Zamicol's Relativity Playground

Name Symbol Value
Gravitational Constant G `6.67408 × 10^-11 m^3 kg^-1 s^-2`
Speed of Light c 299,792,458 m/s
Plank Constant h `6.62607015 × 10^-34 J⋅s`
Fine-structure constant α `\alpha =e^2/(2\varepsilon _{0}hc)` which is `0.00729735 = 1/137.03600`

Special Relativity, Lorentz Transformation, and Velocity Time Dilation

Lorentz factor
`gamma = 1/root()(1-v^2/c^2)`
Relative velocity
`β = v/c`
Relativistic mass
`m = root()(E^2 - (p^2c^2)/c^2`
Rest mass energy equivalence
` E = mc^2`
`E^2 = (pc)^2 + (mc^2)^2`
Time, relativistic mass, kinetic energy, momentum, luminosity, and more
`"time'" = "time"/root()(1-v^2/c^2) " or "t' = gammat`

t' is the moving frame (spaceship)
t is the stationary frame (observer on Earth)
Length
`"ℓ'" = "ℓ"root()(1-v^2/c^2) " or "ℓ' = (ℓ)/gamma`

l' is the length in the moving frame (spaceship)
l is the rest frame length (proper length)(Earth)
`"gravity'" = "gravity"/root()(1-v^2/c^2) " or "g' = gammag`

Velocity Time Dilation Playground

Frame (proper frame)
Time:     
Length:
Energy:
Gravity:

β:   Δ Velocity km/s:     γ :
Slow
Fast
Frame 1 (+v perspective)
Time':     
Length':
Energy':
Gravity':
Frame 2 (-v perspective)
Time':     
Length':
Energy':
Gravity':

General relativity, Gravitational Time Dilation, Escape Velocity, and Schwarzschild radius

Einstein field equations `G_(munu) + Lamdag_(munu) = (8piG)/c^4T_(munu)`
Einstein gravitational constant
`kappa = (8piG)/c^4`
Newton's gravity
`g = (GM)/r^2`
Gravitational Time Dilation
`t' = troot()(1-(2GM)/(rc^2))`
Escape velocity
`v_e = root()((2GM)/r)`
Inverse Square Law
`I = P/(4pir^2)`
Schwarzschild radius
`r_s = (2GM)/c^2`

Gravitational Time Dilation Playground

Outside the gravity field
Time:     
Length:

Object

Units:  

Mass:    
Radius:
Low Mass
High Mass
Small Radius
Large Radius
Inside the gravity field
Time':    1
Length':    1
Gravity: 1
Escape Velocity: 1
Schwarzchild Radius: 1
Schwarzchild Mass: 1

Information theory - Entropy, Bekenstein bound, Landauer's limit, and Kolmogorov complexity

Entropy (Shannon)
`H = -sum plog_2p`
Von Nuemann Entropy
In terms of bits:
`S(\rho) = -\text{Tr}(\rho \log_2 \rho)`

In terms of nats:
`S(\rho) = -\text{Tr}(\rho \ln \rho)`
Boltzmann entropy
In terms of bits:
`S = k_b log_2 W`

In terms of nats:
`S = k_b ln W`
Landauer limit
`E \geq k_b T ln 2`
Black hole surface area entropy
`S = A/(4Gℏ)`
Bekenstein bound
`S ge (2pikRE)/(ℏc)`






Important Concepts

Major Interesting or Surprising Things

Minor Interesting or Surprising Things

Learning Resources

Minor Learning Resources

Tools

Interesting Papers

Find papers using https://scholar.archive.org
Landauer's 1961 paper: Irreversibility and Heat Generation in the Computing Process
"The Thermodynamics of Computation--a Review" Charles H. Bennett, IBM 1981

Alexandre Deur Work, in particular:
Other:

Other Things

Alternatives