstein transformations, it becomes permissible to apply them to all manner of phenomena.. ...temperature, pressure and many other physical magnitudes turned out to be relatives. ...entropy, electric charge and the velocity of light in vacuo were absolutes transcending the observer's motion. ...a number of other entities are found to be absolutes, the most important of which is that abstract mathematical quantity called the Einsteinian interval, which plays so important a part in the fabric of the new objective world of science, the world of four-dimensional space-time.
- In the study of electricity and magnetism we may consider phenomena in which conditions do not vary as time passes by; the electric charges and the magnets remain at rest, and the currents flowing in fixed wires do not vary in intensity. Conditions are then termed stationary [static]; it is as though time played no part. The laws which govern this type of phenomena were discovered empirically over a century ago, and were expressed mathematically in terms of spatial vectors. The problem of ascertaining how electric and magnetic phenomena would behave when conditions ceased to be stationary was one that could not be predicted; further experimental research was necessary before the general laws could be obtained. Even so, the difficulties were considerable, and it needed Maxwell's genius to establish the laws from the incomplete array of experimental evidence then at hand. All this work extended over nearly a century; it was slow and laborious.
Yet, had men realised that our world was one of four-dimensional Minkowskian space-time, and not one of separate space and time, things would have been different. By extending the well-known stationary laws to four-dimensional space-time, through the mere addition of time components to the various trios of space ones, we should have written out inadvertently the laws governing varying fields, or, in other words, we should have constructed Maxwell's celebrated equations. Electromagnetic induction, discovered experimentally by Faraday, the additional electrical term introduced tentatively by Maxwell, radio waves, everything in the electromagnetics of the field, could have been foreseen at one stroke of the pen. A century of painstaking effort could have been saved. We are assuming that a four-dimensional vector calculus would have been in existence; but this is purely a mathematical question.
- Let us revert to the metrical field, as defining the space-time structure. Although Riemann had attributed the existence of the structure, or metrical field, of space to the binding forces of matter, there is not the slightest indication in Einstein's special theory that any such view is going to be developed later on; in fact, it does not appear that Einstein was influenced in the slightest degree by Riemann's ideas. ...in the special theory, the problem of determining whence the structure, or field, arises, what it is, what causes it, is not even discussed in a tentative manner. Space-time, with its flat structure, is assumed to be given or posited by the Creator.
But in the general theory the entire situation changes when Einstein accounts for gravitation, hence for a varying lay of the metrical field, in terms of a varying non-Euclidean structure of space-time around matter. We are then compelled to recognise not only that the metrical field regulates the behaviour of material bodies and clocks, as was also the case in the special theory, but, furthermore, that a reciprocal action takes place and that matter and energy in turn must affect the lay of the metrical field. But we are still a long way from Riemann's view that the field is not alone affected but brought into existence by matter; and it is only when we consider the cosmological part of Einstein's theory that this idea of Riemann's may possibly be vindicated.
And here we come to a parting of the ways with de Sitter and Eddington on one side, Einstein and Thirring on the other, and Weyl somewhere in between the two extremes.
- Willem de Sitter, "Relativity and Modern Theories of the Universe"
- In electromagnetism... the law of the inverse square had been supreme, but, as a consequence of the work of Faraday and Maxwell, it was superseded by the field. And the same change took place in the theory of gravitation. By and by the material particles, electrically charged bodies, and magnets which are the things that we actually observe come to be looked upon only as "singularities" in the field.
- This is the mathematical formulation of the theory of relativity. The metric properties of the four-dimensional continuum are described... by a certain number (ten, in fact) of quantities denoted by gαβ, and commonly called "potentials." The physical status of matter and energy, on the other hand, is described by ten other quantities, denoted by Tαβ, the set of which is called the "material tensor." This special tensor has been selected because it has the property which is mathematically expressed by saying that its divergence vanishes, which means that it represents something permanent. The fundamental fact of mechanics is the law of inertia, which can be expressed in its most simple form by saying that it requires the fundamental laws of nature to be differential equations of the second order. Thus the problem was to find a differential equation of the second order giving a relation between the metric tensor gαβ and the material tensor Tαβ. This is a purely mathematical problem, which can be solved without any reference to the physical meaning of the symbols. The simplest possible equation (or rather set of ten equations, because there are ten g's) of that kind that can be found was adopted by Einstein as the fundamental equation of his theory. It defines the space-time continuum, or the "field." The world-lines of material particles and light quanta are the geodesics in the four-dimensional continuum defined by the solutions gαβ of these field-equations. The equations of the geodesic thus are equivalent to the equations of motion of mechanics. When we come to solve the field-equations and substitute the solutions in the equations of motion, we find that in the first approximation, i.e. for small material velocities (small as compared with the velocity of light), these equations of motion are the same as those resulting from Newton's theory of gravitation. The distinction between gravitation and inertia has disappeared; the gravitational action between two bodies follows from the same equations, and is the same thing, as the inertia of one body. A body, when not subjected to an extraneous force (i.e. a force other than gravitation), describes a geodesic in the continuum, just as it described a geodesic, or straight line, in the absolute space of Newton under the influence of inertia alone.
The field-equations and the equations of the geodesic together contain the whole science of mechanics, including gravitation.
- If we put in the details, the singularities of the field, viz. the galactic systems and the stars, we find that there is... a tendency, called gravitation, to decrease the mutual distances of these "singularities." At short distances, within the confines of a galactic system, this second tendency is by far the strongest, and the galactic systems retain their size independent of the expansion or contraction of the universe...