Hermann Minkowski, Albert Einstein and Four-dimensional Space-time

Is the concept of free-will valid as it relates to humans? A mathematics lecture presented in September of 1908 in Cologne, Germany by Hermann Minkowski not only paved the way for the successful formulation of Albert Einstein’s general theory of relativity in 1916, it also forced us to completely revamp our intuitions regarding the notion of time and space while calling into question the concept of human free-will! Some brief and simplified background is in order.

Prior to Minkowski’s famous lecture concerning Raum Und Zeit (Space and Time), the fabric of our universe was characterized by three-dimensional space accompanied by the inexorable forward flow of time. The concept of time has long been a stubbornly elusive notion, both in philosophy and in physics. From the mid-nineteenth century onward, there had increasingly been problems with our conception of “time.” The difficulties surfaced with the work of James Clerk Maxwell and his mathematical characterization of electromagnetic waves (which include radio waves and even light) and their propagation through space. Maxwell revealed his milestone “Maxwell’s equations” to the world in 1865. His equations have stood the test of time and remain the technical basis for today’s vast communication networks. But there was a significant problem stemming from Maxwell’s work, and that was his prediction that the speed of light propagation (and that of all electromagnetic waves) is constant for all observers in the universe. Logically, that prediction appeared to be implausible when carefully examined. In fact, notice of that implausibility stirred a major crisis in physics during the final decades of the nineteenth century. Einstein, Poincare, Lorentz and many other eminent physicists and mathematicians devoted much of their time and attention to the seeming impasse during those years.

Enter Einstein’s special theory of relativity in 1906

In order to resolve the dilemma posed by Maxwell’s assertion of a constant propagation speed for light and all related electromagnetic phenomena, Albert Einstein formulated his special theory of relativity which he published in 1906. Special relativity resolved the impasse created by Maxwell by introducing one of the great upheavals in the history of science. Einstein posited three key stipulations for the new physics:

A new law of physics: The speed of light is constant as determined by all “observers” in the universe, no matter what their relative motion may be with respect to a light source. This, in concert with the theoretically-based dictate from Maxwell that the speed of light is constant for all observers. Einstein decreed this as a new fundamental law of physics. In order for this new law to reign supreme in physics, two radical concessions regarding space and time proved necessary.
Concession #1: There exists no absolute measure of position and distance in the universe. Stated another way, there exists no reference point in space and no absolute framework for determining distance coordinates. One result of this: consider two observers, each with his own yardstick, whose platforms (habitats, or “frames of reference,” as it were) are moving relatively to one another. At rest with respect to one another, each observer sees the other’s yardstick as identical in length to their own. As the relative velocity (speed) between the two observers and their platforms increases and approaches the constant speed of light (roughly 186,000 miles per second), the other observer’s “yardstick” will increasingly appear shorter to each observer, even though, when at relative rest, the two yardsticks appear identical in length.
Concession #2: There is no absolute time-keeper in the universe. The passage of time depends on one observer’s velocity with respect to another observer. One result of this: consider our same two observers, each with their own identical clocks. At rest with respect to one another, each observer sees the other’s clock as keeping perfect time with their own. As the relative velocity (speed) between the two observers and their platforms increases and approaches the constant speed of light, the other observer’s clock appears increasingly to slow down relative to their own clock which ticks merrily along at its constant rate.

Needless to say, the appearance in 1906 of Einstein’s paper on special relativity overturned many long-held assumptions regarding time and space. Einstein dissolved Isaac Newton’s assumptions of absolute space and absolute time.The new relativity physics of Einstein introduced a universe of shrinking yardsticks and slowing clocks. It took several years for Einstein’s new theory to gain acceptance. Even with all these upheavals, the resulting relativistic physics maintained the notion of (newly-relative) spatial frames defined by traditional coordinates in three mutually perpendicular directions: forward/backward, left/right, and up/down.

Also still remaining was the notion of time as a (newly-relative) measure which still flows inexorably forward in a continuous manner. As a result of the special theory, relativistic “correction factors” were required for space and time for observers and their frames of reference experiencing significant relative, velocities.

This framework of mathematical physics worked splendidly for platforms or “frames of reference” (and their resident observers) experiencing uniform relative motion (constant velocity) with respect to each other.

