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The diagram shows the application of Huygens’s principle to calculating light bending ⌬ for a distance of closest approach ⌬ to a spherically symmetric mass distribution m . The path traveled by a light ray in time ⌬ t is denoted c ( x ) ⌬ t .
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The problem of the deflection of light in a medium with varying refractive index is applied to the motion of light in a weak Schwarzschild gravitational field. In contrast to the standard deviation, the present method is physically transparent, providing a clear reason for the factor-of-2 deviation of the general relativistic result from that of th...
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Citations
... The Schwarzschild black hole has been further discussed in [21][22][23][24][25][26][27][28][29]. The results obtained in this case can be extended in rather straightforward way to other spherically symmetric configurations like Reissner-Nordström metric [16,[30][31][32]. ...
The Fermat principle is used for the analysis of light propagation in the metric of a black hole. It is shown how to apply the Lindstedt-Poincare method of solving perturbatively the nonlinear oscillations to the description of light trajectories in weak deflection regime.
... Based on the general relativity theory, the approximate light path deflection angle, Δϴ is found to be [21], ...
The nonlinear Schrödinger equation is used to show how numerical methods can be used to solve mathematical problems present in nonlinear analysis. The Lanzos-Chevbychev Pseudospectral method is shown to be effective, flexible, and economical to meet various demands in practical applications of mathematical simulations using nonlinear differential equations. The electromagnetic wave propagation through an inhomogeneous, anisotropic, and complex space is used as an example to show how successful mathematical modeling could be used to explain the complex phenomenon of astronomical redshift that is the central issue in the widely debated Hubble tension.
... For this kind of medium we will assume that the refractive index depends only on the radius n(r). Our result is like that one obtained in General Relativity for an isotropic metrics (Lerner, 1997;de Felice, 1971;Simaciu and Ionescu-Pallas, 1996). ...
... Assuming a gravitational field of a body with mass m, the refractive index can be expressed like (3), for the first-order isotropic Schwarzshild metric (Lerner, 1997;de Felice, 1971;Simaciu and Ionescu-Pallas, 1996). Therefore, the maximum deviation, with p = 1 and g Nr in Eq. (21), is g m r R . ...
... This result is just the relativistic deviation (Lerner, 1997;Møller, 1955). That is, an inhomogeneous optic medium with a refractive index of the form (3) behaves like a spherical lens and this lens mimics a gravitational field. ...
Using the formula found by Noorbala and Sepehrinia, the wave
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propagate in the gravitational field, the deduced deviation is identical to that
calculated from General Relativity. The method and their consequences are a
good pedagogical example that verifies the Noorbala-Sepehrinia’s formula as
well as the mechano-optics analogy (Hamilton’s principle/principle of
stationary action and Fermat’s principle) for the bodies movement in the
gravitational field.
... The result of the simplified derivation deviates from the full general-relativistic result by a factor 2; this is the same as for Newtonian calculations of the gravitational deflection of light. 10 Section IV extends the analysis to descriptions using the Schwarzschild metric, comparing different text-book derivations of the Shapiro effect and pointing out ambiguities that can be used to prepare students for the necessity of defining a rigorous framework for post-Newtonian physics. Section V provides the information that will enable students to understand astronomical applications of the gravitational time delay, both in the solar system and for binary pulsars. ...
The gravitational time delay of light, also called the Shapiro time delay, is one of the four classical tests of Einstein's theory of general relativity. This article derives the Newtonian version of the Shapiro time delay from Einstein's principle of equivalence and the Newtonian description of gravity, in a manner that is accessible to undergraduate students and advanced high-school students. The derivation can be used as a pedagogical tool, similar to the way that simplified derivations of the gravitational deflection of light are used in teaching about general relativity without making use of the more advanced mathematical concepts. Next, we compare different general-relativistic derivations of the Shapiro time delay from the Schwarzschild metric, which leads to an instructive example for the challenges of formulating the post-Newtonian limit of Einstein's theory. The article also describes simple applications of the time delay formula to observations within our solar system, as well as to binary pulsars.
