Editing Spacetime (section)

== Technical topics ==

=== Is spacetime really curved? ===
In Poincaré's [[conventionalist]] views, the essential criteria according to which one should select a Euclidean versus non-Euclidean geometry would be economy and simplicity. A realist would say that Einstein discovered spacetime to be non-Euclidean. A conventionalist would say that Einstein merely found it ''more convenient'' to use non-Euclidean geometry. The conventionalist would maintain that Einstein's analysis said nothing about what the geometry of spacetime ''really'' is.<ref name="Murzi">{{cite web|last1=Murzi|first1=Mauro|title=Jules Henri Poincaré (1854–1912)|url=http://www.iep.utm.edu/poincare/#H4|publisher=Internet Encyclopedia of Philosophy (ISSN 2161-0002)|access-date=9 April 2018|archive-date=23 December 2020|archive-url=https://web.archive.org/web/20201223123326/http://www.iep.utm.edu/poincare/#H4|url-status=live}}</ref>

Such being said,
:# Is it possible to represent general relativity in terms of flat spacetime?
:# Are there any situations where a flat spacetime interpretation of general relativity may be ''more convenient'' than the usual curved spacetime interpretation?

In response to the first question, a number of authors including Deser, Grishchuk, Rosen, Weinberg, etc. have provided various formulations of gravitation as a field in a flat manifold. Those theories are variously called "[[bimetric gravity]]", the "field-theoretical approach to general relativity", and so forth.<ref name="Deser1970">{{cite journal|last1=Deser|first1=S.|title=Self-Interaction and Gauge Invariance|journal=General Relativity and Gravitation|date=1970|volume=1|issue=18|pages=9–8|arxiv=gr-qc/0411023|bibcode=1970GReGr...1....9D|doi=10.1007/BF00759198|s2cid=14295121}}</ref><ref name="Grishchuk1984">{{cite journal|last1=Grishchuk|first1=L. P.|last2=Petrov|first2=A. N.|last3=Popova|first3=A. D.|title=Exact Theory of the (Einstein) Gravitational Field in an Arbitrary Background Space–Time|journal=Communications in Mathematical Physics|date=1984|volume=94|issue=3|pages=379–396|url=https://projecteuclid.org/download/pdf_1/euclid.cmp/1103941358|access-date=9 April 2018|bibcode=1984CMaPh..94..379G|doi=10.1007/BF01224832|s2cid=120021772|archive-date=25 February 2021|archive-url=https://web.archive.org/web/20210225061922/https://projecteuclid.org/download/pdf_1/euclid.cmp/1103941358|url-status=live}}</ref><ref name="Rosen1940">{{cite journal|last1=Rosen|first1=N.|title=General Relativity and Flat Space I|journal=Physical Review|date=1940|volume=57|issue=2|pages=147–150|doi=10.1103/PhysRev.57.147|bibcode=1940PhRv...57..147R}}</ref><ref name="Weinberg1964">{{cite journal|last1=Weinberg|first1=S.|title=Derivation of Gauge Invariance and the Equivalence Principle from Lorentz Invariance of the S-Matrix|journal=Physics Letters|date=1964|volume=9|issue=4|pages=357–359|doi=10.1016/0031-9163(64)90396-8|bibcode=1964PhL.....9..357W}}</ref> Kip Thorne has provided a popular review of these theories.<ref name="Thorne1995">{{cite book|last1=Thorne|first1=Kip|title=Black Holes & Time Warps: Einstein's Outrageous Legacy|date=1995|publisher=W. W. Norton & Company|isbn=978-0-393-31276-8}}</ref>{{rp|397–403}}

The flat spacetime paradigm posits that matter creates a gravitational field that causes rulers to shrink when they are turned from circumferential orientation to radial, and that causes the ticking rates of clocks to dilate. The flat spacetime paradigm is fully equivalent to the curved spacetime paradigm in that they both represent the same physical phenomena. However, their mathematical formulations are entirely different. Working physicists routinely switch between using curved and flat spacetime techniques depending on the requirements of the problem. The flat spacetime paradigm is convenient when performing approximate calculations in weak fields. Hence, flat spacetime techniques tend be used when solving gravitational wave problems, while curved spacetime techniques tend be used in the analysis of black holes.<ref name="Thorne1995" />{{rp|397–403}}

=== Asymptotic symmetries ===

{{Main|Bondi–Metzner–Sachs group}}
The spacetime symmetry group for [[Special Relativity]] is the [[Poincaré group]], which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries if any might apply in [[General Relativity]]. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity, ''viz.'', the Poincaré group.

