# spacetime

• The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different
observers will disagree on the length of time between two events (because of time dilation) or the distance between the two events (because of length contraction).

• When the event considered is infinitesimally close to each other, then we may write In a different inertial frame, say with coordinates , the spacetime interval can be written
in a same form as above.

• Many counterintuitive consequences emerge: in addition to being independent of the motion of the light source, the speed of light is constant regardless of the frame of reference
in which it is measured; the distances and even temporal ordering of pairs of events change when measured in different inertial frames of reference (this is the relativity of simultaneity); and the linear additivity of velocities no longer
holds true.

• In special relativity, an observer will, in most cases, mean a frame of reference from which a set of objects or events is being measured.

• [3] In the context of special relativity, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends
on the object’s velocity relative to the observer.

• Spacetime in special relativity Spacetime interval See also: Causal structure In three dimensions, the distance between two points can be defined using the Pythagorean
theorem: Although two viewers may measure the x, y, and z position of the two points using different coordinate systems, the distance between the points will be the same for both (assuming that they are measuring using the same units).

• A position in spacetime is called an event, and requires four numbers to be specified: the three-dimensional location in space, plus the position in time (Fig.

• The magnitude of this scale factor (nearly 300,000 kilometres or 190,000 miles in space being equivalent to one second in time), along with the fact that spacetime is a manifold,
implies that at ordinary, non-relativistic speeds and at ordinary, human-scale distances, there is little that humans might observe which is noticeably different from what they might observe if the world were Euclidean.

• the true distance Likewise, a timelike spacetime interval gives the same measure of time as would be presented by the cumulative ticking of a clock that moves along a given
world line.

• Suppose an observer measures two events as being separated in time by and a spatial distance Then the spacetime interval between the two events that are separated by a distance
in space and by in the -coordinate is: or for three space dimensions, [32] The constant the speed of light, converts time units (like seconds) into space units (like meters).

• In special relativity, however, the distance between two points is no longer the same if measured by two different observers when one of the observers is moving, because of
Lorentz contraction.

• In ordinary space, a position is specified by three numbers, known as dimensions.

• In particular, one notes that if two events are simultaneous in a particular reference frame, they are necessarily separated by a spacelike interval and thus are noncausally
related.

• Each location in spacetime is marked by four numbers defined by a frame of reference: the position in space, and the time (which can be visualized as the reading of a clock
located at each position in space).

• [citation needed] General relativity also provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the
field.

• Although measurements of distance and time between events differ for measurements made in different reference frames, the spacetime interval is independent of the inertial
frame of reference in which they are recorded.

• Spacetime intervals are equal to zero when In other words, the spacetime interval between two events on the world line of something moving at the speed of light is zero.

• These two lines form what is called the light cone of the event O, since adding a second spatial dimension (Fig.

• Until the 20th century, it was assumed that the three-dimensional geometry of the universe (its spatial expression in terms of coordinates, distances, and directions) was
independent of one-dimensional time.

• All observers who measure the time and distance between any two events will end up computing the same spacetime interval.

• The series of events can be linked together to form a line which represents a particle’s progress through spacetime.

• Unlike the analogies used in popular writings to explain events, such as firecrackers or sparks, mathematical events have zero duration and represent a single point in spacetime.

• However, in 1905, Einstein based his work on special relativity on two postulates: • The laws of physics are invariant (i.e., identical) in all inertial systems (i.e., non-accelerating
frames of reference)[citation needed] • The speed of light in vacuum is the same for all inertial observers, regardless of the motion of the light source.

• The squared interval is a measure of separation between events A and B that are time separated and in addition space separated either because there are two separate objects
undergoing events, or because a single object in space is moving inertially between its events.

• [18][15] In 1905, Einstein introduced special relativity (even though without using the techniques of the spacetime formalism) in its modern understanding as a theory of space
and time.

• 1-4), and included a remarkable demonstration that the concept of the invariant interval (discussed below), along with the empirical observation that the speed of light is
finite, allows derivation of the entirety of special relativity.

• The latticework of clocks is used to determine the time and position of events taking place within the whole frame.

• Animation illustrating relativity of simultaneity All observers will agree that for any given event, an event within the given event’s future light cone occurs after the given
event.

• For example, if one observer sees two events occur at the same place, but at different times, a person moving with respect to the first observer will see the two events occurring
at different places, because (from their point of view) they are stationary, and the position of the event is receding or approaching.

• [citation needed] The logical consequence of taking these postulates together is the inseparable joining of the four dimensions—hitherto assumed as independent—of space and
time.

• [19][note 2] In 1900, he recognized that Lorentz’s “local time” is actually what is indicated by moving clocks by applying an explicitly operational definition of clock synchronization
assuming constant light speed.

• Although time comes in as a fourth dimension, it is treated differently than the spatial dimensions.

• Thus, a different measure must be used to measure the effective “distance” between two events.

• The magenta hyperbolae, which cross the x axis, are timelike curves, which is to say that these hyperbolae represent actual paths that can be traversed by (constantly accelerating)
particles in spacetime: Between any two events on one hyperbola a causality relation is possible, because the inverse of the slope—representing the necessary speed—for all secants is less than .

• [33]: 2  The invariance of interval of any event between all intertial frames of reference is one of the fundamental results of special theory of relativity.

• Since spatial distance traversed by any massive object is always less than distance traveled by the light for the same time interval, real intervals are always timelike.

