de Sitter Space¶
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.
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, phi and theta.
phi corresponds to the minimum distance between the receiver and emitter, in other words, the emitter’s closest point of approach.
theta measures the angle of approach.
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 if you go back the Hubble time.
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, 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, the mixing of space and time.
The result when you measure distance in this way is de Sitter Space.
It is the simplest possible model for a universe where special relativity holds, and it matches observations extraordinarily well.
It also explains how redshift naturally occurs forwards through time in space-time as a result of special relativity.
The space is highly symmetric, in time as well as space.
Backwards in time, paths separate exponentially in time too.
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.
So at any time there is a large but finite set of galaxies in the visible universe.
The modulation of a galaxy’s arrival depends on the closest distance it approaches and the angle of approach.
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.
So when we observe through the JWST we may occassionally find galaxies, new arrivals that are not as redshifedt 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 significant.
Part of this belief comes from the theory of acretion models and over-coming the angular-momentum obstruction to acretion.
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.
Some of the distant galaxies we are seeing may in fact be smaller quasars, closer to home. That’s a story for another module.
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.
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.
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.
Now let’s see if we can simulate some of this.
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.
These hyperbolae arise from the Lorentz transformations of special relativity.
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 by about one thousandth the speed of light every few mega-parsecs.
But this is just what we would expect when we do an analysis of light paths taking into account special relativity.
Just as we can naturally divide space time into 3 dimensions of space and one of time, so can the alien on a distant galaxy.
We both measure the same speed of light locally. This is an assumption of special 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.
…
References and Footnotes
[1] The parameter to WMAP9.H allows the cosmology to have different Hubble constants at different redshifts. >>>>>>> 34b35a1aeac7ade2c0edeaf423f6c29c727dcc67
- class gotu.dss.DeSitterSpace[source]¶
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
- async zedten(z=10, theta=0, nearest=1)[source]¶
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.