In 1962 [[Hermann Bondi]], M. G. van der Burg, A. W. Metzner<ref name="bondi etal 1962">{{cite journal |last1=Bondi |first1=H. |last2=Van der Burg |first2=M. G. J. |last3=Metzner |first3=A. |year=1962 |title=Gravitational waves in general relativity: VII. Waves from axisymmetric isolated systems |journal=Proceedings of the Royal Society of London A |volume=A269 |issue=1336 |pages=21–52 |bibcode=1962RSPSA.269...21B |doi=10.1098/rspa.1962.0161 |s2cid=120125096}}</ref> and [[Rainer K. Sachs]]<ref name="sachs1962">{{cite journal |last1=Sachs |first1=Rainer K. |year=1962 |title=Asymptotic symmetries in gravitational theory |journal=Physical Review |volume=128 |issue=6 |pages=2851–2864 |bibcode=1962PhRv..128.2851S |doi=10.1103/PhysRev.128.2851}}</ref> addressed this [[Bondi–Metzner–Sachs group|asymptotic symmetry]] problem in order to investigate the flow of energy at infinity due to propagating [[gravitational wave]]s. Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at lightlike infinity to characterize what it means to say a metric is asymptotically flat, making no ''a priori'' assumptions about the nature of the asymptotic symmetry group—not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields.<ref name=strominger2017>{{cite arXiv|title=Lectures on the Infrared Structure of Gravity and Gauge Theory|eprint=1703.05448|year=2017|last1=Strominger|first1=Andrew|class=hep-th |quote=...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.}}</ref>{{rp|35}}

What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as ''supertranslations''. This implies the conclusion that General Relativity (GR) does ''not'' reduce to special relativity in the case of weak fields at long distances.<ref name=strominger2017/>{{rp|35}}

{{anchor|Riemannian geometry}}

=== Riemannian geometry ===
{{Excerpt|Riemannian geometry|template=-General geometry}}

{{anchor|Curved manifolds}}

=== Curved manifolds ===
{{Main|Manifold|Lorentzian manifold|Differentiable manifold}}

For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected [[Lorentzian manifold]] <math>(M, g)</math>. This means the smooth [[Lorentz metric]] <math>g</math> has [[metric signature|signature]] <math>(3,1)</math>. The metric determines the ''{{vanchor|geometry of spacetime|SPACETIME_GEOMETRY}}'', as well as determining the [[geodesic]]s of particles and light beams. About each point (event) on this manifold, [[coordinate charts]] are used to represent observers in reference frames. Usually, Cartesian coordinates <math>(x, y, z, t)</math> are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light <math>c</math> is equal to 1.<ref name="Pfaffle">{{cite book|last1=Bär|first1=Christian|last2=Fredenhagen|first2=Klaus|title=Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations|date=2009|publisher=Springer|location=Dordrecht|isbn=978-3-642-02779-6|pages=39–58|chapter-url=https://www.springer.com/cda/content/document/cda_downloaddocument/9783642027796-c1.pdf?SGWID=0-0-45-800045-p173910618|access-date=14 April 2017|archive-url=https://web.archive.org/web/20170415201236/http://www.springer.com/cda/content/document/cda_downloaddocument/9783642027796-c1.pdf?SGWID=0-0-45-800045-p173910618|archive-date=15 April 2017|chapter=Lorentzian Manifolds|url-status=dead}}</ref>

A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event <math>p</math>. Another reference frame may be identified by a second coordinate chart about <math>p</math>. Two observers (one in each reference frame) may describe the same event <math>p</math> but obtain different descriptions.<ref name="Pfaffle" />

Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing <math>p</math> (representing an observer) and another containing <math>q</math> (representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a [[Singularity (mathematics)|non-singular]] coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.<ref name="Pfaffle" />

For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event <math>p</math>). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples <math>(x, y, z, t)</math> (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces [[tensors]] into relativity, by which all physical quantities are represented.

Geodesics are said to be timelike, null, or spacelike if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by timelike and null (lightlike) geodesics, respectively.<ref name="Pfaffle" />

{{anchor|Privileged character of 3+1 spacetime}}

=== Privileged character of 3+1 spacetime ===
{{Excerpt|Anthropic principle|Dimensions of spacetime}}