In this post, I will provide the ultra orthodox fringe view (translation: just my view) on the problems that face physics when doing classical problems with gravity. The difference between this blog and a more conventional presentation is that I will emphasize the problems instead of starting off with the leading proposals of the day.
Classical gravity rules when nothing is moving fast and the masses are at a low density. The same mathematical expression has ruled our description of the heavens since the publication of the Principia by Newton in 1687. Modern theoretical research on gravity leave this area untouched. Studies on black holes are for those rare places of enormous mass densities. Quantum gravity rarely finds a stage to play except at the start of the Universe or around black holes, both of which are not classical.
The Good in Newton's Universal Law of Gravitation
The law works great to describe the Moon, the Sun, the planets and tides. That is an impressive list of accomplishments. The equation is simple:
The force of gravity is attractive (the minus sign). It is universal, applying to apples, the Moon, and to stars in different galaxies (a result Newton could not anticipate). The product of one mass by another divided by the distance that separates them squared, an inverse square force law.
The form of the expression is simple enough that it is used in most rockets that must account for gravity as they fly millions of miles around our solar system. The exceptions are those rockets have atomic clocks on board or make other very precise measurements.
The Bad in Newton's Universal Law of Gravitation
There are two kinds of bad: bad in theory and bad in measurement. Newton's law of gravity predicts that changes in a mass density should create a change in the force that travels everywhere in the Universe instantaneously. That sounds bad: everything takes a little time. Light might be quick as anything we know, but it still takes time for the flash of a lightning strike to reach our eyes.
When refined measurements of gravity are made, Newton's gravity law does not pass the tests. Newtonian theory accounts for only half of the bending of light that is seen. The precession of the perihelion of Mercury is created by a dozen different things, for example Jupiter adds 500 arcseconds per century. Yet all the contributions left 43 arcseconds per century unaccounted for. Oops. Nice work was done with a time delay of radar reflections off of planets.
Newton's theory of gravity conserves energy. Binary pulsars are impressively accurate clocks. Yet those clocks slow down. The way to slow down is to lose energy from a gravity wave with a quadrapole moment. Newton's theory of gravity cannot account for the slow down.
The Good in General relativity
Gravity is limited by the speed of light. All the measurements Newton's theory gets wrong, general relativity gets right: light bending, the precession, time delay, and energy loss by gravity waves. None of these issues though would be classified as "classical" in the sense I defined above: either the speed of light is involved or a large mass. General relativity is one of the most impressive accomplishments in physics, but its confirmed improvements do not alter gravity for systems that are slow moving and not dense.
General relativity is elegant minimalism, needing only the Ricci scalar for a gravity field in a vacuum for its Lagrangian.
The Bad in GR
On the theory side, no well-behaved calculations can be done with efforts to quantize general relativity. The equations can be written down, but perturbation calculations do not converge. Ten nonlinear equations have proven to be too difficult to tame.
General relativity has not been integrated into the standard model. Despite letting the brightest minds in each generation work on this problem, I don't think there is anything to report other than the bridge has not been built. Scary.
Neither of these two issues in theory should be classified as classical. Both are in the wheelhouse of quantum mechanics. In the domain of slow speeds and low densities, there is no difference between general relativity and Newtonian gravity theory.
General relativity fails in measurements of every system the size of a galaxy or larger. The failures are either epic or small, depending on one's hypothesis. The problems are all for classical systems, with slow speeds and low mass densities.
The first galaxy scale problem was seen by Jan Oort. He studied our own Milky Way galaxy, realizing that the Sun is not at the center of the galaxy, but outside of it by more than nineteen thousand light years. He observed the orbital velocity of stars in the galaxy and concluded that stars were moving too fast to be accounted for by the visible stars and Newtonian gravity.
The problem was confirmed a year later in a different context. Fritz Zwicky studied the motion of galaxies with each other in the Coma Cluster. He concluded that again things moved too fast to make sense with Newtonian gravity applied to the visible matter.
Not only are the speeds observed faster than one would predict, but Alar Toomre was able to show that a thin disk galaxy is mathematically unstable. If such a galaxy gets a small nudge, it should collapse into a ball. Such a nudge would happen if two galaxies passed by close to each other. Such a near collision has been seen, but the galaxies have not collapsed. For the motion of stars in spiral galaxies, general relativity in the classical domain gets both the speed and stability of the system wrong. Ouch.
A second issue appears if one considers the motion of all the galaxies in the heavens, the biggest big picture one can imagine, and what happens under the effects of the always attractive gravity, the answer should be obvious: galaxies should be slowing down relative to each other. Even if the answer should be obvious, the practice of science involves confirming the obvious. It is a darn hard issue to address with real data. It relied on observations of Type 1a supernovae by teams of astronomers led by Saul Perlmutter, Brian Schmidt and Adam Riess, reported first in 1998. The answer came back with a huge surprise: galaxies are moving away from each other. WTF?