• However, it is not at all clear when Minkowski began to formulate the geometric formulation of special relativity that was to bear his name, or to which extent he was influenced
by Poincaré’s four-dimensional interpretation of the Lorentz transformation.

• But special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time.

• Einstein performed his analysis in terms of kinematics (the study of moving bodies without reference to forces) rather than dynamics.

• A spacetime diagram is typically drawn with only a single space and a single time coordinate.

• The interior of the future light cone consists of all events that are separated from the apex by more time (temporal distance) than necessary to cross their spatial distance
at lightspeed; these events comprise the timelike future of the event O.

• [21][22] He did not pursue the 4-dimensional formalism in subsequent papers, however, stating that this line of research seemed to “entail great pain for limited profit”,
ultimately concluding “that three-dimensional language seems the best suited to the description of our world”.

• By using the mass–energy equivalence, Einstein showed, in addition, that the gravitational mass of a body is proportional to its energy content, which was one of the early
results in developing general relativity.

• 2–4, event O is at the origin of a spacetime diagram, and the two diagonal lines represent all events that have zero spacetime interval with respect to the origin event.

• Spacetime diagram illustrating two photons, A and B, originating at the same event, and a slower-than-light-speed object, C The equation above is similar to the Pythagorean
theorem, except with a minus sign between the and the terms.

• The past light cone contains all the events that could have a causal influence on O.

• (a) Galilean diagram of two frames of reference in standard configuration, (b) spacetime diagram of two frames of reference, (c) spacetime diagram showing the path of a reflected
light pulse To gain insight in how spacetime coordinates measured by observers in different reference frames compare with each other, it is useful to work with a simplified setup with frames in a standard configuration.

• From the reference frame of an observer moving at, the events appear to occur in the order A, B, C. The white line represents a plane of simultaneity being moved from the
past of the observer to the future of the observer, highlighting events residing on it.

• He obtained all of his results by recognizing that the entire theory can be built upon two postulates: The principle of relativity and the principle of the constancy of light
speed.

• [18][15] While his results are mathematically equivalent to those of Lorentz and Poincaré, Einstein showed that the Lorentz transformations are not the result of interactions
between matter and aether, but rather concern the nature of space and time itself.

• We are always concerned with differences of spatial or temporal coordinate values belonging to two events, and since there is no preferred origin, single coordinate values
have no essential meaning.

• Space and Time included the first public presentation of spacetime diagrams (Fig.

• In 1908, Hermann Minkowski—once one of the math professors of a young Einstein in Zürich—presented a geometric interpretation of special relativity that fused time and the
three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space.

• A minor variation is to place the time coordinate last rather than first.

• [8]: 17–22  In this idealized case, every point in space has a clock associated with it, and thus the clocks register each event instantly, with no time delay between an event
and its recording.

• However, Lorentz considered local time to be only an auxiliary mathematical tool, a trick as it were, to simplify the transformation from one system into another.

• Introduction Definitions Non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space, and separate from
space.

• 1-1, imagine that the frame under consideration is equipped with a dense lattice of clocks, synchronized within this reference frame, that extends indefinitely throughout
the three dimensions of space.

• Failure to understand the difference between what one measures/observes versus what one sees is the source of much error among beginning students of relativity.

• By the mid-1800s, various experiments such as the observation of the Arago spot and differential measurements of the speed of light in air versus water were considered to
have proven the wave nature of light as opposed to a corpuscular theory.

• While discussing various hypotheses on Lorentz invariant gravitation, he introduced the innovative concept of a 4-dimensional spacetime by defining various four vectors, namely
four-position, four-velocity, and four-force.

• In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold.

• […] He told me later that it came to him as a great shock when Einstein published his paper in which the equivalence of the different local times of observers moving relative
to each other was pronounced; for he had reached the same conclusions independently but did not publish them because he wished first to work out the mathematical structure in all its splendor.

• In addition, C illustrates the world line of a slower-than-light-speed object.

• In dimensional Minkowski spacetime (having one temporal and one spatial dimension), the points at some constant spacetime interval away from the origin (using the Minkowski
metric) form curves given by the two equations with some positive real constant.

• Although these theories included equations identical to those that Einstein introduced (e.g., the Lorentz transformation), they were essentially ad hoc models proposed to
explain the results of various experiments—including the famous Michelson–Morley interferometer experiment—that were extremely difficult to fit into existing paradigms.

• The opening words of Space and Time include Minkowski’s famous statement that “Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow,
and only some sort of union of the two shall preserve independence.”

• Because of the invariance of the spacetime interval, all observers will assign the same light cone to any given event, and thus will agree on this division of spacetime.

• It is hence possible for event O to have a causal influence on event D. The future light cone contains all the events that could be causally influenced by O.

• Light cone in 2D space plus a time dimension A light (double) cone divides spacetime into separate regions with respect to its apex.

