One of my favorite topics is spacetime and the effects of mass and gravity on light propagation and gravitational lensing. Here are my latest two cents on this fascinating topic.
In 1907, Einstein's former professor, Hermann Minkowski*, made a brilliant breakthrough: he showed that you could conceive of space and time in a single formulation. All at once, he had developed the formalism of spacetime. This provided a stage for particles to move through the Universe (relative to one another) and interact with one another, but it didn't include gravity. The spacetime he had developed -- still today known as
Minkowski space -- describes all of special relativity, and also provides the backdrop for the vast majority of the quantum field theory calculations.
If there were no such thing as the gravitational force, Minkowski spacetime would do everything needed. Spacetime would be simple, uncurved, and would simply provide a stage for matter to move through and interact. The only way you'd ever accelerate would be through an interaction with another particle.
But in our Universe, we have the gravitational force, and it was Einstein's principle of equivalence that explained that so long as you can't see what's accelerating you, gravitation treats you the same as any other acceleration.
It was this revelation, and the development to link this, mathematically, to the Minkowskian concept of spacetime, that led to general relativity. The major difference between special relativity's Minkowski space and the curved space that appears in general relativity is the mathematical formalism known as the
Metric Tensor, sometimes called Einstein's Metric Tensor or the Riemann Metric. Bernhard Riemann** was a pure mathematician in the 19th century (and a former student of Gauss, perhaps the greatest mathematician of them all), and he gave a formalism for how any fields, lines, arcs, distances, etc., can exist and be well-defined in an arbitrarily curved space of any number of dimensions. It took Einstein (and a number of collaborators) nearly a decade to cope with the complexities of the math, but when all was said and done, we had general relativity: a theory that described our three-space-and-one-time dimensional Universe, where gravitation existed.
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Conceptually, the metric tensor defines how much spacetime is curved. Its curvature is directly proportional to the total amount of matter, energy and stresses present within it; the contents of our Universe define its spacetime curvature. By the same token, how our Universe is curved tells us how the matter and energy is going to move through it. Imagine that an object in motion will continue in motion: Newton's first law. Then visualize that as a straight line, but what curved space tells us is that instead an object in motion continuing in motion follows a
geodesic, which is a particularly-curved line that corresponds to unaccelerated motion. Ironically, it's a geodesic, not necessarily a straight line, that is the shortest distance between two points because there are no straight lines as the objects move from gravity well to gravity well. This shows up even on cosmic scales, where the curved spacetime due to the presence of extraordinary masses can curve the background light from behind it, sometimes into multiple images. In the latter cases we have gravitational lenses, see below.
lens1.jpg
Physically, there are a number of different pieces that contribute to the Metric Tensor in general relativity. Think of gravity related to masses: the locations and magnitudes of different masses determine the gravitational force. In general relativity, this corresponds to the mass density and does contribute, but it's one of only 16 components of the Metric Tensor! There are also pressure components (such as radiation pressure, vacuum pressure or pressures created by fast-moving particles) that contribute, which are three additional contributors (one for each of the three spatial directions) to the Metric Tensor. And finally, there are six other components that tell us how volumes change and deform in the presence of masses and tidal forces, along with how the shape of a moving body is distorted by those forces. This applies to everything from a planet like Earth to neutron stars orbiting each other which create massless waves moving through space: gravitational radiation, which has now been found by the Laser Interferometer Gravitational-wave Observatory (LIGO)***.
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The Metric Tensor may be a 4 × 4 entity, but it's symmetric, meaning that there are four "diagonal" components (the density and the pressure components), and six off-diagonal components (the volume/deformation components) that are independent; the other six off-diagonal components are then uniquely determined by symmetry. The metric tells us the relationship between all the matter/energy in the Universe and the curvature of spacetime itself.
The expanding Universe's emergence from the Big Bang and the dark energy-domination that will lead to a cold, empty fate are all only understandable in the context of general relativity, and that means understanding the relationship between matter/energy and spacetime.
The Universe is a play, unfolding every time a particle interacts with another, spacetime is the stage on which it all plays out and we are but fleeting actors on that same stage. The stage isn't a constant backdrop, but it, too, evolves along with the Universe itself. And wherever it is, it evolves along world lines.
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The set of all light speed world lines going through an event defines the light cones of that event: the past light cone and the future light cone. An example of light cones is shown above. Each of us is that observer above.
Through any event in space-time, in any given direction, there is only one world line corresponding to motion solely influenced by gravity.
Advanced coordinates of a point x relative to a world line γ. The advanced time v selects a particular light cone, the unit vector Ωa := xˆa/r selects a particular generator of this light cone, and the advanced distance r selects a particular point on this generator.
