/** This contains functions for relativity-related issues, including
black holes, relativistic travel, etc.
Good resources:
The Relativistic Rocket by John Baez.
http://www.math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html
*/
/** This is the Lorentz factor, often known as gamma[v].
It represents several transformations, such as length contraction or mass
increase when traveling close to the speed of light. The result is
a dimensionless number from 1 (at zero velocity) to infinity (at lightspeed)
that represents the contraction.
*/
Lorentz[v is velocity] := 1 / sqrt[1 - v^2/c^2]
/** This is the inverse of the Lorentz factor. It takes
a dimensionless number from 1 (at zero velocity) to infinity (at lightspeed)
representing the contraction factor and returns a velocity. */
inverseLorentz[d is dimensionless] := c sqrt[1 - 1/d^2]
/** This is the integral of the Lorentz transformation from v0 to v1.
Not quite sure how this is interpreted yet. */
LorentzIntegral[v0 is velocity, v1 is velocity] := c (arcsin[v1/c] - arcsin[v0/c])
/** Relativistic Doppler ratio. This returns a dimensionless number by which
the source frequency is to be multiplied to get the observed frequency.
v is positive when the source is approaching.
*/
relativisticDopplerRatio[v is velocity] :=
{
sqrt[ (1 + v/c) / (1-v/c) ]
}
/** Inverse relativistic Doppler ratio. This takes a dimensionless number
by which the source frequency is to be multiplied to get the observed
frequency and returns a velocity. The velocity is positive when
approaching. */
inverseRelativisticDopplerRatio[r is dimensionless] :=
{
c (r^2 - 1) / (r^2 + 1)
}
/** Relativistic Z factor. The z factor is the ratio by which a frequency
is shifted due to movement. Z = deltaLambda / lambda.
See:
http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/redshf.html#c2
*/
relativisticZ[v is velocity] :=
{
sqrt[ ( 1 + v/c ) / ( 1 - v/c) ] - 1
}
/** Inverse relativistic Z factor.
Returns: a velocity
See:
http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/redshf.html#c2
*/
inverseRelativisticZ[z is dimensionless] :=
{
(c z^2 + 2 c z) / (z^2 + 2z + 2)
}
/** Relativistic velocity addition. This is the simplest case where the motions
lie along a line.
A B -> v C ->
where v is the relative velocity between A (assumed stationary) and B.
B launches a projectile (C).
u is the velocity of the projectile (C) as seen by A
uprime is the velocity of the projectile (C) as seen by B.
To use this function, pass *TWO* of the arguments to this function, leaving
the other as the special value undef. It will solve for the undefined
variable and return the results in the same order [v, u, uprime] with the
unspecified value filled in.
For example,
[v, u, uprime] = addVelocity[.5 c, undef, .5 c]
println[u -> "c"]
returns:
[v, u, uprime]
*/
addVelocity[v, u, uprime] :=
{
if uprime == undef
uprime = (u - v) / (1 - u v / c^2)
else
if v == undef
v = (u - uprime) / (1 - u uprime / c^2)
else
if u == undef
u = (uprime + v) / (1 + uprime v / c^2)
return [v, u, uprime]
}
/** Relativistic rocket equations. See
"The Relativistic Rocket" by John Baez.
http://www.math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html
and Gravitation, Misner, Thorne, and Wheeler, Section 6.2.
This defines the equations using the following conventions:
"First of all, we need to be clear what we mean by continuous acceleration
at 1g. The acceleration of the rocket must be measured at any given
instant in a non-accelerating frame of reference travelling at the same
instantaneous speed as the rocket (see the FAQ article on accelerating
clocks, http://www.math.ucr.edu/home/baez/physics/Relativity/SR/clock.html
), because this is the acceleration that its occupants would physically
feelâ€”and we want them to accelerate at a comfortable rate that has the
effect of mimicking their weight on Earth. We'll call this acceleration a.
The proper time measured by the crew of the rocket (i.e. how much they age)
is called T, and the time measured in the non-accelerating frame of
reference in which they started (e.g. Earth) is t. We assume that the
stars are essentially at rest in this Earth frame. The distance covered by
the rocket as measured in this frame of reference is d, and the rocket's
velocity is v. The time dilation or length-contraction factor at any
instant is the gamma factor gamma.