Works Cited

[‘luminiferous from the Latin lumen, light, + ferens, carrying; aether from the Greek αἰθήρ (aithēr), pure air, clear sky
2. ^ By stating that simultaneity is a matter of convention, Poincaré meant that to talk about time at all, one must have synchronized
clocks, and the synchronization of clocks must be established by a specified, operational procedure (convention). This stance represented a fundamental philosophical break from Newton, who conceived of an absolute, true time that was independent of
the workings of the inaccurate clocks of his day. This stance also represented a direct attack against the influential philosopher Henri Bergson, who argued that time, simultaneity, and duration were matters of intuitive understanding.[19]
3. ^
The operational procedure adopted by Poincaré was essentially identical to what is known as Einstein synchronization, even though a variant of it was already a widely used procedure by telegraphers in the middle 19th century. Basically, to synchronize
two clocks, one flashes a light signal from one to the other, and adjusts for the time that the flash takes to arrive.[19]
4. ^ A hallmark of Einstein’s career, in fact, was his use of visualized thought experiments (Gedanken–Experimente) as a
fundamental tool for understanding physical issues. For special relativity, he employed moving trains and flashes of lightning for his most penetrating insights. For curved spacetime, he considered a painter falling off a roof, accelerating elevators,
blind beetles crawling on curved surfaces and the like. In his great Solvay Debates with Bohr on the nature of reality (1927 and 1930), he devised multiple imaginary contraptions intended to show, at least in concept, means whereby the Heisenberg
uncertainty principle might be evaded. Finally, in a profound contribution to the literature on quantum mechanics, Einstein considered two particles briefly interacting and then flying apart so that their states are correlated, anticipating the phenomenon
known as quantum entanglement.[24]
5. ^ In the original version of this lecture, Minkowski continued to use such obsolescent terms as the ether, but the posthumous publication in 1915 of this lecture in the Annals of Physics (Annalen der Physik)
was edited by Sommerfeld to remove this term. Sommerfeld also edited the published form of this lecture to revise Minkowski’s judgement of Einstein from being a mere clarifier of the principle of relativity, to being its chief expositor.[26]
6. ^
(In the following, the group G∞ is the Galilean group and the group Gc the Lorentz group.) “With respect to this it is clear that the group Gc in the limit for c = ∞, i.e. as group G∞, exactly becomes the full group belonging to Newtonian Mechanics.
In this state of affairs, and since Gc is mathematically more intelligible than G∞, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena actually possess an invariance, not for the group G∞, but rather for
a group Gc, where c is definitely finite, and only exceedingly large using the ordinary measuring units.”[28]
7. ^ For instance, the Lorentz group is a subgroup of the conformal group in four dimensions.[29]: 41–42  The Lorentz group is
isomorphic to the Laguerre group transforming planes into planes,[29]: 39–42  it is isomorphic to the Möbius group of the plane,[30]: 22  and is isomorphic to the group of isometries in hyperbolic space which is often expressed
in terms of the hyperboloid model.[31]: 3.2.3
8. ^ In a Cartesian plane, ordinary rotation leaves a circle unchanged. In spacetime, hyperbolic rotation preserves the hyperbolic metric.
9. ^ Even with no (de)acceleration i.e. using one
inertial frame O for constant, high-velocity outward journey and another inertial frame I for constant, high-velocity inward journey – the sum of the elapsed time in those frames (O and I) is shorter than the elapsed time in the stationary inertial
frame S. Thus acceleration and deceleration is not the cause of shorter elapsed time during the outward and inward journey. Instead the use of two different constant, high-velocity inertial frames for outward and inward journey is really the cause
of shorter elapsed time total. Granted, if the same twin has to travel outward and inward leg of the journey and safely switch from outward to inward leg of the journey, the acceleration and deceleration is required. If the travelling twin could ride
the high-velocity outward inertial frame and instantaneously switch to high-velocity inward inertial frame the example would still work. The point is that real reason should be stated clearly. The asymmetry is because of the comparison of sum of elapsed
times in two different inertial frames (O and I) to the elapsed time in a single inertial frame S.
10. ^ The ease of analyzing a relativistic scenario often depends on the frame in which one chooses to perform the analysis. In this linked image,
