"""de Sitter Space
It's a ball, expanding and contracting, like an Escher drawing.
Are gamma-ray bursts optical illusions?
Robert S Mackay, Colin Rourke.
https://pjm.ppu.edu/paper/247
The paper describes the relationships between pairs of geodesics in de
Sitter Space.
One geodesic corresponds to the path of a distant galaxy and the other
a receiver geodesic, such as our own galaxy.
Each emitter arrives in our visible universe highly blue shifted, then
becomes increasingly red-shifted as time goes by.
The actual blue shift period and full details depend on two
parameters, theta and phi.
theta measures the angle between the asymptotes of the hyperbola of the geodesic.
phi corresponds to the scale of the hyperbolic rotation.
The formulae used in this module are from Appendix F of the Geometry
of the Universe, and present formulae to convert emitter time, u, to
our time t.
Because of curvature it is convenient to work in terms of T = e**t and
U == u**t.
See the code for more information, as well as the paper.
de Sitter Space and the Space Telescope
=======================================
Phillip James E. Peebles recently published a fascinating paper:
Anomalies in Physical Cosmology
https://arxiv.org/abs/2208.05018
It describes the standard, lambda-Cold-Dark-Matter model for the
universe and observations that suggest the model may need some new
insight.
The paper is timely, with our new view on the universe thanks to the JWST.
de Sitter Space is mentioned briefly in the paper, remarking that it
was not consistent with the observations.
I believe Peebles is talking about that sub-space of de Sitter Space,
restricted to bodies with a common origin.
de Sitter Space appears to disappear from consideration as a model for
the universe.
The issue is not de Sitter Space, rather it is the assumption all
matter in the universe was co-located at a big bang some 13.7 billion
years ago.
So what is de Sitter Space? Imagine a universe, of galaxies, as far
as the eye can see, and far, far beyond.
Moving in seemingly random directions, with local velocities at around
one thousandth the speed of light.
To transform time at some distant galaxy, to time at our galaxy, we
have to take account of special relativity, which tells us how to
account of the relative velocity of an emitter and a receiver.
The result when you measure distance in this way is de Sitter Space.
It requires one further ingredient, a constant curvature for
spacetime. The lambda in Einstein's General Relativity equation.
A solution for a universe with no mass, but a constant curvature.
It is the simplest possible model for a universe where special
relativity holds, and it matches observations extraordinarily well, so
long as you look at those observations with the perspective that the
universe has the Perfect Copernican Principle.
It also explains how an asymptotic redshift-distance relationship
arises.
de Sitter Space is highly symmetric, in time as well as space.
Backwards in time, paths separate exponentially, as well as in
forwards time.
Each galaxy that passes through our visible universe, arrives in a
burst of blue shifted light, comes as near as it gets and then
separates exponentially from then, following a hyperbola.
Just as there is a first time that the source is visible, there is a
last time it will be visible, but the observer will have to wait until
the end of time to see that.
With this model for a universe there is an explanation for the
redshift that we see, whilst there being no overall expansion of
space-time, as the redshift is exactly balanced by the blue shift
period of a new arrival.
When we observe through the JWST we should also find galaxies, newer
arrivals that are not as high redshift as they would be expected to
be, given their distance.
There is a further complication, with associating red-shift with
distance. If the light is coming from a place close to a
super-massive black hole, it may be highly redshifted by the local
gravity, following Einstein's general relativity.
The current assumption is that light producing regions are far enough
from any central mass for the gravitational redshift to be
insignificant.
Part of this belief comes from the theory of accretion models and
over-coming the angular-momentum obstruction to accretion.
According to Rourke, once you take account of the frame dragging due
to the rotation of the central mass, the angular momentum problem goes
away: the central black hole can absorb angular momentum of in-flowing
particles.
Distant galaxies we are seeing may in fact be smaller quasars, closer
to home. Resolving cosmological and gravitational redshift is
complex, but the shape of emissions peaks can be helpful here.
See the `.desi` module for more on this.
JWST is also showing us how much dust is scattered across galaxies
and, the beautiful dust spirals that emerge.
