"Starting with Einstein’s general relativity, differential geometry has started playing a major role in physics. General relativity describes the gravitational fields as a metric property of the spacetime manifold. More precisely, spacetime (i.e., the manifold the points of which are events; we may intuitively say that an event is ‘something that happens in a given point in space at a certain time’) is supposed to be endowed with a Lorentzian metric. This means that spacetime has pointwise the same structure as the Minkowski space of special relativity but in general is not flat, as on the contrary Minkowski space is. Indeed, out of the metric tensor one can construct another tensor field, the curvature field, which measures how far the geometry of spacetime is from that of a flat space. The celebrated Einstein equations prescribe how the matter in our universe determines the curvature of spacetime, and in turn the curvature determines how matter (particles, light rays, extended bodies…) moves." (Claudio Bartocci & Ugo Bruzzo, [Claudio Bartocci et al (Eds), "New Trends in Geometry: Their tole in the natural and social sciences"], 2011)
"The Newtonian universe is material in the sense that the world was viewed as being made up of stuff - tangible, real objects. It was argued that even forces like gravity that appear to act across empty stretches of space are conveyed by tiny particles, or corpuscles. Moreover, since the universe is material, its behavior can be predicted or understood. Things are they way they are for a reason or a cause. The Newtonian world is mathematical, in that it was viewed that the regularities or laws that describe or govern the world are mathematical in nature." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"[…] there’s atomic physics - electrons and protons and neutrons, all the stuff of which atoms are made. At these very, very, very small scales, the laws of physics are much the same, but there is also a force you ignore, which is the gravitational force. Gravity is present everywhere because it comes from the entire mass of the universe. It doesn’t cancel itself out, it doesn’t have positive or negative value, it all adds up." (Michael F Atiyah, [interview] 2013)
"The most familiar manifold, however, is the space-time manifold, which has 4 dimensions. It is described by a time coordinate and three spatial coordinates. In addition to being a differentiable manifold, space-time has much more additional structure. It is at the level of this additional structure, which will be the subject of later chapters, that the space-time of Newtonian physics differs from the space-time of special relativity and from the space-times of Einstein’s theory of gravity (also called general relativity)." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"String theory today looks almost fractal. The more closely people explore any one corner, the more structure they find. Some dig deep into particular crevices; others zoom out to try to make sense of grander patterns. The upshot is that string theory today includes much that no longer seems stringy. Those tiny loops of string whose harmonics were thought to breathe form into every particle and force known to nature (including elusive gravity) hardly even appear anymore on chalkboards at conferences." (K C Cole, "The Strange Second Life of String Theory", Quanta Magazine", 2016) [source]
"Granularity is ubiquitous in nature: light is made of photons, the particles of light. The energy of electrons in atoms can acquire only certain values and not others. The purest air is granular, and so, too, is the densest matter. Once it is understood that Newton’s space and time are physical entities like all others, it is natural to suppose that they are also granular. Theory confirms this idea: loop quantum gravity predicts that elementary temporal leaps are small, but finite." (Carlo Rovelli, "The Order of Time", 2018)
"[...] one thing that's worth mentioning, though, it that apart from the dream of understanding physics at a deeper level involving gravity, work in string theory has been useful in shedding lights on more conventional problems in quantum field theory and even in condensed matter physics and as well with applications to mathematics. Apart from its intrinsic interest, those successes are one of the things that tend to give us confidence that we're on the right track. Because, speaking personally, I find it implausible that a completely wrong new physics theory would give rise to useful insights about so many different areas." (Edward Witten, [in "Dirac Conversation: Edward Witten". Interview at Int'l Center for Theoretical Physics], 2024) [source]
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