*/
/** Relativistic rocket. For a constant acceleration, calculate the "fixed"
(Earth) time elapsed given the time experienced by the astronauts.
args:
[a, properTime]
where
a is acceleration (as felt by people in the rocket)
properTime is time (T, the time experienced on board.)
returns:
time elapsed on earth (t)
(This is the first half of Baez eq. 4)
*/
rocketEarthTimeFromProperTime[a is acceleration, properTime is time] :=
{
c / a sinh[a properTime / c]
}
/** Relativistic rocket. For a constant acceleration, calculate the "fixed"
(Earth) time elapsed given the time experienced by the astronauts.
args:
[a, d]
where
a is acceleration (as felt by people in the (rocket)
d is distance (as measured on earth)
returns:
time elapsed on earth (t)
(This is the second half of Baez eq. 4)
*/
rocketEarthTimeFromDistance[a is acceleration, d is length] :=
{
sqrt[(d/c)^2 + 2d/a]
}
/** Relativistic rocket. For a constant acceleration, calculate the "proper"
time experienced by the astronauts given the time elapsed on Earth.
args:
[a, earthTime]
where
a is acceleration (as felt by people in the rocket)
earthTime is time (t, the time experienced on earth.)
returns:
"proper" time experienced by astronauts. (T)
(This is the first half of Baez eq. 5)
*/
rocketProperTimeFromEarthTime[a is acceleration, earthTime is time] :=
{
c / a arcsinh[a earthTime / c]
}
/** Relativistic rocket. For a constant acceleration, calculate the "proper"
time (that is, the time experienced by the astronauts) given the distance
traveled (as seen from Earth.)
args:
[a, d]
where
a is acceleration (as experienced by people in the rocket)
d is distance (as seen from Earth)
returns:
"proper" time experienced by astronauts. (T)
(This is the second half of Baez eq. 5)
*/
rocketProperTimeFromDistance[a is acceleration, d is length] :=
{
c / a arccosh[a d / c^2 + 1]
}
/** Relativistic rocket. For a constant acceleration, calculate the distance
traveled (as seen from Earth) given the "proper" time, that is, the time
as experienced by the astronauts.
args:
[a, properTime]
where
a is acceleration (as experienced by people in the rocket)
properTime is time (T, the time experienced on board.)
returns:
distance traveled, as measured from Earth (d)
(This is the first half of Baez eq. 6)
*/
rocketDistanceFromProperTime[a is acceleration, properTime is time] :=
{
c^2 / a (cosh[a properTime / c] - 1)
}
/** Relativistic rocket. For a constant acceleration, calculate the distance
(as seen from Earth) given the time elapsed on earth.
args:
[a, earthTime]
where
a is acceleration (as experienced by people in the rocket)
earthTime is time (t)
returns:
distance, as measured from Earth (d)
(This is the second half of Baez eq. 6)
*/
rocketDistanceFromEarthTime[a is acceleration, earthTime is time] :=
{
c^2 / a (sqrt[1 + (a earthTime / c)^2] - 1)
}
/** Relativistic rocket. For a constant acceleration, calculate the velocity as
seen from earth, given the "proper time", that is, the time experienced by
the astronauts.
args:
[a, properTime]
where
a is acceleration (as experienced by people in the rocket)
properTime is time (T, the time experienced on board.)
returns:
velocity, as measured from Earth (v)
(This is the first half of Baez eq. 7)
*/
rocketVelocityFromProperTime[a is acceleration, properTime is time] :=
{
c tanh[a properTime / c]
}
/** Relativistic rocket. For a constant acceleration, calculate the velocity as
seen from earth, given the time elapsed on earth.
args:
[a, earthTime]
where
a is acceleration (as experienced by people in the rocket)
earthTime is time (t)
returns:
velocity, as measured from Earth (v)
(This is the second half of Baez eq. 7)
*/
rocketVelocityFromEarthTime[a is acceleration, earthTime is time] :=
{
a earthTime / sqrt[1 + (a earthTime/c)^2]
}
/** Relativistic rocket. For a constant acceleration, calculate the Lorentz
factor gamma given the "proper time", that is, the time experienced by
the astronauts.