The observations we have of our universe, show a place that is very
much in balance, it has had a long time to settle into its current
state of equilibrium.
Once we remove the time limit imposed, due to the big bang, it is
possible to imagine very different galactic timescales and evolution.
It also explains the many observations that indicate a system in high
state of equilibrium, for example as shown by the Cosmic Microwave
Background.
Galaxies evolve over time, matter moves out along the spiral arms,
that journey would take of the order of 15 billion years, with many
super-novae along the way.
It should also be noted, that the conditions close to a galaxy's
central black hole are very similar to those shortly after the big
bang, making the journey of matter along spiral arms an even better
match to the big bang theory.
We see, in the JWST pictures, baby quasars, spinning close to their
parent galaxies.
Galaxies grow from their surrounding dust, and there appears to be
just a steady flow of dust, with wonderful harmonics. Matter moving
out along spiral arms before falling back into the centre.
But the Cosmic Microwave Background, what's that? It's the heat from
billions of billions of distant galaxies, the glowing dust of the
cosmos.
It's all modulated by the lense of de Sitter Space, but why is it the
temperature it is, and what of the harmonics?
First, the Hubble radius is important too in de Sitter Space, it is
the de Sitter Radius, the distance at which things appear and where
they disappear from view.
The universe is not, however, a pure de Sitter Space. It is also full
of roations and spherical harmonics. The 500 million light year
radius of the Lanakea super cluster, the first harmonic of the Cosmic
Microwave Background's spherical harmonics.
Which is why the CMB is the temperature it is. The wobbles mean light
can rarely get more than a handful of Hubble distances from it's
source, like a main of hair, it eventually turns in on itself.
6-7 Hubble distances seems to be about the limit, the CMB is 45 times
brighter than the thermalised energy of all the galaxies in a Hubble
distance radius sphere.
One criticism of de Sitter Space is that it is a vacuum solution to
Einstein's equations. There is no matter and no Mach's Principle.
Now Rourke's proposal of intertial drag from rotation, dropping off as
1/radius, is also problematic.
The Kerr metric is the unique solution to Einstein's equations assuming
space is not a vacuum.
But space is clearly not a vacuum, it is full of dust and microwaves.
When you apply the Sciama Pricnciple to every celestial body, from the
smallest grain of dust to the largest central mass in a galaxy, then I
believe it will be clear why the Sciama Principle applies.
Furthermore, we know from gravitational wave detections that
space-time can carry waves, or rather, space-time itself wobbles, and
those waves propagate according to the Sciama Principle.
There is also a natural link between rotation and curvature.
Hyperbolas
==========
I have been stuck at this part of the journey for a while, looking for
a good way to explain how space time seems to work.
At this point here, we suddenly run into a lot of mathematics.
Conic sections, manifolds, matrices, rotations. Four dimensional
hyperbolic space.
The key observation is that when you plot a distant galaxy's distance
against time we get a rectangular hyperbola. A hyperbola that is
embedded in curved space time.
Most of the sources of light we see are galaxies that are now in the
rapidly receding part of their hyperbola, since that is where each
source spends all but a small finite time of the infinite time it is
visible.
Presumably, under current cosmology, the few exceptions are assumed to
be smaller objects nearer to our galaxy?
To set the scene, consider someone on a planet in a distant galaxy.
It is possible to estimate the movement, relative to the distant fixed
stars. For example, our own galaxy is cruising through space at some
2.1 million km/h. With a little help rom `astropy`:
>>> from astropy import units, constants
>>> milky_way_speed = 2.1e6 * units.kilometer / units.hour
>>> milky_way_speed / constants.c.to(milky_way_speed.unit)
<Quantity 0.00194579>
We see that this is an appreciable fraction of the speed of light. It
is also in line with the speeds for other galaxies in their locality
in the universe.
More generally, due to the Hubble expansion:
>>> from astropy import cosmology
>>> cosmology.WMAP9.H(0)
<Quantity 69.32 km / (Mpc s)>
So local velocities are large enough that Einstein's special
relativity has to be taken into account when mapping the distant
galaxy's space time to our space time. [1]
Further, when Hubble expansion is taken into account, these relative
velocities go up expontenentially, with a horizon at the Hubble
distance.
