Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe force of gravity in Schwarzschild and Gullstrand-Painleve coordinates
91
0
0.0
(
0
)
Authors
C. A. Brannen
Ask ChatGPT about the research
No Arabic abstract
We derive the exact equations of motion (in Newtonian, F=ma, form) for test masses in Schwarzschild and Gullstrand-Painleve coordinates. These equations of motion are simpler than the usual geodesic equations obtained from Christoffel tensors in that the affine parameter is eliminated. The various terms can be compared against tests of gravity. In force form, gravity can be interpreted as resulting from a flux of superluminal particles (gravitons). We show that the first order relativistic correction to Newtons gravity results from a two graviton interaction.
rate research
Read More
We construct an exact solution for the spherical gravitational collapse in a single coordinate patch. To describe the dynamics of collapse, we use a generalized form of the Painleve-Gullstrand coordinates in the Schwarzschild spacetime. The time coordinate of the form is the proper time of a free-falling observer so that we can describe the collapsing star not only outside but also inside the event horizon in a single coordinate patch. We show the both solutions corresponding to the gravitational collapse from infinity and from a finite radius.
We construct a coordinate system for the Kerr solution, based on the zero angular momentum observers dropped from infinity, which generalizes the Painleve-Gullstrand coordinate system for the Schwarzschild solution. The Kerr metric can then be interpreted as describing space flowing on a (curved) Riemannian 3-manifold. The stationary limit arises as the set of points on this manifold where the speed of the flow equals the speed of light, and the horizons as the set of points where the radial speed equals the speed of light. A deeper analysis of what is meant by the flow of space reveals that the acceleration of free-falling objects is generally not in the direction of this flow. Finally, we compare the new coordinate system with the closely related Doran coordinate system.
We perform a numerical study of black hole formation from the spherically symmetric collapse of a massless scalar field. The calculations are done in Painleve-Gullstrand (PG) coordinates that extend across apparent horizons and allow the numerical evolution to proceed until the onset of singularity formation. We generate spacetime maps of the collapse and illustrate the evolution of apparent horizons and trapping surfaces for various initial data. We also study the critical behaviour and find the expected Choptuik scaling with universal values for the critical exponent and echoing period consistent with the accepted values of $gamma=0.374$ and $Delta = 3.44$, respectively. The subcritical curvature scaling exhibits the expected oscillatory behaviour but the form of the periodic oscillations in the supercritical mass scaling relation, while universal with respect to initial PG data, is non-standard: it is non-sinusoidal with large amplitude cusps.
Introducing a set ${alpha_i} in R$ of fractional exponential powers of focal distances an extension of symmetric Cassini-coordinates on the plane to the asymmetric case is proposed which leads to a new set of fractional generalized Cassini-coordinate systems. Orthogonality and classical limiting cases are derived. An extension to cylindrically symmetric systems in $R^3$ is investigated. The resulting asymmetric coordinate systems are well suited to solve corresponding two- and three center problems in physics.
We provide expansions of the Detweiler-Whiting singular field for motion along arbitrary, planar accelerated trajectories in Schwarzschild spacetime. We transcribe these results into mode-sum regularization parameters, computing previously unknown terms that increase the convergence rate of the mode-sum. We test our results by computing the self-force along a variety of accelerated trajectories. For non-uniformly accelerated circular orbits we present results from a new 1+1D discontinuous Galerkin time-domain code which employs an effective-source. We also present results for uniformly accelerated circular orbits and accelerated bound eccentric orbits computed within a frequency-domain treatment. Our regularization results will be useful for computing self-consistent self-force inspirals where the particles worldline is accelerated with respect to the background spacetime.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