The added complications to the picture which result from including accelerated relative motions (the effect of gravity included) complicated Einstein’s task enormously and set the great man on the quest for a general theory of relativity which could also accommodate accelerated motion and gravity.

Einstein labored mightily on this new quest for almost ten years. By 1913, he had approached the central ideas necessary for general relativity, but the difficulties inherent in elegantly completing the task were seriously beginning to affect his health. In fact, the exertion nearly killed Einstein. The mathematics necessary for success was staggering, involving a complex “tensor calculus” which Einstein was insufficiently prepared to deal with. In desperation, he called his old friend from university days, Marcel Grossman, for help. Grossman was a mathematics major at the Zurich Polytechnic, and it was his set of class notes that saved the day for young Einstein on the frequent occasions when Einstein forsook mathematics lectures in favor of physics discussions at the local coffee houses. Grossman’s later assistance with the requisite mathematics provided a key turning point for Einstein’s general theory of relativity.

Enter Hermann Minkowski with Raum Und Zeit

The initial 1909 publication of Raum Und Zeit

On September 8, 1908 in Cologne, Germany, the rising mathematics star, Hermann Minkowski, gave a symposium lecture which provided the elusive concepts and mathematics needed by Einstein to elegantly complete his general theory of relativity. Similar to Einstein’s 1906 special theory of relativity, the essence of Minkowski’s contribution involved yet another radical proposal regarding space and time. Minkowski took the notion of continuously flowing time and melded it together with the three-dimensional coordinates defining space to create a new continuum: four-dimensional space-time which relegated the time parameter to a fourth coordinate point in his newly proposed four-dimensional space-time.

Now, just as three coordinate points in space specify precisely one’s physical location, the four-dimensional space-time continuum is an infinite collection of all combinations of place and time expressed in four coordinates. Every personal memory we have of a specific place and time – each event-instant in our lives – is defined by a “point” in four-dimensional space-time. We can say we were present, in times past, at a particular event-instant because we “traversed-through” or “experienced” a specific four-dimensional coordinate point in space-time which characterizes that particular event-instant. That is very different from saying we were positioned in a specific three-dimensional location at a specific instant of time which flows irresistibly only forward.

What do Minkowski’s mathematics imply about human free-will?

By implication, the continuum of four-dimensional space-time includes not only sets of four coordinate points representing specific events in our past (place and “time”), the continuum must include points specifying the place and “time” for all future events. This subtly suggests a pre-determined universe, where places and “times” are already on record for each of us, and this implies the absence of free-will, the ability to make conscious decisions such as where we will be and when in the future. This is a very controversial aspect of Minkowki’s four-dimensional space-time with distinctly philosophical arguments.

For certain, however, is the great success Minkowski’s mathematics of space-time has enjoyed as a basis for Einstein’s general theory of relativity. Most, if not all, aspects of Einstein’s special and general theories of relativity have been subjected to extensive experimental verification over many decades. There is no instance of any validly conducted experiment ever registering disagreement with Einstein’s special or general theories. That is good news for Hermann Minkowski, as well.

Minkowski’s new reality takes us beyond the two-dimensional world of a flat piece of paper, through the recent universe of three-dimensional space plus time, and into the brave new world of not only four-dimensional space-time, but curved four-dimensional space-time. The nature of curved space-time serves to replace the Newtonian notion of a gravitational force of attraction which enables the celestial ballet of the heavens. For instance, the orbit of earth around the sun is now regarded as the “natural path” of the earth through the curvature of four-dimensional space-time and not due to any force of attraction the sun exerts on the earth. According to the general theory of relativity, the mass of the sun imposes a curvature on the four-dimensional space-time around it, and it is that curvature which determines the natural path of the earth around the sun. Minkowski and his mathematics provided the final, crucial insight Einstein needed to not only radically redefine the nature of gravity, but to also successfully complete his general theory of relativity in 1916. Einstein’s theory and its revelations are generally regarded as the most significant and sublime product ever to emanate from the human intellect. Take a bow, Albert and Hermann.