At that point the galaxy is recessing so fast, we can never see it.
We both measure the same speed of light locally. This is an
assumption of relativity.
However, to map their space time to ours, we need to know our relative velocity.
For distant galaxies, the redshift allows us to calculate a starting
point for this velocity.
Distant galaxies have high red shift, so let's suppose this galaxy is
called Zedten.
Now we want a transformation that preserves distances, and takes
Zedten space time to ours.
What does the path of the Zedten look like through our space-time?
First, let's answer a simpler question.
What does the distance of Zedten look like through our time?
Reduce the three dimensions of space to one dimension, the distance.
So we need to be able to map a clock and a standard ruler to our clock
and standard ruler.
Light lines give the paths of light through space time. Both
ourselves and the people of Zed10, the Zeeten, agree that on these
lines, time stands still, along a light line.
...
References and Footnotes
[1] The parameter to `WMAP9.H` allows the cosmology to have different
Hubble constants at different redshifts.
>>>>>>> 34b35a1aeac7ade2c0edeaf423f6c29c727dcc67
"""
# we are going to need this
import random
import math
import numpy as np
from astropy import constants, units
from scipy import integrate
from traceback import print_exc
from blume import magic
from blume import farm as fm
[docs]
class Dss(magic.Ball):
""" Coming and going in de Sitter Space
An exploration of the equations of the paper.
There is a path from emitter at time u to the receiver at time t if and only if::
-a sinh(t) sinh(u) + d cosh(t) cosh(u) = 1
Here a is cosh(phi) and d is cos(theta)
Working with U=e**u and T=e**t, this constraint becomes::
-ATU + BT/U + CU/T - D/TU = 2
Where A, B, C and D are calculated as shown in `Dss.set_abcd`.
If we multiply by UT and re-arrange we get::
-ATTUU + BTT + CUU - D - 2TU = 0
This can be considered as a quatric in either T or U.
As a quadratic in T::
(B-AUU)TT - 2UT + CUU - D = 0
T = (+2U +/- sqrt(4UU - 4 * (B-AUU)(CUU - D))/(2*(B -AUU)
= (U +/- sqrt(UU - (B-AUU)(CUU - D))/(B -AUU)
= (U +/- sqrt(UU + BD + ACUUUU - ADUU - BCUU ))/(B -AUU)
= (U + sqrt(BD + (1-BC-AD)UU + ACUUUU)/(B - AUU) (1)
Taking the positive square root as the causal solution.
(why not the negative?)
And for U::
(C-ATT)UU - 2UT + BTT - D = 0
U = (T +/- sqrt(TT - (C-ATT)(BTT-D)))/(C-ATT)
U = (T - sqrt(CD + (1-BC-AD)TT + ABTTTT))/(C-ATT) (2)
As U tends to zero, T = sqrt(BD)/B = sqrt(D/B)
So, there is a first time that T sees U, at T = sqrt(D/B). See `Dss.tstar`.
Now examine (2) and see that as T tends to infinity, U tends to sqrt(B/A).
There is a last time U = sqrt(B/A) that the emitter is visible. See `Dss.umax`.
There is an interesting situation at T = +/- sqrt(C/A).
"""
def __init__(self):
""" initialise """
super().__init__()
self.theta = math.pi / 4
self.phi = 0.5
self.size = 50
self.set_abcd()
def set_abcd(self):
self.a, self.b, self.c, self.d = (
math.cosh(self.phi), 0., 0., math.cos(self.theta))
self.A = (self.a + self.b - self.c - self.d) / 2
self.B = (self.a - self.b - self.c + self.d) / 2
self.C = (self.a + self.b + self.c + self.d) / 2
self.D = (self.a - self.b + self.c - self.d) / 2
def constraints(self):
assert(self.a**2 - self.c**2 >= 1.)
