models:general_relativity
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models:general_relativity [2018/03/29 13:37] kennymann |
models:general_relativity [2020/04/03 05:19] (current) 71.197.218.238 There seems to be a typo (slows doesn't fit the context). |
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| The main idea of general relativity is that there is not really a gravitational force, but instead what we perceive as gravity is just a result because spacetime is curved through the presence of energy. | The main idea of general relativity is that there is not really a gravitational force, but instead what we perceive as gravity is just a result because spacetime is curved through the presence of energy. | ||
| - | This means that every object transforms spacetime and all other particles which move through space roll down the slows or up the hills that are created by all other objects. | + | This means that every object transforms spacetime and all other particles which move through space roll down the slopes or up the hills that are created by all other objects. |
| For example, imagine a planet like our earth and a smaller object like a satellite. The heavy earth creates a depression in spacetime. As a result, the smaller satellite runs down this slope created by the earth and thus flies towards the earth's surface. (Take note that satellites can orbit around earth because the movement in a circle prevents that they fall directly onto the earth.) | For example, imagine a planet like our earth and a smaller object like a satellite. The heavy earth creates a depression in spacetime. As a result, the smaller satellite runs down this slope created by the earth and thus flies towards the earth's surface. (Take note that satellites can orbit around earth because the movement in a circle prevents that they fall directly onto the earth.) | ||
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| * [[http://math.ucr.edu/home/baez/einstein/einstein.pdf|The Meaning of Einstein’s Equation]] by John C. Baez and Emory F. Bunn | * [[http://math.ucr.edu/home/baez/einstein/einstein.pdf|The Meaning of Einstein’s Equation]] by John C. Baez and Emory F. Bunn | ||
| * http://people.carleton.edu/~nchriste/PTO000041.pdf | * http://people.carleton.edu/~nchriste/PTO000041.pdf | ||
| + | * [[https://arxiv.org/abs/0910.5167|Gravity from a particle Physicist's perspective]] by Percacci | ||
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| Mathematically general relativity is the study of a differential equation (the [[equations:einstein_equation|Einstein equation]]) satisfied by the Riemann tensor of a Lorentzian metric on a manifold. | Mathematically general relativity is the study of a differential equation (the [[equations:einstein_equation|Einstein equation]]) satisfied by the Riemann tensor of a Lorentzian metric on a manifold. | ||
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| + | Spacetime is a Lorentzian manifold, whereas just space is a Riemannian manifold. | ||
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| + | So formulated a bit differently, general relativity is geometry on a Lorentzian manifold. | ||
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| + | In general relativity, spacetime is an $(3+1)$-dimensional Lorentzian | ||
| + | manifold. A Lorentzian manifold is a a smooth $(3+1)$-dimensional manifold $Q$ | ||
| + | with a __Lorentzian metric__ $g$. | ||
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| + | The metric is defined by the conditions: | ||
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| + | * For each $x\in Q$, we have a bilinear map \begin{align*} g(x):T_x Q\times T_x Q &\longrightarrow\mathbb{R} \\ (v,w) &\longmapsto g(x)(v,w) \end{align*} or shorter: $g(v,w)$. | ||
| + | * With respect to some basis of $T_x Q$ we have \begin{align*} g(v,w) &= g_{ij}v^iw^j \\ g_{ij} &=\left(\begin{array}{cccc} 1 & 0 & \ldots & 0 \\ 0 & -1 & & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \ldots & -1 \end{array}\right)\end{align*} Of course we can write $g(v,w)=g_{ij}v^iw^j$ in any basis, but for different bases $g_{ij}$ will have a different form. | ||
| + | * $g(x)$ varies smoothly with $x$. | ||
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| ---- | ---- | ||
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| * [[http://xavirivas.com/cloud/Differential%20Geometry/Eguchi,%20Gilkey,%20Hanson%20-%20Gravitation,%20Gauge%20Theories%20And%20Differential%20Geometry%20(Pr%201980).pdf|Gravitation, Gauge Theories and Differential Geometry by Tohru Eguchi et. al.]] | * [[http://xavirivas.com/cloud/Differential%20Geometry/Eguchi,%20Gilkey,%20Hanson%20-%20Gravitation,%20Gauge%20Theories%20And%20Differential%20Geometry%20(Pr%201980).pdf|Gravitation, Gauge Theories and Differential Geometry by Tohru Eguchi et. al.]] | ||
| * Principles of Relativity Physics by James L. Anderson | * Principles of Relativity Physics by James L. Anderson | ||
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
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| <tabbox FAQ> | <tabbox FAQ> | ||
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| So what really took him so long was not finding the correct equations, but rather understanding why they work although they are generally covariant. | So what really took him so long was not finding the correct equations, but rather understanding why they work although they are generally covariant. | ||
| + | <-- | ||
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| + | -->Can we understand General Relativity as a gauge theory?# | ||
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| + | <blockquote>"It is ironic that General Relativity (GR), although being historically the mother of all gauge theories, is the one whose gauge structure is the least understood. The question, "What is the gauge group of gravity?", has been raised innumerable times and has received a variety of answers that do not seem to be mutually exclusive. The Lorentz group, the Poincare group, the group of translations, and the deSitter group have all been proposed as gauge group for gravity. On the other hand, some physicists have always maintained that the gauge group of GR is the group of general coordinate transformations (or equivalently its active counter par, the group of diffeomorphisms of spacetime.) It has to be admitted that no fundamental new insight has emerged from these investigations. So the situation is quite discouraging. Yet, the question about the gauge group of gravity is not an idle one, for the attitude that we assume in this respect will determine the kind of questions that we are going to ask about the theory later." | ||
| + | <cite>[[http://cds.cern.ch/record/165379/?ln=de|Role of Soldering Gravity Theory]] by Roberto Percacci</cite></blockquote> | ||
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| <-- | <-- | ||
| <tabbox Interpretation> | <tabbox Interpretation> | ||
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| <tabbox Research> | <tabbox Research> | ||
| + | For a great overview see the chapter 7.6 Modifications and Extensions of/to General Relativity in the book Symmetries in Fundamental Physics by Sundermeyer. | ||
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| + | **Gauge Gravity** | ||
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| + | * [[https://arxiv.org/abs/1010.5822|Gauge Gravity: a forward-looking introduction]] by Randono | ||
| + | * [[http://dec1.sinp.msu.ru/~panov/Lib/Papers/GRG/1210.3775v1.pdf|Gauge Theories of Gravity - A reader with commentary]] by Hehl et. al. | ||
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| **Spacetime theories beyond General Relativity** | **Spacetime theories beyond General Relativity** | ||
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| --> Curvature# | --> Curvature# | ||
| + | {{ :theories:curvature.png?nolink&600 |}} | ||
| - | A manifold has a nonzero curvature if a vector that gets parallel transported along a loop is not the same. | + | A manifold has a nonzero curvature if a vector that gets parallel transported along two differents loops is not the same. |
| - | {{ :theories:classical_theories:curvature.jpg?nolink&200 |}} | ||
| <-- | <-- | ||
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| **Recommended Resources:** | **Recommended Resources:** | ||
| + | |||
| * http://www.mpiwg-berlin.mpg.de/Preprints/P264.PDF | * http://www.mpiwg-berlin.mpg.de/Preprints/P264.PDF | ||
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