Catégorie

Relativité générale

La relativité générale est une théorie relativiste de la gravitation, c'est-à-dire qu'elle décrit l'influence de la présence de matière, et plus généralement d'énergie, sur le mouvement des astres en tenant compte des principes de la relativité restreinte. La relativité générale englobe et supplante la théorie de la gravitation universelle d'Isaac Newton qui en représente la limite aux petites vitesses (comparées à la vitesse de la lumière) et aux champs gravitationnels faibles. Elle est principalement l'œuvre d'Albert Einstein, qui l'a élaborée entre 1907 et 1915, et est considérée comme sa réalisation majeure. Cette attribution est généralement admise, bien que les chercheurs en histoire des sciences estiment que d'autres savants ont aussi apporté des avancées substantielles à la même époque en rapport avec ces théories. Le , il soumet son manuscrit de la théorie de la relativité générale à la section de mathématique et de physique de l'Académie royale des sciences de Prusse, qui le publie le . Les noms de Marcel Grossmann et de David Hilbert lui sont également associés, le premier ayant aidé Einstein à se familiariser avec les outils mathématiques nécessaires à la compréhension de la théorie (la géométrie différentielle), le second ayant franchi conjointement avec Einstein les dernières étapes menant à la finalisation de la théorie après que ce dernier lui en eut présenté les idées générales dans le courant de l'année 1915. La relativité générale est fondée sur des concepts radicalement différents de ceux de la gravitation newtonienne. Elle énonce notamment que la gravitation n'est pas une force, mais la manifestation de la courbure de l'espace (en fait de l'espace-temps), courbure elle-même produite par la distribution de l'énergie, sous forme de masse ou d'énergie cinétique, qui diffère suivant le référentiel de l'observateur. Cette théorie relativiste de la gravitation prédit des effets absents de la théorie newtonienne mais vérifiés, comme l'expansion de l'Univers, les ondes gravitationnelles et les trous noirs.

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Exercices associés (32)

Chute avec force de frottement visqueux

A body of mass m in a gravitation field g is initially at rest. It is subjected to a friction force F that is proportional to its velocity v, i.e. Fu2004=u2004−u2006bvu200au2001whereu200au2001bu2004>u20040xa0.

Determine the expression of the norm of the terminal speed v_(l) in terms of m, b and g.
Show that the speed v(t) is given by $v\left(t\right) = v_l\left(1-\exp\left(-\frac{t}{\tau}\right)\right)\ ,$ where τ is the relaxation time, and determine the expression of τ in terms

Voyage au centre de la terre

We assume that there is a cylindrical tunnel along the earth’s diameter, going from one location on the earth’s surface to another location on the opposite side by passing through the centre of the earth. An object of mass m is dropped initially into this tunnel from the earth’s surface. The earth has a radius R. We neglect any kind of friction. Determine the position equation using the expression of the norm of the gravitational field

g\left(r\right) = \frac{4}{3}\,\pi\,G\,\rho\,r

inside a homogeneous sp

Trajectoire elliptique

A material point of mass m is moving in the plane defined by the orthonormal coordinate system Oxy such that its position vector is given by r⃗u2004=u2004Acosu2006(ωt)u2006îu2005+u2005Bsinu2006(ωt)u2006ĵ where A, B and ω are positive constants and î and ĵ are the unitary vectors of axes Ox and Oy.

Show that the material point’s path is elliptical. Sketch the velocity v⃗ and acceleration a⃗ vectors along the trajectory. Show that if Au2004≠u2004B, then vectors r⃗(t) and v⃗(t) are, generally,

La Fusée

A rocket contains a fuel mixture that is ejected with a relative speed u with respect to the nozzle. We assume that the fuel escapes vertically towards the bottom of the nozzle. We neglect the air friction and the variation of the gravitational field with height (i.e. we treat g as a constant). The fuel mass is m, and the total mass of the rest of the rocket (tanks, accessories, etc...) is M.

Show that the rocket can only take off if the flow of burned fuel (i.e. the variation of fuel mass per unit

Champ de gravitation dans la terre

We consider the earth as a compact set of concentric spheres.