we present alternative views of the transverse Doppler shift scenario where source and receiver are at their closest approach to each other. (a) If we analyze the scenario in the frame of the receiver, we find that the analysis is more complicated
than it should be. The apparent position of a celestial object is displaced from its true position (or geometric position) because of the object’s motion during the time it takes its light to reach an observer. The source would be time-dilated relative
to the receiver, but the redshift implied by this time dilation would be offset by a blueshift due to the longitudinal component of the relative motion between the receiver and the apparent position of the source. (b) It is much easier if, instead,
we analyze the scenario from the frame of the source. An observer situated at the source knows, from the problem statement, that the receiver is at its closest point to him. That means that the receiver has no longitudinal component of motion to complicate
the analysis. Since the receiver’s clocks are time-dilated relative to the source, the light that the receiver receives is therefore blue-shifted by a factor of gamma.
11. ^ Not all experiments characterize the effect in terms of a redshift. For
example, the Kündig experiment was set up to measure transverse blueshift using a Mössbauer source setup at the center of a centrifuge rotor and an absorber at the rim.
12. ^ Rapidity arises naturally as a coordinates on the pure boost generators
inside the Lie algebra algebra of the Lorentz group. Likewise, rotation angles arise naturally as coordinates (modulo 2π) on the pure rotation generators in the Lie algebra. (Together they coordinatize the whole Lie algebra.) A notable difference
is that the resulting rotations are periodic in the rotation angle, while the resulting boosts are not periodic in rapidity (but rather one-to-one). The similarity between boosts and rotations is formal resemblance.
13. ^ In relativity theory, proper
acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being
measured.
14. ^ Newton himself was acutely aware of the inherent difficulties with these assumptions, but as a practical matter, making these assumptions was the only way that he could make progress. In 1692, he wrote to his friend Richard Bentley:
“That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro’ a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to
another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it.”
15. ^ More precisely, the gravitational field couples to itself. In Newtonian gravity, the potential
due to two point masses is simply the sum of the potentials of the two masses, but this does not apply to GR. This can be thought of as the result of the equivalence principle: If gravitation did not couple to itself, two particles bound by their
mutual gravitational attraction would not have the same inertial mass (due to negative binding energy) as their gravitational mass.[56]: 112–113
16. ^ This is because the law of gravitation (or any other inverse-square law) follows from
the concept of flux and the proportional relationship of flux density and field strength. If N = 3, then 3-dimensional solid objects have surface areas proportional to the square of their size in any selected spatial dimension. In particular, a sphere
of radius r has a surface area of 4πr2. More generally, in a space of N dimensions, the strength of the gravitational attraction between two bodies separated by a distance of r would be inversely proportional to rN−1.
17. Different reporters viewing
the scenarios presented in this figure interpret the scenarios differently depending on their knowledge of the situation. (i) A first reporter, at the center of mass of particles 2 and 3 but unaware of the large mass 1, concludes that a force of repulsion
exists between the particles in scenario A while a force of attraction exists between the particles in scenario B. (ii) A second reporter, aware of the large mass 1, smiles at the first reporter’s naiveté. This second reporter knows that in reality,
the apparent forces between particles 2 and 3 really represent tidal effects resulting from their differential attraction by mass 1. (iii) A third reporter, trained in general relativity, knows that there are, in fact, no forces at all acting between
the three objects. Rather, all three objects move along geodesics in spacetime.
18. Rowe, E.G.Peter (2013). Geometrical Physics in Minkowski Spacetime (illustrated ed.). Springer Science & Business Media. p. 28. ISBN 978-1-4471-3893-8. Archived
from the original on 17 January 2023. Retrieved 10 September 2020. Extract of page 28 Archived 17 January 2023 at the Wayback Machine
19. ^ Rynasiewicz, Robert (12 August 2004). “Newton’s Views on Space, Time, and Motion”. Stanford Encyclopedia
of Philosophy. Metaphysics Research Lab, Stanford University. Archived from the original on 11 December 2015. Retrieved 24 March 2017.
20. ^ Davis, Philip J. (2006). Mathematics & Common Sense: A Case of Creative Tension. Wellesley, Massachusetts:
A.K. Peters. p. 86. ISBN 978-1-4398-6432-6.
