First order description of static black holes and the Hamilton–Jacobi equation

AbstractObtaining solutions of the Einstein field equations describing spinning compact bodies is typically challenging. The Newman–Janis algorithm provides a procedure to obtain rotating spacetimes from a static, spherically symmetric, seed metric. It is not guaranteed, however, that the resulting rotating spacetime solves the same field equations as the seed. Moreover, the former may not be circular, and thus expressible in Boyer–Lindquist-like coordinates. Amongst the variations of the original procedure, a modified Newman–Janis algorithm (MNJA) has been proposed that, by construction, originates a circular, spinning spacetime, expressible in Boyer–Lindquist-like coordinates. As a down side, the procedure introduces an ambiguity, that requires extra assumptions on the matter content of the model. In this paper we observe that the rotating spacetimes obtained through the MNJA always admit separability of the Hamilton–Jacobi equation for the case of null geodesics, in which case, moreover, the aforementioned ambiguity has no impact, since it amounts to an overall metric conformal factor. We also show that the Hamilton–Jacobi equation for light rays propagating in a plasma admits separability if the plasma frequency obeys a certain constraint. As an illustration, we compute the shadow and lensing of some spinning black holes obtained by the MNJA.

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The Hamilton-Jacobi Equations for a Relativistic Charged Particle

Canadian Mathematical Bulletin ◽

10.4153/cmb-1963-028-4 ◽

1963 ◽

Vol 6 (3) ◽

pp. 341-350 ◽

Cited By ~ 1

Author(s):

J. R. Vanstone

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Jacobi Equation ◽

Classical Particle ◽

Quantum Mechanical ◽

Order Partial Differential Equation ◽

First Order ◽

Relativistic Charged Particle ◽

Jacobi Equations ◽

Hamilton Jacobi Equation

In the problem of finding the motion of a classical particle one has the choice of dealing with a system of second order ordinary differential equations (Lagrange's equations) or a single first order partial differential equation (the Hamilton-Jacobi equation, henceforth referred to as the H-J equation). In practice the latter method is less frequently used because of the difficulty in finding complete integrals. When these are obtainable, however, the method leads rapidly to the equations of the trajectories. Furthermore it is of fundamental theoretical importance and it provides a basis for quantum mechanical analogues.

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Variational and viscosity operators for the evolutionary Hamilton–Jacobi equation

Communications in Contemporary Mathematics ◽

10.1142/s0219199718500189 ◽

2019 ◽

Vol 21 (04) ◽

pp. 1850018 ◽

Cited By ~ 1

Author(s):

Valentine Roos

Keyword(s):

Jacobi Equation ◽

A Priori ◽

Initial Condition ◽

First Order ◽

Momentum Variable ◽

Lipschitz Estimates ◽

The Cauchy Problem ◽

Local Lipschitz Estimates ◽

Variational Operator ◽

Hamilton Jacobi Equation

We study the Cauchy problem for the first-order evolutionary Hamilton–Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarily convex in the momentum variable and not a priori compactly supported. We build and study an operator giving a variational solution of this problem, and get local Lipschitz estimates on this operator. Iterating this variational operator we obtain the viscosity operator and extend the estimates to the viscosity framework. We also check that the construction of the variational operator gives the Lax–Oleinik semigroup if the Hamiltonian is convex or concave in the momentum variable.

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Hamiltonian Mechanics for Functionals Involving Second-Order Derivatives

Journal of Applied Mechanics ◽

10.1115/1.1505626 ◽

2002 ◽

Vol 69 (6) ◽

pp. 749-754 ◽

Cited By ~ 1

Author(s):

B. Tabarrok ◽

C. M. Leech

Keyword(s):

Differential Equations ◽

Generating Functions ◽

Jacobi Equation ◽

Equations Of Motion ◽

Second Order ◽

Hamiltonian Mechanics ◽

Energy Functionals ◽

First Order ◽

Independent Variable ◽

Hamilton Jacobi Equation

Hamilton’s principle was developed for the modeling of dynamic systems in which time is the principal independent variable and the resulting equations of motion are second-order differential equations. This principle uses kinetic energy which is functionally dependent on first-order time derivatives, and potential energy, and has been extended to include virtual work. In this paper, a variant of Hamiltonian mechanics for systems whose motion is governed by fourth-order differential equations is developed and is illustrated by an example invoking the flexural analysis of beams. The variational formulations previously associated with Newton’s second-order equations of motion have been generalized to encompass problems governed by energy functionals involving second-order derivatives. The canonical equations associated with functionals with second order derivatives emerge as four first-order equations in each variable. The transformations of these equations to a new system wherein the generalized variables and momenta appear as constants, can be obtained through several different forms of generating functions. The generating functions are obtained as solutions of the Hamilton-Jacobi equation. This theory is illustrated by application to an example from beam theory the solution recovered using a technique for solving nonseparable forms of the Hamilton-Jacobi equation. Finally whereas classical variational mechanics uses time as the primary independent variable, here the theory is extended to include static mechanics problems in which the primary independent variable is spatial.

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Weak KAM from a PDE point of view: viscosity solutions of the Hamilton–Jacobi equation and Aubry set

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210550000064 ◽

2012 ◽

Vol 142 (6) ◽

pp. 1193-1236 ◽

Cited By ~ 9

Author(s):

Albert Fathi

Keyword(s):

Prior Knowledge ◽

Jacobi Equation ◽

Kam Theory ◽

General Setting ◽

Point Of View ◽

Weak Kam Theory ◽

First Order ◽

Aubry Set ◽

The Subject ◽

Hamilton Jacobi Equation

We introduce the notion of a viscosity solution for the first-order Hamilton–Jacobi equation, in the more general setting of manifolds, to obtain a weak KAM theory using only tools from partial differential equations. This work should be accessible to people with no prior knowledge of the subject.

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