Mach’s Principle – Relating Gravity to the Universe

432px-Ernst_Mach_01What’s the origin of the inertia of masses? This is one of the most profound questions in physics. Let’s talk about this aspect of Mach’s principle, named after the Viennese physicist and philosopher Ernst Mach (1838-1916). Mach suggested that the weakness of gravity was due to the universe’s enormous size. Of course, the first estimates for the size and mass of the universe were not available before 1930, after Hubble’s discovery of the expanding universe. Sir Arthur Eddington first noticed that the gravitational constant may be numerically interrelated with the properties of the universe. By dividing c2by G, he obtained roughly the same value as dividing the universe’s mass by its radius:

c2/G ≈Mu/Ru. This is quite extraordinary.

Astonishingly, Erwin Schrödinger had already considered this possibility in 1925 (Ann. Phys. 382, p. 325 ff.), though he couldn’t know about the size of the universe!

Mach, around 1887, could not even have dreamed of such measurements, but his aspiration to formulate all the laws of dynamics by means of relative movements (with respect to all other masses!) turned out to be visionary. Einstein was guided to general relativity by that very idea. Although Einstein gave Mach due credit, he didn’t ultimately incorporate Mach’s principle in his theory. Mach’s central idea, that inertia is related to distant objects in the universe, does not appear in general relativity.

One possibility to realize Mach’s principle is to make up a formula where the gravitational constant is related to the mass and size of the universe. Because the universe is so huge, G would be very tiny. This radical idea was advocated by the British-Egyptian cosmologist Dennis Sciama in 1953ii:

c2/G Σ mi/ri,

Sciama2the sum taken over all masses i in the universe. Or, in other words, the gravitational potential of all masses in the universe amounts just to the square of the speed of light – intriguing!

Later, very few researchers pursued the idea, among them the cosmology “outlaw” Julian Barbour.iii During the inauguration of the Dennis Sciama building at the University of Portsmouth in England in June 2009, everybody talked about all the work that Sciama had done… except for Sciama’s reflections on inertia. What a pity!

It is idle to speculate whether Einstein would have warmed up to the idea. In any case, the size of the universe could only be guessed fifteen years after the completion of general relativity. So today, Mach’s principle has a miserable reputation and is sometimes even dismissed as numerological hokum. To the arrogant type of theorists who consider the question of the origin of inertia as obsolete chatter from Old Europe, I recommend they to look up “Inertia and fathers”, by Richard Feynman, on YouTube. In this video, Feynman wonders about one big question. Where does inertia actually come from? A truly fundamental problem. For more, see chap. 5 of “Bankrupting Physics”.

iiSciama, MNRAS 113 (1953), p.34.

iiiBarbour, gr-qc/0211021.

Can Dimensionful Constants Have a Fundamental Meaning?

WhittOf Course!

As with many of the thoughts in this blog, this is contrary to common wisdom, but I think it particularly weird how the perceived wisdom that “only dimensionless constants can have fundamental meaning” has been established. Not only has this idea become representative of a methodology that has replaced thinking by calculating, but the full ignorance of the statement reveals itself only if we look at the history of physics, for example, with the book “A History of the Theories of Aether and Electricity” by Sir E. Whittaker.

First, let’s recall how the idea came to be. It doesn’t require exceptional intelligence to realize that our definitions of the meter and second, etc., are arbitrary, and it is just as evident that dimensionful constants, such as h, c, G, and others, are expressed with such arbitrary units. This discussion has attracted some attention in the debate over theories about the variable speed of light, and it has been claimed (e.g., by Ellis) that every dimensionful quantity can be set to unity using an appropriate reference. While this is a possible mathematical formulation, the question remains whether such a procedure makes sense physically. It doesn’t!

If we think of a temperature map, then according to the above logic, as temperature is a dimensionful quantity, it could be set to unity at every point – the analogy is one-to-one, but the number of people appreciating a forecast with unit temperature would appear limited.

This is probably why Einstein didn’t mind pondering the variation of a dimensionful quantity when he considered a variable speed of light in 1911. Are we to understand from Ellis’ critique that Einstein didn’t have a clue about the basics of his own theory? As one of the few reasonable people discussing the subject, John Duffield, has recently pointed out by reference to original quotes, that Einstein’s attempts around 1911 were a sound approach to describe the phenomenology of general relativity with a variable speed of light. In the meantime, other researchers have shown that even general relativity can be formulated in terms of a variable speed of light.

