Introduction to general relativity, black holes and cosmology. The fermat principle in general relativity and applications. On a conformally stationary spacetime, fermats principle, and hence the equations of motion of light rays, can be formulated in threedimensional purely spatial terms. In this paper we use a general version of fermats principle for light rays in general relativity and a curve shortening method to write the morse relations for light rays joining an event with a smooth timelike curve in a lorentzian manifold with boundary. Thus s power as power flow is also required by fermats principle. Mimicking the description of spinning particles in general relativity, the fermat principle is extended to spinning photons. Critique of the formula derivation for light deflection by. Fermats principle and the geometric mechanics of ray. Fermats principle in general relativity springerlink. Einsteins original theory applied only to positive real numbers for special relativity. The theory of special relativity and its role in the. Fermats principle, in optics, statement that light traveling between two points seeks a path such that the number of waves the optical length between the points is equal, in the first approximation, to that in neighbouring paths.
It yields applications to the gravitational lens effect. Introduction to general relativity lewis ryder a studentfriendly style, over 100 illustrations, and numerous exercises are brought together in this textbook for advanced undergraduate and beginning graduate students in physics and mathematics. Chapter 3 the variational principle harvey mudd college. A timelike extension of fermats principle in general. Three tests of general relativity as shortwavelength. The hamiltonian formulation begins with general relativity and makes clear. Introduction to general relativity lewis ryder download. In this paper we use a general version of fermats principle for light rays in general relativity and a curve. Fermat took up this approach and hypothesized and demonstrated in 1662 that light follows a path of quickest time, rather than shortest distance.
Analytic mechanics structure of quantum mechanics and relativity spacetime wu xinzhong shanghai jiaotong university, school of history and culture of science email. Pdf the fermat principle in general relativity and. The genesis of the theory of relativity olivier darrigol cnrs. General relativity for di erential geometers with emphasis on world lines rather than space slices philadelphia, spring 2007 hermann karcher, bonn contents p. A ljusterrlikscllllirelmall theory for light rays, preprint 1995. Rehseis 83, rue broca 750 paris the most famous of albert einsteins papers of 1905 is undoubtedly the one concerning the theory of relativity. From fermats principle, one can derive a the law of reflection the angle of incidence is equal.
Ray optics, fermats principle, and applications to. The lensing mass is assumed to be confined to a single plane between the source and the observer, and a time delay is associated with each position in the sky of a potential image. In other words, fermats principle says that only trajectories that have an extremal value of the total time travelled are possible total time travelled is the action. Starting from the field equations of general relativity, light deflection can be calculated by studying geodesic curves. Fermats principle and the laws of reflection and refraction fermats principle states that light travels between two points along the path that requires the least time, as compared to other nearby paths. The fermat principle in general relativity and applications j. The morse relations are obtained under the most general assumptions and one can apply them to have a mathematical.
In this paper an infinite dimensional morse theory for lightlike geodesics joining a point with a timelike curve on a class of lorentzian manifolds is developed under intrinsic assumptions. Chapter 3 the variational principle while newton was still a student at cambridge university, and before he had. We show that the latter leads to a particular hamiltonjacobi formalism. Application of fermats principle to magnetic spectrometers. Electromagnetic power flow, fermats principle, and. Volume m61 of the lecture notes in physics mono graphs.
Ray optics, fermats principle, and applications to general relativity. In this paper we use a general version of fermat s principle for light rays in general relativity and a curve shortening method to write the morse relations for light rays joining an event with a smooth timelike curve in a lorentzian manifold with boundary. Fermat s principle in general re lativity springerlink. In a simplified demonstration of the time delay of radar sounding caused by a massive body 4, t. This simple application of fermats principle gives the same result as general relativity. Lectures on classical mechanics university of california. The reader is assumed to have some basic knowledge of general relativity and some familiarity with differential geometry. Fermats principle, caustics, and the classification of. Theory included a complex number in fourdimensional space in the minkowski metric by attempting to include the three dimensions of spacetime which are irrelevant to the special theory. The extrema of this time surface then give the true positions of the images. It might be interesting to point out that, if s p s e u. As a physical meaning, one can apply the morse relations to have a mathematical description of the gravitational lens effect in a very. Analytic mechanics structure of quantum mechanics and. Fermats principle is a minimum on the line integral.
Ray optics, fermats principle, and applications to general relatively author. In optics, fermat s principle or the principle o least time is the principle that the path taken atween twa pynts bi a ray o licht is the path that. Any modern physicist knows that this theory imposes a strict and general constraint on the laws of nature. Pdf kovner has observed that fermats principle can be used to describe the motion of light rays in arbitrary gravitational fields, not just. Can fermats principle be applied to gravitational lensing.
The classical fermat principle of least time, and the classical hamilton principle of least action, are expressed in terms of a tetrad multiplied by a phase factor expis, where sis the action in general relativity. As far as the light is concerned, it is always propagating straight ahead by this principle. The fermat principle has to be satisfied because it follows from the wave equation that. Get pdf 316 kb abstract in this paper we use a general version of fermats principle for light rays in general relativity and a curve shortening method to write the morse relations for light rays joining an event with a smooth timelike curve in a lorentzian manifold with boundary. A scalar description of gravitational lensing based on fermats principle is described. Fermats principle and the laws of reflection and refraction. This reduction from fourdimensional spacetime to threedimensional space is very similar to the well known reduction formalism of kaluzaklein theory.
Fermats principle can not be the cause, for otherwise we would be attribut. Fermats principle says that the path taken between two points by a ray of light is the path that can be traversed in the least time it occurred to me today that maybe the path is actually the one that covers the smallest distance through spacetime, and that in flattish places the distance would be measured using the minkowski metric. Volker perlick published by springer berlin heidelberg isbn. Phase fronts are not needfully normal to the poynting vectors. Abstract the least action principle is the basic principle of the newtonian mechanics and geometrical optics. This is the same situation as in an anisotropic crystal. Introduction to part i light propagation in linear dielectric and permeable media light propagation in other kinds of media introduction to. Wave or quantum mechanics emerges from these classical principles of general relativity for all matter and radiation.
Pdf fermats principle in general relativity researchgate. We derive fermats principle from the causal structure of spacetime, as well as from an appropriate variational principle. Some of the results are published here for the first time, e. On a conformally stationary spacetime, fermats principle, and hence the. The optical fermat principle applies timeinvariant mediums. Instead an ad hoc principle is used that violates the basic tenets of relativity in order to force an agreement with observations. Newtonian gravity can be fully described without differential equations, using the principle of least action, yet it appears that general relativity cannot because fermats principle would predict that light deflect away from mass if anything towards less. A fermat principle for stationary spacetimes and applications to light rays, j. A ljusternikschnirelmann theory for light rays giannoni, fabio. The first relativistic formulation of the principle, valid in the case of a static spacetime, is due to weyl see 33. The fermat principle in general relativity and applications article pdf available in journal of mathematical physics 431. Landau and lifshitz 1 and in ordinary optics see e. The morse relations are obtained under the most general assumptions and one can apply them to have a mathematical description of the. A ljusternikschnirelmann theory for light rays on a fermat principle in general relativity.
Integrating fermats principle of least time for light and maupertuis. The completed formulation of the general theory of relativity was published in 1916 fig. The sections on variation principles and on applications are now at the end of part ii, in which a. We derive fermat s principle from the causal structure of spacetime, as well as from an appropriate variational principle. Kovner has observed that fermats principle can be used to describe the motion of light rays in arbitrary gravitational fields, not just stationary ones. Another way of stating this principle is that the path taken by a ray of light in traveling between two points requires either a minimum or a maximum.
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