My eulogy to Hermann Minkowski

Albert Einstein is assuredly the most recognized individual in human history – both the name and the image, and that is very understandable and appropriate. Very few in the public realm not involved with mathematics and physics have ever even heard the name, “Hermann Minkowski,” and that is a shame, for he was a full participant in Einstein’s milestone achievement, general relativity. Minkowski’s initial 1907 work on Raum Und Zeit came to Einstein’s attention early-on, but its mathematics were well beyond Einstein’s comprehension in that earlier time frame. It was not until several years later, that Einstein and Marcel Grossman began to recognize Minkowski’s gift to general relativity in the form of his mathematics of four-dimensional curved space-time.

Hermann Minkowski delivered his by-then polished lecture on space-time at Cologne, Germany, in September, 1908. Tragically, he died suddenly in January, 1909, at the young age of forty-four – from a ruptured appendix. His latest findings as presented in the Cologne lecture were published in January, 1909, days after his death, sadly.

The “lazy dog” has the last bark

Albert Einstein and Hermann Minkowski first crossed paths during Einstein’s student days at the Zurich Polytechnic, where Minkowski was teaching mathematics to young Einstein. Noting Einstein’s afore-mentioned irregular attendance at lectures in mathematics, the professor reportedly labeled the student Einstein as, “a lazy dog.” Rarely in the annals of human history has such an unpromising prospect turned out so well! I noted with great interest while researching this post that Einstein long regarded mathematics as merely a necessary tool for the advancement of physics, whereas Minkowski and other fine mathematicians of the past tended to consider mathematics as a prime mover in the acquisition and advancement of knowledge, both theoretical and practical; they viewed physics as the fortunate beneficiary of insights that mathematics revealed.

In the late years, Einstein came to appreciate the supremely important role that mathematics plays in the general advancement of science. As proof, I will only add that the great physicist realized his dependence on the mathematicians Grossman and Minkowski in the nick of time to prevent his theory of general relativity from going off the rails, ending on the scrap heap, and leaving Albert Einstein a completely spent physicist.

Note: For a detailed tour and layperson’s explanation of Einstein’s relativity theories, click on the image of my book: The Elusive Notion of Motion – The Genius of Kepler, Galileo, Newton, and Einstein – available on Amazon

THINK. Thinking is Hard Work

The history of IBM, the International Business Machine Corporation is as storied as any the world has seen. In recent times, Apple Computer had its iconic guru, Steve Jobs, to pave its pathway to fame and fortune. In earlier times, IBM’s Thomas J. Watson served much the same role in building his company into the tech giant it was to become. Watson coined the famous admonition, THINK – his way of spurring on the company’s workforce to bigger and brighter contributions. I recall as a youngster seeing his famous single-word motto displayed in such diverse places as banks, schools, and other institutions.

Photo: IBM Archives

IBM headquarters at Endicott, New York, 1935. Note the “THINK” motto emblazoned on the building. Pictured are 25 female college graduates, newly trained for three months as IBM system service women. Their role: after assignment to IBM branch offices, they assisted salesmen in assessing customer requirements and training customers on the use of IBM equipment. Their three male instructors are also pictured.

I find Watson’s admonition at once simple, yet profound. What does constitute the notion of “thinking,” and why is that a very non-trivial exercise? Critical thinking is important across all life-disciplines. I would venture, however, that science and engineering are more viable as gateways to understanding the process of critical thinking than most activities in which we humans are involved. Recall the oft-used phrase: “Its not exactly rocket science!”

My acquaintance with the subject derives from my educational and career background as an electrical engineer, here, in Silicon Valley, California. Anyone who has studied chemistry, physics, and mathematics at the college level can truly appreciate the notion of critical thinking. During my undergrad and graduate level years, I can recall, more than I care to admit, the long hours (even nights) spent on a concept or a homework problem that just would not submit to standard perusal.

Such incidents would call for sweeping aside the current method of attack in favor of a fresh new visualization of the problem. Often, this nasty situation occurred late at night while working under pressure to complete a homework assignment due the next day. The scenario just described demands what Thomas Watson so unabashedly promoted as his corporate motto: THINK. When persistence coupled with a fresh approach saved the day for me as a student, and later as working engineer, the joy of sudden insight and mastery of the issue at hand was sweet, indeed. That very joy and satisfaction serve to fuel the desire of science and engineering students to keep on studying and learning, despite the prospect of new and greater challenges ahead. One soon realizes that learning is primarily about harnessing the ability to think!