args:
[a, properTime]
where
a is acceleration (as experienced by people in the rocket)
properTime is time (T, the time experienced on board.)
returns:
Lorentz factor gamma
(This is the first third of Baez eq. 8)
*/
rocketGammaFromProperTime[a is acceleration, properTime is time] :=
{
cosh[a properTime / c]
}
/** Relativistic rocket. For a constant acceleration, calculate the Lorentz
factor gamma, given the time elapsed on earth.
args:
[a, earthTime]
where
a is acceleration (as experienced by people in the rocket)
earthTime is time (t)
returns:
Lorentz factor gamma
(This is the second third of Baez eq. 8)
*/
rocketGammaFromEarthTime[a is acceleration, earthTime is time] :=
{
sqrt[1 + (a earthTime / c)^2]
}
/** Relativistic rocket. For a constant acceleration, calculate the Lorentz
factor gamma, given the distance traveled as seen from Earth.
args:
[a, d]
where
a is acceleration (as experienced by people in the rocket)
d is distance (as seen from Earth)
returns:
Lorentz factor gamma
(This is the third third of Baez eq. 8)
*/
rocketGammaFromDistance[a is acceleration, d is length] :=
{
a d / c^2 + 1
}
/** Relativistic rocket. For a constant acceleration, calculate the "proper
time", that is, the time experienced by the astronauts, required to travel
to a distant object, and *stop*.
args:
[a, d]
where
a is acceleration (as experienced by people in the rocket)
d is distance (as measured from earth)
returns:
proper time T experienced by the astronauts.
*/
rocketProperTimeToStop[a is acceleration, d is length] :=
{
2 c / a arccosh[a d / (2 c^2) + 1]
}
/** This section contains black hole calculations. Much of it is based on
the nice resource (collected in one place) here:
https://www.vttoth.com/CMS/physics-notes/311-hawking-radiation-calculator
*/
/** Calculate the radius of a black hole given its mass. */
blackHoleRadius[M is mass] := 2 M G / c^2
/** Calculate the mass of a black hole given its radius. */
blackHoleMassFromRadius[r is length] := 1/2 r c^2 / G
/** Calculate the surface area of a black hole given its mass. */
blackHoleArea[M is mass] := M^2 16 pi G^2 / c^4
/** Calculate the mass of a black hole given its surface area. */
blackHoleMassFromArea[a is area] := 1/4 G^-1 a^(1/2) c^2 pi^(-1/2)
/** Calculate the "surface" gravity of a black hole given its mass. */
blackHoleGravity[M is mass] := c^4 / (4 G M)
/** Calculate the mass of a black hole given its surface gravity. */
blackHoleMassFromGravity[a is acceleration] := 1/4 c^4 / (a G)
/** Calculate the tides at the surface of a black hole given its mass. The
results have dimensions of acceleration/distance, that is, how much
gravity changes as you move away. */
blackHoleTides[M is mass] := c^6 / (M^2 4 G^2)
/** Calculate the mass of a black hole given tides at its surface. The
argument has dimensions of acceleration/distance or s^-2 */
blackHoleMassFromTide[tide] := 1/2 c^3 / (G sqrt[tide])
/** Calculate the (dimensionless) Bekenstein-Hawking entropy of a black hole
given its mass. To get the "chemist's" entropy, this is multiplied by
the Boltzmann constant k. */
blackHoleDimensionlessEntropy[M is mass] := M^2 4 pi G / (hbar c)
/** Calculate the "chemist's" entropy of a black hole
given its mass. This is just the dimensionless entropy multiplied by
the Boltzmann constant k. */
blackHoleEntropy[M is mass] := k M^2 4 pi G / (hbar c)
/** Calculate the black hole's temperature given its mass. */
blackHoleTemperature[M is mass] := hbar c^3 / (M 8 pi k G)
/** Calculate the black hole's mass given its temperature. */
blackHoleMassFromTemperature[T is temperature] := 1/8 c^3 hbar / (G T k pi)
/** Calculate the black hole's luminosity (radiated power) given its mass. */
blackHolePower[M is mass] := hbar c^6 / (M^2 15360 pi G^2)
/** Calclulate the black hole's mass given its radiated power. */
blackHoleMassFromPower[p is power] := c^3 sqrt[hbar / (15 pi p)] / (32 G)
/** Calculate a black hole's lifetime given its mass. */
blackHoleLifetime[M is mass] := M^3 5120 pi G^2 / (hbar c^4)
/** Calculate a black hole's mass given its lifetime. */
blackHoleMassFromLifetime[t is time] := (c^4 hbar t / (5120 G^2 pi))^(1/3)
/** Calculate a black hole's effective density given its mass. */
blackHoleDensity[M is mass] := 3 c^6 / (32 pi G^3 M^2)
/** Calculate a black hole's mass given its effective density. */
blackHoleMassFromDensity[d is mass_density] := sqrt[3 c^6 / (32 d G^3 pi)]
/** Calculate the maximum proper time it takes to fall to the singularity from
the Schwarzschild radius. The use of any rocket decreases this time! */
blackHoleTimeToSingularity[M is mass] := pi G M / c^3
/** Calculate the mass of a black hole given the time it takes to fall to the
singularity. */
blackHoleMassFromTimeToSingularity[t is time] := c^3 t / (pi G)
/** Relativistic escape velocity for an object with a given mass and radius.