21. ^ Lawden, D. F. (1982). Introduction to Tensor Calculus, Relativity and Cosmology (3rd ed.). Mineola, New York: Dover Publications. p. 7. ISBN 978-0-486-42540-5.
22. ^ Jump up to:a b c d e Collier,
Peter (2017). A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity (3rd ed.). Incomprehensible Books. ISBN 978-0-9573894-6-5.
23. ^ Rowland, Todd. “Manifold”. Wolfram Mathworld. Wolfram Research.
Archived from the original on 13 March 2017. Retrieved 24 March 2017.
24. ^ Jump up to:a b French, A.P. (1968). Special Relativity. Boca Raton, Florida: CRC Press. pp. 35–60. ISBN 0-7487-6422-4.
25. ^ Jump up to:a b c d e f g Taylor, Edwin F.;
Wheeler, John Archibald (1992). Spacetime Physics: Introduction to Special Relativity (2nd ed.). San Francisco, California: Freeman. ISBN 0-7167-0336-X. Retrieved 14 April 2017.
26. ^ Scherr, Rachel E.; Shaffer, Peter S.; Vokos, Stamatis (July 2001).
“Student understanding of time in special relativity: Simultaneity and reference frames” (PDF). American Journal of Physics. College Park, Maryland: American Association of Physics Teachers. 69 (S1): S24–S35. arXiv:physics/0207109. Bibcode:2001AmJPh..69S..24S.
doi:10.1119/1.1371254. S2CID 8146369. Archived (PDF) from the original on 28 September 2018. Retrieved 11 April 2017.
27. ^ Hughes, Stefan (2013). Catchers of the Light: Catching Space: Origins, Lunar, Solar, Solar System and Deep Space. Paphos,
Cyprus: ArtDeCiel Publishing. pp. 202–233. ISBN 978-1-4675-7992-6. Archived from the original on 17 January 2023. Retrieved 7 April 2017.
28. ^ Williams, Matt (28 January 2022). “What is Einstein’s Theory of Relativity?”. Universe Today. Archived
from the original on 3 August 2022. Retrieved 13 August 2022.
29. ^ Stachel, John (2005). “Fresnel’s (Dragging) Coefficient as a Challenge to 19th Century Optics of Moving Bodies.” (PDF). In Kox, A. J.; Eisenstaedt, Jean (eds.). The Universe of
General Relativity. Boston: Birkhäuser. pp. 1–13. ISBN 0-8176-4380-X. Archived from the original (PDF) on 13 April 2017.
30. ^ “George Francis FitzGerald”. The Linda Hall Library. Archived from the original on 17 January 2023. Retrieved 13 August
2022.
31. ^ “The Nobel Prize in Physics 1902”. NobelPrize.org. Archived from the original on 23 June 2017. Retrieved 13 August 2022.
32. ^ Jump up to:a b c d e Pais, Abraham (1982). “”Subtle is the Lord–”: The Science and the Life of Albert Einstein
(11th ed.). Oxford: Oxford University Press. ISBN 0-19-853907-X.
33. ^ Born, Max (1956). Physics in My Generation. London & New York: Pergamon Press. p. 194. Retrieved 10 July 2017.
34. ^ Darrigol, O. (2005), “The Genesis of the theory of relativity”
(PDF), Séminaire Poincaré, 1: 1–22, Bibcode:2006eins.book….1D, doi:10.1007/3-7643-7436-5_1, ISBN 978-3-7643-7435-8, archived (PDF) from the original on 28 February 2008, retrieved 17 July 2017
35. ^ Jump up to:a b c Miller, Arthur I. (1998). Albert
Einstein’s Special Theory of Relativity. New York: Springer-Verlag. ISBN 0-387-94870-8.
36. ^ Jump up to:a b c Galison, Peter (2003). Einstein’s Clocks, Poincaré’s Maps: Empires of Time. New York: W. W. Norton & Company, Inc. pp. 13–47. ISBN
0-393-02001-0.
37. ^ Poincare, Henri (1906). “On the Dynamics of the Electron (Sur la dynamique de l’électron)”. Rendiconti del Circolo Matematico di Palermo. 21: 129–176. Bibcode:1906RCMP…21..129P. doi:10.1007/bf03013466. hdl:2027/uiug.30112063899089.
S2CID 120211823. Archived from the original on 11 July 2017. Retrieved 15 July 2017.
38. ^ Zahar, Elie (1989) [1983], “Poincaré’s Independent Discovery of the relativity principle”, Einstein’s Revolution: A Study in Heuristic, Chicago: Open Court
Publishing Company, ISBN 0-8126-9067-2
39. ^ Jump up to:a b Walter, Scott A. (2007). “Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910”. In Renn, Jürgen; Schemmel, Matthias (eds.). The Genesis of General Relativity,
Volume 3. Berlin: Springer. pp. 193–252. Archived from the original on 15 July 2017. Retrieved 15 July 2017.
40. ^ Einstein, Albert (1905). “On the Electrodynamics of Moving Bodies ( Zur Elektrodynamik bewegter Körper)”. Annalen der Physik. 322
(10): 891–921. Bibcode:1905AnP…322..891E. doi:10.1002/andp.19053221004. Archived from the original on 6 November 2018. Retrieved 7 April 2018.
41. ^ Isaacson, Walter (2007). Einstein: His Life and Universe. Simon & Schuster. pp. 26–27, 122–127,
145–146, 345–349, 448–460. ISBN 978-0-7432-6473-0.
42. ^ Jump up to:a b c d e f g h i j k l m n o p q r s t Schutz, Bernard (2004). Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity (Reprint ed.). Cambridge: Cambridge
University Press. ISBN 0-521-45506-5. Archived from the original on 17 January 2023. Retrieved 24 May 2017.
43. ^ Jump up to:a b Weinstein, Galina (2012). “Max Born, Albert Einstein and Hermann Minkowski’s Space–Time Formalism of Special Relativity”.
arXiv:1210.6929 [physics.hist-ph].
44. ^ Galison, Peter Louis (1979). “Minkowski’s space–time: From visual thinking to the absolute world”. Historical Studies in the Physical Sciences. 10: 85–121. doi:10.2307/27757388. JSTOR 27757388.