See:
https://arxiv.org/pdf/gr-qc/0606093.pdf
See:
https://www.forbes.com/sites/startswithabang/2017/01/28/ask-ethan-what-is-spacetime/?sh=2008633250bd
See:
https://www.forbes.com/sites/startswithabang/2017/01/28/ask-ethan-what-is-spacetime/?sh=2008633250bd
Space and time in Einstein's universe are no longer flat (as implicitly assumed by Newton, where gravity and light act instantly at all distances.) but can pushed and pulled, stretched and warped by matter. Gravity feels strongest where spacetime is most curved, and it vanishes where spacetime is flat. This is the core of Einstein's theory of general relativity, which is often summed up in words as follows:
"matter tells spacetime how to curve, and curved spacetime tells matter how to move
". A standard way to illustrate this idea is to place a bowling ball (representing a massive object such as the sun) onto a stretched rubber sheet (representing spacetime). If a marble is placed onto the rubber sheet, it will roll toward the bowling ball, and may even be put into "orbit" around the bowling ball. This occurs, not because the smaller mass is "attracted" by a force emanating from the larger one, but because it is traveling along a surface which has been deformed by the presence of the larger mass. In the same way gravitation in Einstein's theory arises not as a force propagating
through spacetime, but rather as a feature of spacetime itself.
Matter can act on spacetime in a manner that is very much in the spirit of Mach's principle. Calculations by Hans Thirring (1888-1979), Josef Lense (1890-1985) and others have shown that a large rotating mass will "drag" an observer's inertial reference frame around with it. This is the phenomenon of frame-dragging, whose existence Gravity Probe B detected****. The same calculations suggest that, if the entire contents of the universe were to rotate, our local inertial frame would undergo "perfect dragging" — that is, we would not notice it, because we would be rotating too! In that sense, general relativity is indeed nearly as relational as Ernst Mach***** might have wished. Some physicists (such as Julian Barbour) have gone further and asserted that general relativity is in fact perfectly Machian. If one goes beyond classical physics and into modern quantum field theory, then questions of absolute versus relational spacetime are rendered anachronistic by the fact that even "empty space" is populated by matter in the form of virtual particles, zero-point fields and more. Within the context of Einstein's universe, however, the majority view is perhaps best summed up as follows: spacetime behaves relationally but exists absolutely.
Hartmann352
* Hermann Minkowski (born June 22, 1864, Aleksotas,
Russian Empire [now in
Kaunas, Lithuania]—died Jan. 12, 1909, Göttingen, Germany), German mathematician who developed the geometrical theory of numbers and who made numerous contributions to
number theory,
mathematical physics, and the theory of relativity. His idea of combining the three dimensions of physical
space with that of time into a four-dimensional “Minkowski space”—
space-time—laid the mathematical foundations for
Albert Einstein’s
special theory of relativity.
The son of German parents living in Russia, Minkowski returned to
Germany with them in 1872 and spent his youth in the royal Prussian city of Königsberg. A gifted prodigy, he began his studies at the
University of Königsberg and the
University of Berlin at age 15. Three years later he was awarded the “Grand Prix des Sciences Mathématiques” by the
French Academy of Sciences for his paper on the representation of numbers as a sum of five squares. During his teenage years in Königsberg he met and befriended another young mathematical prodigy,
David Hilbert, with whom he worked closely both at Königsberg and later at the
University of Göttingen.
After earning his doctorate in 1885, Minkowski taught
mathematics at the Universities of
Bonn (1885–94), Königsberg (1894–96),
Zürich(1896–1902), and Göttingen (1902–09). Together with Hilbert, he pursued research on the electron theory of the Dutch physicist
Hendrik Lorentz and its modification in Einstein’s special theory of relativity. In
Raum und Zeit (1907; “Space and Time”) Minkowski gave his famous four-dimensional geometry based on the
group of
Lorentz transformations of special relativity theory. His major work in number theory was
Geometrie der Zahlen (1896; “Geometry of Numbers”). His works were collected in David Hilbert (ed.),
Gesammelte Abhandlungen, 2 vol. (1911; “Collected Papers”).
See:
https://www.britannica.com/biography/Hermann-Minkowski
** Bernhard Riemann, in full Georg Friedrich Bernhard Riemann, (born September 17, 1826, Breselenz,
Hanover [Germany]—died July 20, 1866, Selasca, Italy), German mathematician whose profound and novel approaches to the study of
geometry laid the mathematical foundation for
Albert Einstein’s theory of
relativity. He also made important contributions to the theory of functions,
complex analysis, and
number theory.