Use a geometric argument to show that the field vanishes inside a spherical shell. Thus, for a given point, only the spherical shell for which the point is in the exterior contributes to the field at that point.
Compute the gravitational field inside the earth and outside as a function of the distance to the centre.
A volume element of a sphere is expressed in spherical coordinates as (c.f. figure): dVu2004=

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Notebooks associés (32)

Exploring Quantization and Oversampling through One-Bit Music

One-Bit Music[1]¶ In this notebook we will talk about quantization and oversampling and we will do so by taking a trip down memory lane to revisit the early from monophonic square waves, introduce polyphony by way of pulse-width modulation and finish with the basics of sigma-delta quantization. pacman In [1]: import matplotlib import matplotlib.pyplot as plt import numpy as np import scipy.signal as sp from scipy.io import wavfile import IPython In [2]: import sys ! wget -q -O mus

Modeling Epidemics with Signal Processing Techniques

Epidemics and Signal Processing[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In [1]: import matplotlib.pyplot as plt import numpy as np import scipy.signal as sp import seaborn as sns sns.set() In [2]: plt.rcParams["figure.figsize"] = (14,4) The basic reproduction number[4]¶ The propagation mechanism of an epidemic such as the one caused by SARS-CoV-2 virus can be roughly modeled, at least in its inital phase, as a process in which each infected individ

Understanding and Plotting the Discrete Fourier Transform (DFT)

Plotting the DFT[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In this notebook we will look at the practical issues associated to plotting the DFT and in particular the DFT of real-world signals. We will examine how to map the DFT coefficients to real-world frequencies and we will investigate the frequency resolution of the DFT and the effects of zero padding. In [1]:

first our usual bookkeeping

%matplotlib inline import matplotlib import matplotlib.p

Whistle-Activated Keylock System Implementation

Whistling Keylock[1]¶ In this notebook we will implement a simple "whistle-activated" keylock, that is, an access control system where the password is a pre-defined melody. To program the lock, you record your musical passphrase and then, to open the lock, you whistle it back. Although the implementation will be simple and skip most of the fine details that would be necessary to produce a truly robust system, we need to take into account at least the following points: * The syste

Voice Transformation Techniques Using Digital Signal Processing

A Gallery of Voice Transformers[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In this notebook we will explore various ways to implement a system that changes the voice of a speaker. Typical applications for these techniques, strictly sorted by plausibility, include international spy missions and prank telephone calls, and devices such as the one displayed here can be easily bought on the web... or in a novelty shop. No description has been provided for t

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Exercices associés (32)

Chute avec force de frottement visqueux

Determine the expression of the norm of the terminal speed v_(l) in terms of m, b and g.
Show that the speed v(t) is given by $v\left(t\right) = v_l\left(1-\exp\left(-\frac{t}{\tau}\right)\right)\ ,$ where τ is the relaxation time, and determine the expression of τ in terms

Voyage au centre de la terre

g\left(r\right) = \frac{4}{3}\,\pi\,G\,\rho\,r

inside a homogeneous sp

Trajectoire elliptique

Show that the material point’s path is elliptical. Sketch the velocity v⃗ and acceleration a⃗ vectors along the trajectory. Show that if Au2004≠u2004B, then vectors r⃗(t) and v⃗(t) are, generally,

La Fusée

Show that the rocket can only take off if the flow of burned fuel (i.e. the variation of fuel mass per unit

Champ de gravitation dans la terre

We consider the earth as a compact set of concentric spheres.

Use a geometric argument to show that the field vanishes inside a spherical shell. Thus, for a given point, only the spherical shell for which the point is in the exterior contributes to the field at that point.
Compute the gravitational field inside the earth and outside as a function of the distance to the centre.
A volume element of a sphere is expressed in spherical coordinates as (c.f. figure): dVu2004=

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Relativité générale

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Exotic matter and MOND as special cases of a more general solution in pure general relativity

Constraints on the VHE Counterpart of Two Binary Black Hole Mergers Observed by the MAGIC and CTAO LST-1 Telescopes

BlackTHUNDER: Evidence of three massive black holes in a z ∼ 5 galaxy

first our usual bookkeeping

first our usual bookkeeping

Exotic matter and MOND as special cases of a more general solution in pure general relativity

Constraints on the VHE Counterpart of Two Binary Black Hole Mergers Observed by the MAGIC and CTAO LST-1 Telescopes

BlackTHUNDER: Evidence of three massive black holes in a z ∼ 5 galaxy