Galactic Velocities, Falling Toward Black Hole
Date: Fall 2012
I comprehend that a circular black hole surrounding the universe (easy to say, hard to imagine) would not be a good model. I have talked to my son and I have been trying to get him to reply to you, but he is a little shy and in ninth grade. I guess the fairest thing is to ask you for help developing a model I am programming of his idea until observation rules it out (or validates it).
In a nutshell, my son is trying to explain dark energy (which came along after I graduated in 1975) with a gigantic external directional gravitational gradient that permeates the entire observed universe. He attributes this external directional gravitational gradient to a very distant, huge, black hole.
His idea is inspired by the Shoemaker-Levy 9 (SL-9) impact with Jupiter (a YouTube favorite), where the comet was drawn onto a sequence of fragments prior to entering Jupiter's atmosphere. When I went over the event with him I explained how a fragment in the middle of the series of fragments was accelerated at some average gravitational attraction, while a fragment just a little closer to Jupiter was experiencing a somewhat greater attractive gravitational
force, and one further away from Jupiter a somewhat lesser attractive force, and over time the sum of these differences results in the comet being "smeared".
From this he worked out that an observer on the fragment in the middle of the smear of fragments would see the leading fragments drawing away and also would see the trailing fragments being drawn away. Hence to this observer, the smear of SL-9 fragments would appear to be accelerating away.
So what he is asking is if the entire known universe could be in an external gravitational gradient (off to the side somewhere) and the small difference in gravitational attraction across the vast distances of the universe is that is what is giving the impression of an internally powered expanding universe (comparing the universe expanding away in all directions to the observer sitting mid-smear on SL-9).
I started to work with him to build a model of such a universe, discussing what the observations would be, and teaching him bits of programming along the way. I only included Newtonian physics and, needless to say, the Newtonian model of the universe resulted in a smear of galaxies.
My son asserts that if we included relativistic corrections into the model, his model would fit observation. I do not recall how to how to work through building relativity into the model, nor have I been able to work it out from web sources.
On the surface, I cannot just toss out his model: If you assumed a recession speed of 30,000KM/sec, accumulated over 13 billion years for some galaxy, the galaxy has been accelerating away at an average acceleration of about 10**-12 KM/sec*sec. This does not seem like an impossible number. This can be modeled by assigning a gradient with minus 10**-12 KM/sec*sec on one edge of a cube, zero for the gradient in the middle (where the observer is), and plus 10**-12 KM/sec*sec at the other edge of the cube, leaving the four edges of the cube perpendicular to the gravitational gradient zero, and interpolating over the cube for all points inside, which, using Newton’s physics, results in a SL-9 like smear for the model for the observed universe.
To pose a problem I can work out with my son: Two points are separated by a light year and are both traveling in the same direction and velocity. Taking the direction of travel as a bearing of zero, the two points would have initial bearings of 90 and 270 respectively.
Over the course of a year they both accelerated, in the direction of their travel, to half the speed of light. Where would they appear to be relative to each other?
So far as I understand it, a year later, traveling .5 c faster, the bearings would be 104 and 284, since the received light as emitted a year ago and the acceleration has carried both .25 of a light year. The two points would also be red shifted with respect to each other. If you can tell me what errors I have made, I can then correct the model of my son's hypothesized universe.
This result qualitatively fits my son’s expectation (Newton gives a zero red shift), but I am not sure if I am accounting for relativistic corrections properly. Once the model is corrected for relativity, we can test his model against the observed speeds of recession.
You have a number of issues that I am not sure whether or not you have taken into account. First, Newton’s law of gravity applies to the effect of all bodies with respect to all other bodies (in the Universe). The simplification only occurs when, for example, two bodies are closer or larger than all the other pairs of bodies in the Universe.
For example, in the case of the Earth, Moon, and Sun we have an isolated three body (almost). But careful observations have revealed small perturbations due to Mars, and I believe Jupiter. In the case of the SL-9 impact with Jupiter the simple calculation refers to the net effect of the center of mass of Jupiter and the center of mass of the collection of the comet fragments. That remains conserved even if the contributions of the various fragments may change.
Your son’s assertion that including relativity would “fix everything up” cannot be accepted without proof. You have also assumed some speeds, e.g. 30,000 KM/sec, but do not specify with respect to what.
Finally, you need to explain and quantify what dark energy (which came along after I graduated in 1975) with a gigantic external directional gravitational gradient that permeates the entire observed universe. He attributes this external directional gravitational gradient to a very distant, huge, black hole. How such a black hole “explains” dark energy? I am not sure I see the connection.\
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Update: December 2011