Riemann was born into a poor Lutheran pastor’s family, and all his life he was a shy and introverted person. He was fortunate to have a schoolteacher who recognized his rare mathematical ability and lent him advanced books to read, including
Adrien-Marie Legendre’s
Number Theory (1830). Riemann read the book in a week and then claimed to know it by heart. He went on to study
mathematics at the
University of Göttingen in 1846–47 and 1849–51 and at the University of Berlin (now the
Humboldt University of Berlin) in 1847–49. He then gradually worked his way up the academic profession, through a succession of poorly paid jobs, until he became a full professor in 1859 and gained, for the first time in his life, a measure of financial security. However, in 1862, shortly after his marriage to Elise Koch, Riemann fell seriously ill with
tuberculosis. Repeated trips to
Italy failed to stem the progress of the disease, and he died in Italy in 1866.
Riemann’s visits to Italy were important for the growth of modern mathematics there;
Enrico Betti in particular took up the study of Riemannian ideas. Ill health prevented Riemann from publishing all his work, and some of his best was published only posthumously—e.g., the first edition of Riemann’s
Gesammelte mathematische Werke (1876; “Collected Mathematical Works”), edited by
Richard Dedekind and Heinrich Weber.
See:
https://www.britannica.com/biography/Bernhard-Riemann
*** Laser Interferometer Gravitational-wave Observatory (LIGO) has made the first direct observation of gravitational waves with an instrument on Earth. The researchers detected the gravitational waves on September 14, 2015, at 5:51 a.m. EDT, using the twin LIGO interferometers, located in Livingston, Louisiana and Hanford, Washington.
See:
https://news.mit.edu/2016/ligo-first-detection-gravitational-waves-0211
**** Gravity Probe B; See:
https://www.nasa.gov/mission_pages/gpb/gpb_results.html
***** Ernst Mach (born February 18, 1838, Chirlitz-Turas,
Moravia, Austrian Empire [now Brno-Chrlice, Czech Republic]—died February 19, 1916, Haar, Germany), Austrian physicist and philosopher who established important principles of
optics,
mechanics, and
wave dynamics and who supported the view that all knowledge is a
conceptual organization of the data of sensory experience (or observation).
Mach was educated at home until the age of 14, then went briefly to gymnasium (high school) before entering the
University of Vienna at 17. He received his doctorate in physics in 1860 and taught mechanics and
physics in
Vienna until 1864, when he became professor of
mathematicsat the University of Graz. Mach’s interests had already begun to turn to the
psychology and
physiology of
sensation, although he continued to identify himself as a physicist and to conduct physical research throughout his career. During the 1860s he discovered the physiological phenomenon that has come to be called Mach’s bands, the tendency of the
human eye to see bright or dark bands near the boundaries between areas of sharply differing illumination.
Mach left Graz to become professor of experimental physics at the
Charles University in
Prague in 1867, remaining there for the next 28 years. There he conducted studies on kinesthetic sensation, the feeling associated with
movement and acceleration. Between 1873 and 1893 he developed optical and
photographic techniques for the
measurement of
sound waves and
wave propagation. In 1887 he established the principles of supersonics and the
Mach number—the ratio of the velocity of an object to the velocity of sound.
In
Beiträge zur Analyse der Empfindungen (1886;
Contributions to the Analysis of the Sensations, 1897), Mach advanced the concept that all knowledge is derived from sensation; thus, phenomena under scientific investigation can be understood only in terms of experiences, or “sensations,” present in the observation of the phenomena. This view leads to the position that no statement in natural
science is admissible unless it is empirically verifiable. Mach’s exceptionally rigorous
criteriaof verifiability led him to reject such
metaphysical concepts as absolute time and space, and prepared the way for the
Einstein relativity theory.
Mach also proposed the physical principle, known as
Mach’s principle, that
inertia (the tendency of a body at rest to remain at rest and of a body in motion to continue in motion in the same direction) results from a relationship of that object with all the rest of the matter in the
universe. Inertia, Mach argued, applies only as a function of the interaction between one body and other bodies in the universe, even at enormous distances. Mach’s inertial theories also were cited by Einstein as one of the inspirations for his theories of relativity.
Mach returned to the University of Vienna as professor of inductive
philosophy in 1895, but he suffered a stroke two years later and retired from active research in 1901, when he was appointed to the Austrian parliament. He continued to lecture and write in retirement, publishing
Erkenntnis und Irrtum (“Knowledge and Error”) in 1905 and an autobiography in 1910.
See:
https://www.britannica.com/biography/Ernst-Mach