assert(self.a > 0)
a, b, c, d = self.A, self.B, self.C, self.D
assert((a * b - c * d)**2 <= (a*a - c*c - 1) * (b*b - d*d +1))
assert((b*b - d*d +1) >= 0)
assert(a >= 0)
assert(b >= 0)
assert(c >= 0)
assert(d >= 0)
[docs]
def tb(self):
""" Blue shift time """
return self.blue_shift_time()
[docs]
def blue_shift_time(self, alpha=None, delta=None):
""" How long the blue shift period is. """
a = alpha or self.a
d = delta or self.d
sqrt = math.sqrt
etb = sqrt((1+a)/(a+d)) + sqrt((1-d)/(a-d))
return math.log(etb)
[docs]
def tofu(self, u):
""" Return t for u
See `.uoft` for more.
Some things to note.
"""
U = math.e ** u
a, b, c, d = self.A, self.B, self.C, self.D
xx = (b * d) + ((U**2) * (1 - b*c - a*d)) + (a * c * U**4)
T = (U + math.sqrt(xx)) / (b - a * U**2)
# check equation
print(U,T)
check2 = (-a*T*U) + (b*T/U) + (c*U/T) - (d/(T*U))
print('check should be 2:', check2)
t = math.log(T)
u = math.log(U)
sh = math.sinh
ch = math.cosh
a, b, c, d = self.a, self.b, self.c, self.d
check = - a * sh(t) * sh(u) + d * ch(t) * ch(u)
print('check should be 1:', check)
return t
[docs]
def uoft(self, t):
""" Return u for t
As T -> infinity, U -> sqrt(B/A)
So there is a last time, umax, that we see the emitter.
umax = sqrt(B/A)
There is also a discontinuity at T=sqrt(C/A).
This is the first time, tstar, that the source is visible.
"""
T = math.e ** t
a, b, c, d = self.A, self.B, self.C, self.D
xx = (c * d) + ((T**2) * (1 - b*c - a*d)) + (a * b * T**4)
U = (T - math.sqrt(xx)) / (c - a * T**2)
# check equation
print(U,T)
check2 = (-a*T*U) + (b*T/U) + (c*U/T) - (d/(T*U))
print('check should be 2:', check2)
t = math.log(T)
u = math.log(U)
sh = math.sinh
ch = math.cosh
a, b, c, d = self.a, self.b, self.c, self.d
check = - a * sh(t) * sh(u) + d * ch(t) * ch(u)
print('check should be 1:', check)
return u
def tstar(self):
return math.log(math.sqrt(self.D/self.B))
def umax(self):
return math.log(math.sqrt(self.B/self.A))
[docs]
def time_until_red_shift_matches_expected_for_distance(self, error=0):
""" Curious how this value varies with phi and theta """
raise NotImplemented
[docs]
def time_until_red_shift_turns_light_into_microwaves(self, error=0):
""" Curious how this value varies with phi and theta """
raise NotImplemented
def deSitter(self):
pass
[docs]
def cmb(self):
"""Show why there is gravitational fog.
What would the distant light from a universe a few orders of
magnitude larger and older than the Hubble distance look like?
Curvature means that light that appears to come from a
specific, very distant point, is actually a mixture of light
from all points on the sphere at that radius.
At least that's how I think de Sitter Space works.
It is not a pure de Sitter Space, since space itself appears to wobble.
The goal here is to explore what we might expect to see.
Bonus marks for a model which produces the spherical harmonics
observed in the cmb.
Perhaps restrict to a circle of radius phi and use 0 theta for the
angle around the circle.
Update
======
We should expect the CMB and the oscillations of space time to be in high equilibrium.
As a photon travels through space time it gains energy from the waves from the wobbles
of space time, the Rees-Sciama effect.
At the same time a wave goes out from each photon, that adds to the wobble.
I believe there are 4 billion microwave photons in each meter cubed of space time.
