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2 - Particle Motion under Relativistic Gravity
from Part I - Relativity and Geometry
Piljin Yi, Korea Institute for Advanced Study
Book:

Gravitation for Theorists

Published online:

01 March 2026

Print publication:

19 March 2026, pp 18-28
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Summary

Once the proper time is recognized as the only viable notion of time, relativistic gravity as an external force arises naturally via the analogy of how one introduces the metric in Newtonian dynamics in curvilinear coordinates. The resulting action principle comes with a key property that the time parameter choice should be entirely irrelevant to the dynamics, which is, in turn, used to simplify the action by choosing the parameter to be the proper time of the particle in question. With the metric supplied later by the gravitational field equation, we discover that the Kepler problem elevates to a fully relativistic one straightforwardly. This chapter closes with the application of all these to the light-bending phenomena.

5 - Parallel transport and geodesic lines
from Part I - Elements of differential geometry
Jerzy Plebanski, National Polytechnic Institute of Mexico, Andrzej Krasinski, Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences
Book:

An Introduction to General Relativity and Cosmology

Published online:

30 May 2024

Print publication:

06 June 2024, pp 31-33
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Summary

Parallel transport of vectors and tensor densities along curves is defined using the covariant derivative. A geodesic is defined as such a curve, along which the tangent vector, when parallely transported, is collinear with the tangent vector defined at the endpoint. Affine parametrisation is introduced.

42 - Deflection of Light by the Sun and Comparison with General Relativity
from Part III - Other Spins or Statistics; General Relativity
Horaƫiu Năstase, Universidade Estadual Paulista, São Paulo
Book:

Classical Field Theory

Published online:

04 March 2019

Print publication:

14 March 2019, pp 396-404
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Summary

We calculate “deflection of light by the Sun." First, we define a first-order action for a massless particle moving in a gravitational field, and then we calculate the motion of light on a geodesic as motion of light in a medium with a small, position-dependent index of refraction, giving the light deviation for small angles. Then we redo the calculation from the Hamilton–Jacobi formalism by first defining the Hamilton–Jacobi equation for light motion and then solving it. This gives the nonperturbative light deviation that matches the previous calculation at small angles. We end by comparison with the deflection of light by the Sun in special relativity, which is different by a factor of 2.