Today’s physicists are not only ignorant about these ideas, but they are actively distributing the ideology that “only dimensionless constants can have fundamental meaning”, just like three theorists summarizing their discussion in the CERN cafeteria. Oh, had Einstein had the opportunity to listen to their half-assed thoughts, while having a cappuccino there! Surely, he would suddenly have understood how misguided his 1911 attempts on a variable speed of light were. (cf. Whittaker II, p. 153). There is more to tell, but to see the full absurdity of the “only dimensionless constants are fundamental” argument, look at Whittaker’s treatise on the development of electrodynamics. (vol. I, p. 232). Kirchhoff, Weber, and Maxwell would never have discovered the epochal relation ε0 μ0 =1/c2, and had they not assigned a meaning to the above “dimensionful” constants – our civilization would never have been bothered by electromagnetic waves.

What Einstein Said About Fundamental Constants


Reality and Scientific Truth: Discussions with Einstein,von Laue and Planck is a collection of conversations between Einstein and Ilse Rosenthal-Schneider, a lady who took a PhD in philosophy in Berlin around 1920. After fleeing the Nazi regime in 1938 and settling in Australia, she continued her discussions with Einstein by letter. The book is a unique source of the views Einstein held about the fundamental constants of physics. It is worthwhile to compare them with some modern opinions on the subject.

In a letter dated May 11th, 1945, Einstein wrote that he believed that numbers “arbitrarily chosen by God” do not exist and that their “alleged existence relies on our incomplete understanding”. Similarly, he contended in a letter of March 24th 1950 that “dimensionless constants in the laws of nature, which, from a rational point of view, could have other values as well, shouldn’t exist”.

Such a statement obviously refers to numbers such as the inverse of the fine structure constant (which is about 137), which is reminiscent of Richard Feynman, who forty years later wrote in a very similar vein: “It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say, ‘the hand of God’ wrote this number, and we don’t know how He pushed his pencil.” At the same time, Feynman advised his colleagues that: “All good theoretical physicists put this number up on their wall and worry about it.”

It is interesting that both Einstein and Feynman seemed to be convinced that this was a puzzle to be tackled while at the same time being aware that such a conviction was not testable in a strict sense. Einstein, in another letter from Oct 13th, 1945, wrote: “Obviously, I can’t prove that. But I cannot imagine a reasonable unified theory which contains a number that could have been chosen differently by a whim of the creator.”

Since then, the attitude of physicists toward fundamental constants seems to have changed. Modern cosmology, for example, has terrific data, and Michael Turner of the University of Chicago explained in 2010 in Munich how these observations are accommodated using a couple of parameters by the current ‘concordance model’ of cosmology. After the talks at the Siemens foundation, there is always an ample possibility for discussion (and a very nice buffet). Turner, as he admitted frankly to me, is not very interested in the initial conditions of the universe such as density, photon to baryon ratio or similar stuff. He would be happy to find the correct equations that would make the world go, rather than bothering with how it got started. No doubt, however, these initial conditions (although Einstein would have been delighted by the data) are numbers he would have sought to explain using the theory. All these numbers make the world a little more complicated than it should be in principle. “A theoretical construction has little chance to be successful, unless it is very simple” Einstein wrote on April 23rd, 1949, and here he was certainly in agreement with Isaac Newton’s credo: “Truth is ever to be found in simplicity, and not in the multiplicity and confusion of things.”

Antiblog: The Current vs the Relevant

Blogging is largely against my own convictions. Fundamental physics has a slow pace, and there is no need to comment on its progress on a daily basis. Rather I believe that overestimating the news of the day, a habit that was called “arrogance of the presentˮ by the Swiss writer Max Frisch, is a serious problem in physics, if not in all sciences. The danger is that important questions may fall into oblivion, rather than that something fundamental will be overloooked in the stream of news. You probably won’t miss very much if you don’t scan the arXiv for a couple of months.

Secondly, and probably owing to the fact I’ve written three books now, I believe that blogs are on average a superficial kind of text. By definition, they miss the maturation of ideas that comes after several rounds of rewriting and correcting. A book is a much more elaborate and refined piece of work. Somebody has invested to maximize the ouput for you, the reader. The bestselling author Nassim Taleb (The Black Swan, Antifragile) doesn’t read any newspapers or journals (and I infer, no blogs either), just books. He says that the job of a scholar is to ignore insignificant current affairs. So, you may ask, why this blog?

The purpose is to focus on the relevant, not on the current, to dig out the forgotten gems of physics, rather than commenting on fashionable events. Physics is no sport where we have to rush about. Einstein, in his memoir The world as I see it, wrote on the development of general relativity:

“The years of anxious searching in the dark, with their intense longing, alternating between confidence and fatigue, and their eventual breakthrough to truth – only those who have experienced it can understand that.”

Maxwell, Newton and Kepler needed decades to arrive at their revolutional findings. It’s worth looking at how they got there. The history of physics is not boring stuff, you need to deal with if you want to evaluate the current state of affairs. Thus there will be a lot of history, intriguing problems, and hopefully stuff that will get you to reflect on the fundamental questions of physics.