Thinking is hard, and most of us do not spend enough time doing it. At my advanced age and despite an active curiosity in earlier years, I still find myself formulating questions about all matter of things which I had never questioned before. Often my questions have to do with things financial. For instance: “Why is a rising stock price beneficial to the corporation involved since the corporation generally does not sell its stock directly to traders and investors? Ordinary folks outside the corporation who own shares as investors would seem to be the primary beneficiaries of such gains, and, yet, the mechanisms of corporate finance somehow bestow significant rewards to the corporation as well. How, exactly, does that work?” For a business major, that probably seems a naïve question, but, then again, how many business professionals have thought deeply about Einstein’s theory of special relativity? For us non-business types, it is quite easy to participate successfully as an investor in the complex equities market without really understanding what goes on “behind the curtain.” Ease of use leads to complacency, and complacency is ever the enemy of informative curiosity, it seems.

I worry about the younger generation, so many of whom seem to be satisfied with accumulating “factoids,” little isolated bits of information from the internet and social media. Thomas Watson understood that “to think” meant forming often non-obvious connections between seemingly isolated concepts and bits of information…and that is the hard part of thinking. The resulting “whole” of the picture which emerges by connecting the dots often proves the key to great scientific progress or profitable business opportunities.

Thinking was hard work even for history’s greatest minds. Isaac Newton stated the belief that his greatest personal asset was the ability to hold a particularly intractable problem clearly in his mind’s eye for days and weeks on-end while his conscious and sub-conscious mind churned toward a solution. Newton was clearly aware that such discipline and capability was not an attribute possessed by the rest of us. While attempting to apply his newly created laws of celestial mechanics to the complex motions of our own moon, Newton confessed to experiencing excruciating “headaches” over his difficulties with the moon’s motion. Thinking was hard, even for the greatest mind in recorded history! Certainly, the problems tackled by Newton were of a complexity far beyond our own everyday challenges. Albert Einstein attributed the essence of his genius to “merely” a combination of raging curiosity and the mule-like persistence which he brought to bear when uncovering nature’s most guarded secrets. Thinking and discovery were hard work for Einstein, as well.

The self-stated attributes of these two towering intellects have, as their common foundation, the willingness and the ability to THINK – to think long and hard about difficult problems and critical relationships in the physical world. I concur with Thomas J. Watson: although operating on a much lower plane than Newton and Einstein, we all need to THINK more deeply than ever about the world around us and about who we are. Consider the legacies left to us by Newton and Einstein – all the result of unbridled curiosity and the willingness to think deeply in search of answers to their own questions.

The Essence of Einstein’s Brainchild

       

