Generalized transformations and coordinates for static spherically symmetric general relativity

Area of Science:

• Theoretical physics
• General relativity
• Gravitational physics

Background:

• Standard derivations of the Schwarzschild solution rely on specific coordinate transformations and dynamical assumptions.
• These assumptions limit the scope of possible solutions within general relativity.
• The Schwarzschild solution, while fundamental, possesses a coordinate singularity.

Purpose of the Study:

• To explore alternative solutions to the empty space field equations of general relativity.
• To develop a formalism that circumvents the limitations of standard Schwarzschild solution derivations.
• To investigate a solution that avoids the coordinate singularity inherent in the Schwarzschild metric.

Main Methods:

• Examining a static, spherically symmetric solution using a non-orthogonal line element.
• Relaxing convenient coordinate transformations and dynamical assumptions used in standard derivations.
• Obtaining a more suitable coordinate chart to describe the spacetime geometry.

Main Results:

• A new general relativity solution is derived, encompassing the Schwarzschild solution as a special case.
• The new solution successfully avoids the coordinate singularity.
• The solution contains two arbitrary constants, one linked to the Newtonian gravitational potential.

Conclusions:

• The derived solution offers a more general framework for understanding static, spherically symmetric spacetimes.
• It provides a pathway to generalize the Eddington-Finkelstein and Kruskal-Szekeres transformations.
• This work opens new avenues for exploring gravitational physics beyond the standard Schwarzschild model.