string: Convert to and from string coordinates

In RobinHankin/schwarzschild: General Relativity in R

Convert to and from string coordinates

Description

\loadmathjax

Convert to and from string coordinates

Usage

u0(r, K = 0)
u1(r, K = 0)
u2(r, K = 0)
u3(r, K = 0)
u0_inv(u)
u1_inv(u)
u2_inv(u)
u3_inv(u)

Arguments

r, K	In the forward functions, Schwarzschild radius and constant of integration respectively
u	String coordinate

Details

Convert between Schwarzschild (circumferential) r and string coordinates u.

Function u1(R) is the proper length of a piece of string stretching from the event horizon (\mjseqnr=1) to R. Symbolically

\mjdeqn

u=\int_s=1^r\fracds\sqrt1-1/s = \sqrtr(r-1) + \log\left(\sqrtr + \sqrtr-1\right) omitted; see PDF

(there is an unavoidable notational infelicity here because the symbol “\mjseqnr” has two non-negotiable interpretations: it is both the integration variable, and the upper limit of the definite integral).

Documentation for u2() and u3() to be added.

The R idiom for functions u1() through u3() uses an algebraic trick to ensure that only a single square root is taken, for reasons of efficiency.

Value

All forward functions are vectorised

Author(s)

Robin K. S. Hankin

References

Hankin 2020

See Also

eddington_stringcoords

Examples

u1(1)
u1(1.001)

RobinHankin/schwarzschild documentation built on Nov. 13, 2024, 12:58 p.m.