... For this kind of medium we will assume that the refractive index depends only on the vector of position, ( ) n r . Our result is like that one obtained in General Relativity for an isotropic metrics [6,7,8]. ...
... Assuming a gravitational field of a body with mass m , the refractive index can be expressed like (3), for the first-order isotropic Schwarzshild metric [6,7,8]. Therefore, the maximum deviation, with 1 p = , g N r = in equation (21), [16] , is ...
... This result is just the relativistic deviation [6,9]. That is, an inhomogeneous optic medium with a refractive index of the form (3) behaves like a spherical lens and this lens mimics a gravitational field. ...
Using the formula found by Noorbala and Sepehrinia, the wave deviation in an inhomogeneous medium with continuous variation of propagation velocity is deduced. For electromagnetic waves (light) that propagate in the gravitational field, the deduced deviation is identical to that calculated from General Relativity. The method and its consequences are a good example that verifies the Noorbala-Sepehrinia's formula as well as the mecano-optics analogy (Hamilton's principle / principle of stationary action and Fermat's principle) for the bodies movement in the gravitational field.
... Only a few years later, as part of the complete theory of general relativity [5,6], Einstein obtained the correct value of this deviation, i.e. the double of the previous result. Many authors have discussed the reason of the doubling of the Newtonian result in GR [7]. In the third part of this paper, we propose a new light to interpret this doubling. ...
In this paper we review the derivation of light bending obtained before the discovery of General Relativity (GR). It is intended for students learning GR or specialist that will find new lights and connexions on these historic derivations. Since 1915, it is well known that the observed light bending stems from two contributions : the first one is directly deduced from the equivalence principle alone and was obtained by Einstein in 1911; the second one comes from the spatial curvature of spacetime. In GR, those two components are equal, but other relativistic theories of gravitation can give different values to those contributions. In this paper, we give a simple explanation, based on the wave-particle picture of why the first term, which relies on the equivalence principle, is identical to the one obtained by a purely Newtonian analysis. In this context of wave analysis, we emphasize that the dependency of the velocity of light with the gravitational potential, as deduced by Einstein concerns the phase velocity. Then, we wonder whether Einstein could have envisaged already in 1911 the second contribution, and therefore the correct result. We argue that considering a length contraction in the radial direction, along with the time dilation implied by the equivalence principle, could have led Einstein to the correct result.
... In 1997, L. Lerner discussed the problem of the deflection of light in a medium with varying refractive index applied to the motion of light in a weak Schwarzschild gravitational field [3,4]. ...
The present research paper derives a formula for gravitational force acting between the black hole and light particle passing near the radius of event horizon of black holes and calculates also their values of different test black holes existing in only X-ray binaries (XRBs).
... The index of refraction (110) is known since Weyl[432]. It was employed for calculating lightlike Schwarzschild geodesics, exactly or approximately, e.g., in[16,296,134,258]. This index of refraction can be modeled by a fluid flow[358]. ...
The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations. More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime. It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images. The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others. Comment: 120 pages, 31 figures; submitted to Living Reviews in Relativity; updated version of http://www.livingreviews.org/lrr-2004-9
Space has a complex structure and investigations of any electromagnetic wave transmission theory need to consider the inhomogeneous and anisotropic nature. We have selected two cases for our investigations: regions of pulse energy changes and gravitational deflection. Numerical methods have been developed and examples given to show that these conditions do have their localized effects. But, since the total length of those regions are insignificant in comparison with the total transmission distance involved, their inclusion does not significantly alter the linear relationship between wavelength change and distance travelled. The possible exception is the case of gravitational deflection when the waves have passed through densely populated regions of space. Our findings could be of interest to the current debate on Hubble tension.
Scattering is an important component of any quantum mechanics course. However, the scattering amplitude in the case of a general potential is often calculated using the simple Born approximation, which does not embed general properties such as unitarity or analyticity. We show that a relatively simple extension, the eikonal approximation, offers a significant improvement and demonstrate this in the case of the electromagnetic and gravitational interactions.