This approximates the Newtonian escape velocity which is
v = sqrt[2 G M / r]
See:
http://www.mrelativity.net/MBriefs/Relativistic%20Escape%20Velocity%20using%20Special%20Relativity.htm
args:
[mass, radius]
*/
relativisticEscapeVelocity[M is mass, r is length] :=
{
phi = G M / r // Gravitational potential energy
return sqrt[2 phi - (phi / c)^2]
}
/** Relativistic gravitational potential energy relative to an object (planet,
star, black hole, with mass M) and a smaller object with rest mass m0 at
radial distance r from the more massive object, traveling at velocity v.
If the velocity is zero, this turns into the Newtonian potential energy
which is given by:
PE = - G M m0 / r
See:
http://www.mrelativity.net/MBriefs/Relativistic%20Escape%20Velocity%20using%20Special%20Relativity.htm
args:
[M, m0, r, v=0 m/s]
*/
relativisticPotentialEnergy[M is mass, m0 is mass, r is length, v is velocity = 0 m/s] :=
{
- G M m0 / (r sqrt[1 - v^2/c^2])
}
/** Relativistic kinetic energy for an object with rest mass m0 at
traveling at velocity v. This is valid for all velocities but has numerical
issues if evaluated naively for low velocities in the given form
(m0 c^2) (Lorentz[v] - 1)
so, for low velocities, we expand it to a Taylor series which is closer to
the classical 1/2 m v^2 (which you can actually see in the equation) but
with interesting correction terms..
This would be better with an arbitrary-precision square root routine like
in sqrtWayne.frink. The sample program RelativisticKineticEnergyTest.frink
tests the boundary where the naive algorithm and the Taylor series should
be used (in the absence of an arbitrary-precision square root algorithm,
which is what should really be used.)
As of the 2020-04-01 release of Frink, there is now an arbitrary-precision
sqrt routine sqrt[n, digits] guaranteed to be always available in both Frink
and Frink:The Next Generation (although it will be orders of magnitude faster
in the latter) and the Taylor series can probably be retired, although it's
very interesting and shows how even the simplest relativistic calculations
can quickly split into an infinite series of higher-order terms, or can be
approximated pretty well by a single low-order term in the classical
approximation.
See:
http://www.mrelativity.net/MBriefs/Relativistic%20Escape%20Velocity%20using%20Special%20Relativity.htm
args:
[m0, v]
where:
m0: The rest mass of the object
v: The velocity
*/
relativisticKineticEnergy[m0 is mass, v is velocity] :=
{
if v < 0.13 c // Threshold found empirically, use Taylor series
return 1. * m0 ( (v^2 / 2) + (3 v^4 / (8 c^2)) + (5 v^6 / (16 c^4)) + (35 v^8 / (128 c^6)) + (63 v^10 / (256 c^8)) + (231 v^12 / (1024 c^10)) + (429 v^14 / (2048 c^12)) + (6435 v^16 / (32768 c^14)) )
else
return (m0 c^2) (1/sqrt[1-v^2/c^2] - 1)
}