45. ^ Jump
up to:a b Minkowski, Hermann (1909). “Raum und Zeit” [Space and Time]. Jahresbericht der Deutschen Mathematiker-Vereinigung. B.G. Teubner: 1–14. Archived from the original on 28 July 2017. Retrieved 17 July 2017.
46. ^ Jump up to:a b Cartan, É.;
Fano, G. (1955) [1915]. “La théorie des groupes continus et la géométrie”. Encyclopédie des Sciences Mathématiques Pures et Appliquées. 3 (1): 39–43. Archived from the original on 23 March 2018. Retrieved 6 April 2018. (Only pages 1–21 were published
in 1915, the entire article including pp. 39–43 concerning the groups of Laguerre and Lorentz was posthumously published in 1955 in Cartan’s collected papers, and was reprinted in the Encyclopédie in 1991.)
47. ^ Kastrup, H. A. (2008). “On the
advancements of conformal transformations and their associated symmetries in geometry and theoretical physics”. Annalen der Physik. 520 (9–10): 631–690. arXiv:0808.2730. Bibcode:2008AnP…520..631K. doi:10.1002/andp.200810324. S2CID 12020510.
48. ^
Ratcliffe, J. G. (1994). “Hyperbolic geometry”. Foundations of Hyperbolic Manifolds. New York. pp. 56–104. ISBN 0-387-94348-X.
49. ^ Curtis, W. D.; Miller, F. R. (1985). Differential Manifolds and Theoretical Physics. Academic Press. p. 223. ISBN
978-0-08-087435-7. Archived from the original on 17 January 2023. Retrieved 16 January 2018.
50. ^ Landau, L. D. Lifshitz E,M. (2013). The classical theory of fields (Vol. 2).
51. ^ Curiel, Erik; Bokulich, Peter. “Lightcones and Causal Structure”.
Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Archived from the original on 17 May 2019. Retrieved 26 March 2017.
52. ^ Savitt, Steven. “Being and Becoming in Modern Physics. 3. The Special Theory of Relativity”.
The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Archived from the original on 11 March 2017. Retrieved 26 March 2017.
53. ^ Jump up to:a b c d e f Schutz, Bernard F. (1985). A first course in general relativity.
Cambridge, UK: Cambridge University Press. p. 26. ISBN 0-521-27703-5.
54. ^ Jump up to:a b c d e f g Weiss, Michael. “The Twin Paradox”. The Physics and Relativity FAQ. Archived from the original on 27 April 2017. Retrieved 10 April 2017.
55. ^
Mould, Richard A. (1994). Basic Relativity (1st ed.). Springer. p. 42. ISBN 978-0-387-95210-9. Retrieved 22 April 2017.
56. ^ Lerner, Lawrence S. (1997). Physics for Scientists and Engineers, Volume 2 (1st ed.). Jones & Bartlett Pub. p. 1047. ISBN
978-0-7637-0460-5. Retrieved 22 April 2017.
57. ^ Jump up to:a b c d e f g h i j k l m n o Bais, Sander (2007). Very Special Relativity: An Illustrated Guide. Cambridge, Massachusetts: Harvard University Press. ISBN 978-0-674-02611-7.
58. ^ Forshaw,
Jeffrey; Smith, Gavin (2014). Dynamics and Relativity. John Wiley & Sons. p. 118. ISBN 978-1-118-93329-9. Retrieved 24 April 2017.
59. ^ Jump up to:a b c d e f g h i j k l m n o p q Morin, David (2017). Special Relativity for the Enthusiastic Beginner.
CreateSpace Independent Publishing Platform. ISBN 978-1-5423-2351-2.
60. ^ Landau, L. D.; Lifshitz, E. M. (2006). The Classical Theory of Fields, Course of Theoretical Physics, Volume 2 (4th ed.). Amsterdam: Elsevier. pp. 1–24. ISBN 978-0-7506-2768-9.
61. ^
Jump up to:a b Morin, David (2008). Introduction to Classical Mechanics: With Problems and Solutions. Cambridge University Press. ISBN 978-0-521-87622-3.
62. ^ Rose, H. H. (21 April 2008). “Optics of high-performance electron microscopes”. Science
and Technology of Advanced Materials. 9 (1): 014107. Bibcode:2008STAdM…9a4107R. doi:10.1088/0031-8949/9/1/014107. PMC 5099802. PMID 27877933.
63. ^ Griffiths, David J. (2013). Revolutions in Twentieth-Century Physics. Cambridge: Cambridge University
Press. p. 60. ISBN 978-1-107-60217-5. Retrieved 24 May 2017.
64. ^ Byers, Nina (1998). “E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws”. arXiv:physics/9807044.
65. ^ Nave, R. “Energetics of Charged Pion
Decay”. Hyperphysics. Department of Physics and Astronomy, Georgia State University. Archived from the original on 21 May 2017. Retrieved 27 May 2017.
66. ^ Thomas, George B.; Weir, Maurice D.; Hass, Joel; Giordano, Frank R. (2008). Thomas’ Calculus:
Early Transcendentals (Eleventh ed.). Boston: Pearson Education, Inc. p. 533. ISBN 978-0-321-49575-4.
67. ^ Taylor, Edwin F.; Wheeler, John Archibald (1992). Spacetime Physics (2nd ed.). W. H. Freeman. ISBN 0-7167-2327-1.
68. ^ Jump up to:a b
Gibbs, Philip. “Can Special Relativity Handle Acceleration?”. The Physics and Relativity FAQ. math.ucr.edu. Archived from the original on 7 June 2017. Retrieved 28 May 2017.
69. ^ Franklin, Jerrold (2010). “Lorentz contraction, Bell’s spaceships,
and rigid body motion in special relativity”. European Journal of Physics. 31 (2): 291–298. arXiv:0906.1919. Bibcode:2010EJPh…31..291F. doi:10.1088/0143-0807/31/2/006. S2CID 18059490.
70. ^ Lorentz, H. A.; Einstein, A.; Minkowski, H.; Weyl, H.