"""
raise NotImplemented
async def run(self):
size = self.size
epsilon = 1e-3
img = np.zeros((size, size))
for row in range(1, size+1):
delta = math.cos(math.pi * row/(size+1))
for col in range(1, size+1):
alpha = math.cosh(self.phi * col/(size+1))
try:
img[row-1][col-1] = self.blue_shift_time(
alpha or epsilon, delta or epsilon)
except:
print_exc()
print(alpha, delta)
raise
#print(img[row-1])
await magic.sleep(0)
ax = await self.get()
extent = [0, math.pi, 0, self.phi]
aximg = ax.imshow(
img,
cmap=magic.random_colour(),
extent=extent,
)
#ax.figure._axstack.bubble(ax.delegate)
#ax.figure.colorbar(aximg)
#ax.axis('off')
ax.show()
[docs]
class DeSitterSpace(magic.Ball):
"""Another go at de Sitter Space.
Suppose that the universe is just an endless stream of galaxies,
like the billions
The idea is to consider a distant galaxy, as it arrives in our
visible universe
"""
def __init__(self):
super().__init__()
self.k = 1.
self.x = 0.
self.dt = 0.01
self.t = 0.
self.v = 1 / random.randint(5, 10)
async def uvxt(self):
""" """
[docs]
async def zedten(self, z=10, theta=0, nearest=1):
"""a galaxy at zed ten
z: redshift, optional, default 10
theta: angle of approach
nearest: point of closest approach
plots future and past of a galaxy at zed10
if nearest is zero, then it is a big bang universe.
but what if nearest is one or more? Where one is the Hubble distance.
The answer is a hyperbolic rotation, but how to get there?
Focus on the intersection of our timeline with the future
light cone of `Zed10`.
The plots below are an attempt to follow the arguments on page
163 of `gotu`.
"""
ax = await self.get()
ax.set_ylabel('x')
ax.set_xlabel('t')
# light lines in our co-ordinates
for c in range(10):
t = np.arange(-5, 5)
# x = t + c, positive distance
ax.plot(t + c, t, c='blue')
# negative distance x = -t + c
ax.plot(-t + c, t, c='b')
#ax.show()
#ax = await self.get()
# now think about transformations that preserve `distance`
# distance_squared = (x1-x2)**2 - (t1-t2)**2
# on light lines, dx = dt
# u * v = k, u = x - t, v = x + t
k = self.k
for ix in range(20):
x = np.arange(k**-2, 10, 0.1)
# x**2 - t**2 = k
#t**2 = x**2 - k
t = ((x**2) - k)**0.5
ax.plot(x, t)
ax.plot(x, -t)
k *= 1.5
k = self.k * 2
for trial in range(5):
u = random.random() + random.randint(1, 5)
v = random.random() + random.randint(1,5)
print([(u+v)/2, (u/k + k*v)/2])
print([(v-u)/2, ((v * k) - u/k)/2])
ax.plot((u+v)/2, (v-u)/2, 'ro')
ax.plot((u/k + k*v)/2, ((v * k) - u/k)/2, 'bo')
ax.show()
async def run(self):
""" """
await self.zedten()
return
[docs]
async def lorentz(self):
""" FIXME """
# increment t
self.x += self.v * self.dt
self.t += self.dt
# we see
gamma = 1/math.sqrt(1 - (self.v * self.v))
dr = gamma * self.v * self.dt # ???
dt = self.dt / gamma
v = dr/dt
print(v, self.v)
self.v = v
async def run():
dss = Dss()
dss.set_abcd()
dss.constraints()
farm = fm.Farm()
dss2 = DeSitterSpace()
farm.add(dss2)
farm.add(dss)
await farm.start()
await farm.run()
if __name__ == '__main__':
# theta distributed as cos(theta) * delta_theta
theta = math.cos(random.random() * math.pi) * math.pi
# phi distributed as sinh ** 2 * delta_phi
phi = None
thetas = np.array([math.cos(theta * math.pi * xx) for xx in range(1, 1000)])
shines = np.array([math.sinh(x/100) ** 2 for x in range(1,1000)])
weights = integrate.cumulative_trapezoid(shines, initial=0)
#plt(weights)
#plt.show()
#0/1
wm = False
magic.run(run())