[Paper Review] Cosmological versions of Vaidya's radiating stellar exterior, an accelerating reference frame, and Kinnersley's photon rocket

M. L. McClure, Kaem Anderson+1 more|ArXiv.org|Sep 20, 2007

History and Developments in Astronomy1 references3 citations

TL;DR

This paper derives cosmological extensions of Vaidya's radiating stellar exterior and Kinnersley's photon rocket using a conformal transformation of Minkowski spacetime, yielding two-fluid solutions with a null fluid and a perfect/imperfect fluid. The key result is that the photon rocket spacetime provides a model of absolute acceleration in general relativity, where observers directly feel inertial forces, contradicting Machian notions of acceleration as relative.

ABSTRACT

The spacetimes for Vaidya's radiating stellar exterior and Kinnersley's photon rocket in a cosmological background are obtained by performing the same conformal transformation as is used to obtain the Robertson-Walker metric from Minkowski spacetime. In the case of the cosmological radiating stellar exterior, a two-fluid solution is found that obeys all of the energy conditions and consists of a null fluid and a perfect fluid that asymptotically falls off to the standard cosmological values for pressure and density at infinite radius. For the cosmological photon rocket, the massless case is first interpreted to obtain a solution for an accelerating cosmological reference frame, and then the general case is interpreted: in both cases, a two-fluid solution is found that consists of a null fluid and an imperfect fluid that possesses heat conduction and anisotropic stress. The imperfect fluid appears to contain an inhomogeneous dark energy component that acts to accelerate the matter through space via a pressure gradient, but this component has negative energy density on the trailing side of the rocket, meaning only the leading side of the rocket is guaranteed to satisfy the weak and dominant energy conditions. Unlike spacetimes that have rotation but no acceleration, the cosmological photon rocket can serve as an example of a spacetime that contradicts Mach's notion of acceleration, since an observer would see empirical evidence of acceleration even though the matter does not accelerate relative to the universe's background matter distribution.

Motivation & Objective

• To extend Vaidya’s radiating stellar exterior and Kinnersley’s photon rocket to cosmological backgrounds using conformal transformations.
• To investigate whether cosmological solutions for radiating white holes can satisfy energy conditions and remain physically viable.
• To explore whether acceleration in general relativity can be empirically detectable without reference to distant matter, challenging Machian principles.
• To analyze the energy-momentum structure of these solutions, particularly the emergence of imperfect fluids with heat conduction and anisotropic stress.
• To determine whether the background matter in the photon rocket spacetime accelerates with the rocket or remains inertial, and the implications for energy conditions.

Proposed method

• Applying a time-dependent conformal transformation to the Kerr-Schild forms of Vaidya’s and Kinnersley’s metrics, analogous to the method that generates Robertson-Walker spacetime from Minkowski space.
• Using the Redten package and Reduce computer algebra system to compute the Einstein tensor and energy-momentum tensor components for the transformed spacetimes.
• Interpreting the resulting energy-momentum tensors as two-fluid models: a null fluid and a perfect fluid (for Vaidya) or an imperfect fluid with heat conduction and anisotropic stress (for Kinnersley).
• Analyzing the energy conditions (weak, dominant, strong) across spacetime, particularly focusing on radial and angular dependence.
• Examining the causal structure and ensuring no gravitational radiation is introduced, preserving the white hole nature of the solutions.
• Deriving the effective equation of state and identifying components such as a potential inhomogeneous dark energy fluid with negative energy density on the trailing side of the rocket.

Experimental results

Research questions

• RQ1 Can Vaidya’s radiating stellar exterior be consistently extended to a cosmological background while satisfying all energy conditions?
• RQ2 Does the conformal transformation method yield physically viable two-fluid solutions for cosmological white holes, particularly in the presence of radiation and matter?
• RQ3 Can Kinnersley’s photon rocket spacetime in a cosmological context serve as a model of absolute acceleration, independent of distant matter?
• RQ4 What is the role of anisotropic stress and heat conduction in the imperfect fluid component of the cosmological photon rocket?
• RQ5 Why does the energy density become negative on the trailing side of the rocket, and what are the implications for the weak and dominant energy conditions?

Key findings

• The cosmological Vaidya solution is a two-fluid model consisting of a null fluid and a perfect fluid, with asymptotic density and pressure matching standard Friedmann-Lemaître-Robertson-Walker values at infinity.
• The cosmological photon rocket solution is a two-fluid model with a null fluid and an imperfect fluid exhibiting heat conduction and anisotropic stress, arising from the rocket's mass and acceleration.
• On the leading side of the rocket, all energy conditions are satisfied except the strong energy condition, which is violated at late times due to a lambda-type fluid component.
• On the trailing side of the rocket, the weak and dominant energy conditions are violated due to negative energy density in the imperfect fluid, which is interpreted as an inhomogeneous dark energy component.
• The spacetime demonstrates absolute acceleration: observers feel unbalanced forces due to a pressure gradient, providing empirical evidence of acceleration independent of distant matter, thus contradicting Machian principles.
• The conformal transformation preserves causal structure and does not introduce gravitational radiation, maintaining the white hole nature of the solutions despite the presence of cosmological matter.