lorentz-package: The Lorentz Transform in Relativistic Physics

In RobinHankin/gyrogroup: The Lorentz Transform in Relativistic Physics

The Lorentz Transform in Relativistic Physics

Description

The Lorentz transform in special relativity; also the gyrogroup structure of three-velocities. Performs active and passive transforms and has the ability to use units in which the speed of light is not unity. Includes some experimental functionality for celerity and rapidity. For general relativity, see the 'schwarzschild' package.

Details

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Author(s)

Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>)

Maintainer: Robin K. S. Hankin <[e-mail]>

References

• Ungar 2006. “Thomas precession: a kinematic effect...”. European Journal of Physics, 27:L17-L20.

• https://www.youtube.com/watch?v=9Y9CxiukURw&index=68&list=PL9_n3Tqzq9iWtgD8POJFdnVUCZ_zw6OiB

Examples

u <- as.3vel(c(0.3,0.6,-0.1))  # u is a three-velocity
gam(u)                         # relativistic gamma term for u
U <- as.4vel(u)                # U is a four-velocity
B1 <- boost(u)                 # B1 is the Lorentz transform matrix for u
B1 %*% c(1,0,0,0)              # Lorentz transform of zero 4-velocity (=-u)

B2 <- boost(as.3vel(c(-0.1,0.8,0.3)))  
B3 <- boost(as.3vel(c(-0.1,0.1,0.9)))  # more boosts

Bi <- B1 %*% B2  # Bi is the boost for successive Lorentz transforms


pureboost(Bi)      # Decompose Bi into a pure boost...
orthog(Bi)         # and an orthogonal matrix

Bj <- B2 %*% B1    # B1 and B2 do not commute...

(B1 %*% B2) %*% B3 
B1 %*% (B2 %*% B3)    # ...but composition *is* associative



## Three velocities and the gyrogroup

## Create some random three-velocities:

u <- r3vel(10)
v <- r3vel(10)
w <- r3vel(10)

u+v
v+u        # Three-velocity addition is not commutative...

u+(v+w)   # ... nor associative
(u+v)+w 

RobinHankin/gyrogroup documentation built on July 19, 2024, 6:37 p.m.