A third issue is the big bang theory, the theory theory, not the TV show. We know that everything was traveling at the same speed 400,000 years after that bang to one part in 100k through our observation of the cosmic background radiation. Yet the big bang predicts that most regions of space are space-like separated from each other (pointed out by Charles Misner, known as the horizon problem). This means that they could exchange no information, such as what speed they should all go.
The big bang theory theory also has an issue with stability (found by Robert Dicke and known as the flatness problem). Using the math of classical general relativity on the entire Universe at the beginning, the two stable solutions mathematically end up with either the Universe not getting out of bed, or flying out so fast nothing of consequence can form. Just make sure parameters are just so to one part in ten to the sixty or so, and we can get the nearly flat Universe we live in. Boy, was that lucky.
Failures Big or Small?
Consider the motions of galaxies. If the way to fix the model is dark matter, the failure is epic in both theory and practice. The standard model is marketed as the complete zoo of particles we can see. The no-see-'ems/dark matter is approximately five times the amount of stuff. And so far, we don't know what it is. A bunch of things have been ruled out. I don't keep track of how many things have lost out.
If something more along the lines of the Modification of Newtonian Dynamics (MOND) works out, then the issue is a one-liner, involving a new equation to govern gravity, no new mass needed. When gravity is super-wimpy, literally ten orders of magnitude smaller that it is on the Earth, MOND proposes that the form of the equation changes from an inverse distance squared form to an inverse distance form. Why gravity has such a mood swing is not clear. Using one parameter, it does fit quite a number of disk galaxies (I don't know its full report card).
One prediction of MOND is that wherever the mass is, so must be the gravitational potential. For a bullet galaxy passing through another galaxy, observations have shown that there is a separation between the visible mass and its gravitational potential.
What about the accelerating Universe? This is a one hypothesis town known as dark energy. The cosmological constant rises from the status as Einstein's greatest blunder to save the day. It adds more stuff than the visible stuff or dark master combined. There are defenders of dark energy that will tell you that if you don't get it, well, basically you are dumb.
I confess, I am dumb. I am not proud of that by the way, it is frustrating. A vacuum is, well, nothing. I have done the calculation of how the variation of level of energy in a vacuum is not zero, but the energy, its average energy, is zero. The variation does have measurable effects on observations. The average energy does not. Acceleration requires work to be done. Call me ultra-orthodox, but I don't see how the vacuum could accomplish anything material like accelerating galaxies. That would require an awesome work engine to do anything to 10^43 kg that makes up a typical galaxy.
Here is a look at the data:
This looks, well, subtle. I appreciate all the effort to eliminate possible errors. It matters that two separate teams independently ruled out all kinds of possible errors.
On the final front, to fix the big bang, there is only one game in town: inflation. The inflation period is driven by stuff with insane properties that do not fit in the standard model. Fine-tuning how long the inflation period goes on replaces the fine-tuning of the big bang which doesn't seem like an improvement. A professional critique of inflation is delivered by Paul Steinhardt, the Albert Einstein Professor of Physics at Princeton. We have a simple Universe, so we should have a simple theory to describe it. Prof. Steinhardt does not think inflation is like that. He trying to make alternative proposals.
Interlude: The Planet Vulcan
This is not a reference to Star Trek. The ultra-orthodox fringe doesn't watch the series since the science is so wrong. Space is treated like Manhattan: get on the subway and there is a short journey from Wall Street, to Chinatown, to Harlem. Space is extraordinarily more empty than screenwriters can deal with ("...and ten thousand generations later they arrived at the destination which had no life").
Urbain Jean Joseph Le Verrier was French, the head of the Paris Observatory. I still find it hard to believe that in the time before computers and light bulbs, he was able to do the calculations needed to predict to within one degree where the planet Neptune was located due to a perturbation in the orbit of Uranus. He tried to get things exactly right, accounting for perturbation issues to seventh order even though that required more than 400 terms in his work. All by hand. Insane.
The one problem was Mercury. The observations said there was a precession (wobble) of 5600 arcseconds per century, yet his calculations said there should be only 5557. Off by 43 arcseconds. Because he had succeeded with Neptune, Le Verrier predicted there was another planet, the planet he called Vulcan, the god of the fire in volcanoes.
Finding such a planet is exceptionally difficult due to the overpowering brightness of the Sun. With the great due diligence of many amateur astronomers, the planet was discovered many times in Le Verrier's lifetime. He died believing there was a new planet. Other astronomers trusted the calculations, but not the observations of a new no-see-'em planet.
The two problems in theory have to do with quantum gravity. The three problems in observations are for classical systems - slow moving and low density (at least one year after the big bang). General relativity saved the day when there was only one problem in theory and one observation (the precession of the perihelion of Mercury). Given the historical pattern, it is not unreasonable to dream for new equations for both classical and quantum gravity. Our research money should still flow to the reasonable hypotheses in town: dark matter, dark energy, and inflation.
Note to commenters:
This blog will be kept on the problems with gravity and the three mainstream efforts to fix those problems. Should you have a new fix, congratulations. I will however remove comments that promote the new effort as off-topic for this focus of this particular blog. You are free to write your own blog on this very site.