(1952). The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity. Dover Publications. ISBN 0-486-60081-5.
71. ^ Jump up to:a b c Mook, Delo E.; Vargish, Thoma s (1987). Inside Relativity. Princeton,
New Jersey: Princeton University Press. ISBN 0-691-08472-6.
72. ^ Mester, John. “Experimental Tests of General Relativity” (PDF). Laboratoire Univers et Théories. Archived from the original (PDF) on 18 March 2017. Retrieved 9 June 2017.
73. ^
Jump up to:a b Carroll, Sean M. (2 December 1997). “Lecture Notes on General Relativity”. arXiv:gr-qc/9712019.
74. ^ Le Verrier, Urbain (1859). “Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette
planète”. Comptes rendus hebdomadaires des séances de l’Académie des Sciences. 49: 379–383.
75. ^ Worrall, Simon (4 November 2015). “The Hunt for Vulcan, the Planet That Wasn’t There”. National Geographic. Archived from the original on 24 May
2017.
76. ^ Levine, Alaina G. (May 2016). “May 29, 1919: Eddington Observes Solar Eclipse to Test General Relativity”. This Month in Physics History. APS News. American Physical Society. Archived from the original on 2 June 2017.
77. ^ Hobson,
M. P.; Efstathiou, G.; Lasenby, A. N. (2006). General Relativity. Cambridge: Cambridge University Press. pp. 176–179. ISBN 978-0-521-82951-9.
78. ^ Thorne, Kip S. (1988). Fairbank, J. D.; Deaver, B. S. Jr.; Everitt, W. F.; Michelson, P. F. (eds.).
Near zero: New Frontiers of Physics (PDF). W. H. Freeman and Company. pp. 573–586. S2CID 12925169. Archived from the original (PDF) on 28 July 2017.
79. ^ Feynman, R. P.; Leighton, R. B.; Sands, M. (1964). The Feynman Lectures on Physics, vol. 2
(New Millenium ed.). Basic Books. pp. 13–6 to 13–11. ISBN 978-0-465-02416-2. Archived from the original on 17 January 2023. Retrieved 1 July 2017.
80. ^ Williams, R. K. (1995). “Extracting X rays, Ύ rays, and relativistic e−–e+ pairs from supermassive
Kerr black holes using the Penrose mechanism”. Physical Review D. 51 (10): 5387–5427. Bibcode:1995PhRvD..51.5387W. doi:10.1103/PhysRevD.51.5387. PMID 10018300.
81. ^ Williams, R. K. (2004). “Collimated escaping vortical polar e−–e+ jets intrinsically
produced by rotating black holes and Penrose processes”. The Astrophysical Journal. 611 (2): 952–963. arXiv:astro-ph/0404135. Bibcode:2004ApJ…611..952W. doi:10.1086/422304. S2CID 1350543.
82. ^ Kuroda, Takami; Kotake, Kei; Takiwaki, Tomoya (2012).
“Fully General Relativistic Simulations of Core-Collapse Supernovae with An Approximate Neutrino Transport”. The Astrophysical Journal. 755 (1): 11. arXiv:1202.2487. Bibcode:2012ApJ…755…11K. doi:10.1088/0004-637X/755/1/11. S2CID 119179339.
83. ^
Wollack, Edward J. (10 December 2010). “Cosmology: The Study of the Universe”. Universe 101: Big Bang Theory. NASA. Archived from the original on 14 May 2011. Retrieved 15 April 2017.
84. ^ Jump up to:a b Bondi, Hermann (1957). DeWitt, Cecile M.;
Rickles, Dean (eds.). The Role of Gravitation in Physics: Report from the 1957 Chapel Hill Conference. Berlin: Max Planck Research Library. pp. 159–162. ISBN 978-3-86931-963-6. Archived from the original on 28 July 2017. Retrieved 1 July 2017.
85. ^
Crowell, Benjamin (2000). General Relativity. Fullerton, CA: Light and Matter. pp. 241–258. Archived from the original on 18 June 2017. Retrieved 30 June 2017.
86. ^ Kreuzer, L. B. (1968). “Experimental measurement of the equivalence of active and
passive gravitational mass”. Physical Review. 169 (5): 1007–1011. Bibcode:1968PhRv..169.1007K. doi:10.1103/PhysRev.169.1007.
87. ^ Will, C. M. (1976). “Active mass in relativistic gravity-Theoretical interpretation of the Kreuzer experiment”. The
Astrophysical Journal. 204: 224–234. Bibcode:1976ApJ…204..224W. doi:10.1086/154164. Archived from the original on 28 September 2018. Retrieved 2 July 2017.
88. ^ Bartlett, D. F.; Van Buren, Dave (1986). “Equivalence of active and passive gravitational
mass using the moon”. Phys. Rev. Lett. 57 (1): 21–24. Bibcode:1986PhRvL..57…21B. doi:10.1103/PhysRevLett.57.21. PMID 10033347.
89. ^ “Gravity Probe B: FAQ”. Archived from the original on 2 June 2018. Retrieved 2 July 2017.
90. ^ Gugliotta, G.
(16 February 2009). “Perseverance Is Paying Off for a Test of Relativity in Space”. New York Times. Archived from the original on 3 September 2018. Retrieved 2 July 2017.
91. ^ Everitt, C.W.F.; Parkinson, B.W. (2009). “Gravity Probe B Science Results—NASA
Final Report” (PDF). Archived (PDF) from the original on 23 October 2012. Retrieved 2 July 2017.
92. ^ Everitt; et al. (2011). “Gravity Probe B: Final Results of a Space Experiment to Test General Relativity”. Physical Review Letters. 106 (22):
221101. arXiv:1105.3456. Bibcode:2011PhRvL.106v1101E. doi:10.1103/PhysRevLett.106.221101. PMID 21702590. S2CID 11878715.
93. ^ Ciufolini, Ignazio; Paolozzi, Antonio Rolf Koenig; Pavlis, Erricos C.; Koenig, Rolf (2016). “A test of general relativity
using the LARES and LAGEOS satellites and a GRACE Earth gravity model”. Eur Phys J C. 76 (3): 120. arXiv:1603.09674. Bibcode:2016EPJC…76..120C. doi:10.1140/epjc/s10052-016-3961-8. PMC 4946852. PMID 27471430.