Albert Einstein – the name conjures-up a multitude of images. Along with the familiar and numerous likenesses born of camera and pen, are the mental images of genius, perhaps best exemplified in the public mind by his famous theories of relativity. Everyone knows of these, yet relatively few know much about them. It is my goal in this post to “explain” in two brief and simple statements, the essence of his famous 1905 theory of special relativity. Discussion of his 1916 general theory is deferred for perhaps another post. Yes, special relativity can be very effectively summarized in the following two premises: Premise #1: There is no reference point in space itself which allows for the measurement of absolute motion through space. Accordingly, measurements of motion and velocity made from one “frame of reference” (such as earth or a spacecraft) are meaningful only relative to other similarly “non-privileged” distinct frames of reference (such as the sun, another planet, a distant star, an orbiting spacecraft, etc.). Premise #2: The speed of light is constant for all observers no matter what the motion/velocity between light source and observer. Congratulations, you now understand the essence of special relativity. That was easy! Premise #1 states that there are no markers in empty space, no coordinate grid-lines which allow for determining position “in space” or velocity “through space.” How fast is the earth traveling “through space?” This, we cannot determine; motion “through space” has no validity. Motion relative to another distinct body (another frame of reference) is a valid concept, and subject to measurement. Premise #2 of special relativity relates Einstein’s pronouncement of a new and completely counter-intuitive universal law of nature. Here is the simple but amazing ramification: Imagine a sender of light and a receiver of that light, and both are stationary with respect to the earth. The sender measures the departing speed of his flash of light at 186,000 miles per second as it departs in a straight line toward the receiver who measures its arrival at 186,000 miles per second – as expected. Now the receiver accelerates until he is traveling toward the still-stationary sender at 93,000 miles per second – half the speed of light. Both the sender and the receiver continue to measure the light beam at exactly 186,000 miles per second with respect to themselves! Amazing, but true, and it required no less than the drastic re-structuring of time and space in the special theory to accommodate the constant speed of light as just exemplified in the above example. The two simple assertions by Einstein in premise #1 and premise #2 fomented an upheaval in physics which is virtually unprecedented. Not since Isaac Newton published his monumental book, The Principia, in 1687, had physics seen anything like the revolution Einstein created with his mundane-appearing, but bombshell paper, On the Electrodynamics of Moving Bodies which appeared in the 1905 German scientific journal, Annalen Der Physik. Recall from my earlier post on Einstein (Feb.21, 2013) that he was, at the time, an unknown, part-time, kitchen-table physicist doing physics on weekends and during evenings! With the easy task of stating the essence of special relativity behind us, the difficulty begins for those who wish to probe more deeply into relativity. That difficulty is not inherent in deciphering complex mathematics as might be pre-supposed. Rather, the difficulty is rooted in the task of accepting the necessary but extremely counter-intuitive implications that result from the twin premises of special relativity. These implications, that Einstein insisted must be true and real, include slowing clocks and shrinking rulers resident in one frame of reference (on earth, for example) when viewed from another frame of reference (an uber-speed rocket, for example) passing by at a significant percentage of the constant speed of light, 186,000 miles per second! Coming to terms with such counter-intuitive ideas of non-absolute space and non-absolute time requires considerable imagination and logic. We won’t go there in this post – see my book, The Elusive Notion of Motion, for the details. It took considerable courage on Einstein’s part to insist that the abandonment of Newton’s absolute space and absolute time be considered not merely hazy implications of the twin premises, but, rather, dictates of physical reality. It took well over ten years for even the learned physics establishment to grudgingly fall in line with Einstein’s radical suggestions. In the many years since 1905, not one of the numerous high-level experiments conducted to prove (or disprove) special relativity has countered Einstein’s assertions!

The Big Question and the Key Clue

Now that you “know” the special theory, you might be interested in the simple central question that led Einstein to his dramatic proposals in 1905: Is there a “medium” in space which is required to support the propagation of light (as electromagnetic waves) as postulated by James Clerk Maxwell in 1865? When we reflect upon the fact that air is required to propagate sound waves and water is necessary to produce the spreading wave-ripples in a pond when we drop a stone, we can see the logic of the question. Which answer did physics wish to see? If yes, then the seemingly common-sense need for a supporting medium for any wave propagation could be extended to light-waves. That would be the “good” news of a “yes” answer. Now the “bad” news: Such a proposed medium extending throughout space, tentatively called an “ether sea” at the time, would dictate the speed of light through it and make the observed speed of light dependent on the observer’s motion/speed relative to the ether sea – contrary to important contemporary astronomical observations and laboratory experiments indicating that the speed of light seemed constant to all observers in all states of motion!   The famous Michelson-Morley physics experiments in 1881 and 1887 disproved the conceptually difficult concept of an “ether sea” and supported the prevailing belief in the constant speed of light. Today, physics accepts the fact that “empty” space can nevertheless support the transverse electromagnetic waves which comprise light without having the kind of physical nature exemplified by air or water. I hope you will take-away the following from this post: That Einstein revolutionized physics through special relativity (and in other instances) by asking simple, real-world questions and doggedly pursuing the answers to them. By his own admission, he asked the kind of curiosity-based questions that children would ask and adults might be afraid to – a marvelous aspect of Einstein, the man and the scientist! His focus on the important question of a propagation medium for light (and, ultimately, its non-existence) led him to his sublime and beautiful theory of relativity. The question was simple, and the twin premises of the outcome, special relativity, are easily stated. The devil is in the details, and there the difficulty resides – the difficulty of questioning, of thinking outside the box, of overturning the familiar, the comfortable, the accepted. Therein, in the daring application of simple but precise logic while doggedly pursuing nature’s “truths,” lies the true genius of an Albert Einstein.