94. ^ Iorio, L. (February 2017). “A
comment on “A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model. Measurement of Earth’s dragging of inertial frames,” by I. Ciufolini et al”. The European Physical Journal C. 77 (2): 73. arXiv:1701.06474.
Bibcode:2017EPJC…77…73I. doi:10.1140/epjc/s10052-017-4607-1. S2CID 118945777.
95. ^ Cartlidge, Edwin (20 January 2016). “Underground ring lasers will put general relativity to the test”. physicsworld.com. Institute of Physics. Archived from
the original on 12 July 2017. Retrieved 2 July 2017.
96. ^ “Einstein right using the most sensitive Earth rotation sensors ever made”. Phys.org. Science X network. Archived from the original on 10 May 2017. Retrieved 2 July 2017.
97. ^ Murzi,
Mauro. “Jules Henri Poincaré (1854–1912)”. Internet Encyclopedia of Philosophy (ISSN 2161-0002). Archived from the original on 23 December 2020. Retrieved 9 April 2018.
98. ^ Deser, S. (1970). “Self-Interaction and Gauge Invariance”. General Relativity
and Gravitation. 1 (18): 9–8. arXiv:gr-qc/0411023. Bibcode:1970GReGr…1….9D. doi:10.1007/BF00759198. S2CID 14295121.
99. ^ Grishchuk, L. P.; Petrov, A. N.; Popova, A. D. (1984). “Exact Theory of the (Einstein) Gravitational Field in an Arbitrary
Background Space–Time”. Communications in Mathematical Physics. 94 (3): 379–396. Bibcode:1984CMaPh..94..379G. doi:10.1007/BF01224832. S2CID 120021772. Archived from the original on 25 February 2021. Retrieved 9 April 2018.
100. ^ Rosen, N. (1940).
“General Relativity and Flat Space I”. Physical Review. 57 (2): 147–150. Bibcode:1940PhRv…57..147R. doi:10.1103/PhysRev.57.147.
101. ^ Weinberg, S. (1964). “Derivation of Gauge Invariance and the Equivalence Principle from Lorentz Invariance of
the S-Matrix”. Physics Letters. 9 (4): 357–359. Bibcode:1964PhL…..9..357W. doi:10.1016/0031-9163(64)90396-8.
102. ^ Jump up to:a b Thorne, Kip (1995). Black Holes & Time Warps: Einstein’s Outrageous Legacy. W. W. Norton & Company. ISBN 978-0-393-31276-8.
103. ^
Bondi, H.; Van der Burg, M.G.J.; Metzner, A. (1962). “Gravitational waves in general relativity: VII. Waves from axisymmetric isolated systems”. Proceedings of the Royal Society of London A. A269 (1336): 21–52. Bibcode:1962RSPSA.269…21B. doi:10.1098/rspa.1962.0161.
S2CID 120125096.
104. ^ Sachs, R. (1962). “Asymptotic symmetries in gravitational theory”. Physical Review. 128 (6): 2851–2864. Bibcode:1962PhRv..128.2851S. doi:10.1103/PhysRev.128.2851.
105. ^ Strominger, Andrew (2017). “Lectures on the Infrared
Structure of Gravity and Gauge Theory”. arXiv:1703.05448 [hep-th]. …redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft
theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. To be published Princeton University Press, 158 pages.
106. ^ maths.tcd.ie
107. ^ Jump up to:a
b c d Bär, Christian; Fredenhagen, Klaus (2009). “Lorentzian Manifolds” (PDF). Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations. Dordrecht: Springer. pp. 39–58. ISBN 978-3-642-02779-6. Archived from the original (PDF)
on 13 April 2017. Retrieved 14 April 2017.
108. ^ Skow, Bradford (2007). “What makes time different from space?” (PDF). Noûs. 41 (2): 227–252. CiteSeerX 10.1.1.404.7853. doi:10.1111/j.1468-0068.2007.00645.x. Archived from the original (PDF) on 24
August 2016. Retrieved 13 April 2018.
109. ^ Leibniz, Gottfried (1880). “Discourse on metaphysics”. Die philosophischen schriften von Gottfried Wilhelm Leibniz. Vol. 4. Weidmann. pp. 427–463. Retrieved 13 April 2018.
110. ^ Ehrenfest, Paul (1920).
“How do the fundamental laws of physics make manifest that space has 3 dimensions?”. Annalen der physik. 61 (5): 440–446. Bibcode:1920AnP…366..440E. doi:10.1002/andp.19203660503.. Also see Ehrenfest, P. (1917) “In what way does it become manifest
in the fundamental laws of physics that space has three dimensions?” Proceedings of the Amsterdam academy 20:200.
111. ^ Weyl, H. (1922). Space, time, and matter. Dover reprint: 284.
112. ^ Tangherlini, F. R. (1963). “Atoms in higher dimensions”.
Nuovo Cimento. 14 (27): 636. doi:10.1007/BF02784569. S2CID 119683293.
113. ^ Jump up to:a b c Tegmark, Max (April 1997). “On the dimensionality of spacetime” (PDF). Classical and Quantum Gravity. 14 (4): L69–L75. arXiv:gr-qc/9702052. Bibcode:1997CQGra..14L..69T.
doi:10.1088/0264-9381/14/4/002. S2CID 15694111. Retrieved 16 December 2006.
114. ^ Feng, W.X. (3 August 2022). “Gravothermal phase transition, black holes and space dimensionality”. Physical Review D. 106 (4): L041501. arXiv:2207.14317. Bibcode:2022PhRvD.106d1501F.
doi:10.1103/PhysRevD.106.L041501. S2CID 251196731.
115. ^ Scargill, J. H. C. (26 February 2020). “Existence of life in 2 + 1 dimensions”. Physical Review Research. 2 (1): 013217. arXiv:1906.05336. Bibcode:2020PhRvR…2a3217S. doi:10.1103/PhysRevResearch.2.013217.
S2CID 211734117.
116. ^ “Life could exist in a 2D universe (according to physics, anyway)”. technologyreview.com. Retrieved 16 June 2021.
Photo credit: https://www.flickr.com/photos/jms2/14103897845/’]