Tag Archives: gravity

Einstein was Right All Along

General relativity has a reputation for being very difficult. I think the reason is that it is very difficult.

Leonard Susskind, General Relativity: The Theoretical Minimum

Two events have left me obsessed with the concept of gravity. Such an obsession is fundamentally unhealthy, I’ll admit it. I will nonetheless explain it to you.

First, let us back up a little. While I have an educational background in science and engineering, higher-level math was never my strength. Thus while Physics 101 was both interesting and fairly easy for me, when it came to further courses in Physics, my sailing was not quite so smooth. Electromagnetism, quantum mechanics, relativity, and light all left the realm where my own intuition could no longer help me along and where my tenuous grasp on matrix math and advanced calculus came around to kick me in the tush.

Despite both coursework and a subsequent lifetime spent reading, through which I’ve sought to understand all of these concepts, my self evaluation is that my understanding remains fairly superficial.

The two events that sent me down the path to insanity took place in 2019. The first was that I viewed the TV show The Expanse, which had become available via streaming. The show’s attempt to get right certain aspects of space travel made it stand out from its science fiction peers and got me thinking about the details. Around about the same time, I read the book The Perfect Theory, which helped me cogitate upon some of these same issues.

It took a few years, but these thoughts ultimately began to consume me. I found an old social media post, from the fall of 2021, where I described waking up at night convinced that I could do the requisite math in my head (and, therefore, must before going back to sleep) to solve complex calculations for orbital mechanics. By 2022, I had mulled on this enough to share with my readers here (in another post).

That now-three-year-old post was framed as a question but I had already begun working on creating some answers. The problem was I could not develop for myself, based on intuition, any answers… and no wonder! From the beginning of his conception of General Relativity, Einstein himself struggled to define experiments that could distinguish between “real” gravity (caused by proximity to a very large, massive object) and “apparent” gravity, caused by an externally-driven acceleration.

Let me catch you up with what has happened, inside my own head, since then.

Once again, a science fiction series has driven me to move on this. Since that last post, I had the opportunity to finish the First Formic War trilogy (now almost 10 years old) by Orson Scott Card. In this set of books, there is extensive use of anti-gravity technology to drive several major plot arcs. The way that technology is used strikes me as wrong; both from the engineering and the physics standpoints. Because I’ve spent so much energy dwelling on this, the wrongness is almost painful to me.

I’ll contrast this to the original Ender’s Game. Like many science fiction works that preceded it, Ender’s world sees mankind having developed some control of gravity. The book (and if you haven’t read it, maybe you should) takes place on a space station where the future officers in a space defense force are trained. Ender notes that, in some cases, the gravity on the station defies physical logic but no explanation is given. To a reader, this is satisfying. Maybe the inter-species space warfare imparted some knowledge upon mankind or maybe it is a result of many decades of technological advancement. The fact that gravity and anti-gravity are common in space sci-fi books and, especially, films makes it easy to accept.

The Formic Wars (starting with the novel Earth Unaware) takes place long before Ender’s Game. In it, we find out that the development of gravitation “lensing” will be an entirely human achievement. Through judicious manipulation of the gravity field in the vicinity of an aircraft, the military will develop a rotorless helicopter with near-infinite lift capacity. At the same time, a private company has developed a gravity laser that can disrupt a local gravitation field so as to eliminate the integrity of solid masses.

The flaw in this has to do with balance. There are concepts such as conservation of energy that can be calculated and could, I suspect, throw a big monkey wrench into this concept. It also attempts a technical explanation without considering the equations for the fields that are being disrupted. Some not-so-simple math, one might think, would explain to me what is so troublesome about this as a near-future technology. Unfortunately, one of my problems, apparent for several years now, is that lack of the required mathematical ability.

Non-fiction, scientific books on the subject of gravitation seem to either either with “imagine you’re floating in space” or “here is the equation for a Lorentzian manifold; you should be thoroughly familiar with this math from your prior course work.” What I really needed was something in between. The good news is that I found that something.

The book General Relativity: The Theoretical Minimum by Leonard Susskind provides a reasonably happy medium between those two extremes. It starts its readers off with a minimum of assumptions about their mathematical background. It suggests that the reader should already have gone through some of the earlier books in his series but I found that, for the most part, that wasn’t necessary. A basic university-level understanding of math (even if decades out of use) plus a couple of internet searches seemed more than sufficient to follow along.

The book goes on to illustrate how a physicist might manipulate advanced math concepts using the shorthand representation for vector and tensor equations. Doing so allows one to evaluate the implications of such equations without actually having to solve them. Susskind does, in fact, encourage the reader to work through the math by hand (for simpler solutions) to aid in understanding but I did not do so. I found I could accept the assertions of the author without having to prove them for myself and was comfortable following along with the abstract notions.

Realize, too, that I was reading this book at night, in bed. The last thing I wanted to do was to get up and start doing lengthy math computations when I really should have been sleeping.

The second big impression General Relativity made on me was about black holes. In particular, it answered the question as to why there is so much focus on black holes in nearly everything I’ve read so far.

Indeed, this book does focus on black holes. Heavily. It introduces early a discussion about “falling into a black hole” with then a promise that you can come back to that after establishing a foundation. The second half of the book (more than half, really) is then dedicated to further analysis of black holes and their effects. Surely there are far more interesting things in this universe than the (admittedly very interesting) black holes?

The answer has to do with the math. One way to untangle the incredible complexity is to find situations where the terms of these equations go to zero. Doing so will allow the solutions to be extracted from otherwise unsolvable differential equations. One way to usefully zero out terms is to imagine the space/time in the vicinity of a mass approaching the infinite and a size approximating a dimensionless point.

Further investigation into the nature of this mathematical “singularity” began producing interesting properties. For example, the mass need not be infinite – just very large, and the size is not “zero1.” This prompted astrophysicists to speculate on how such an entity might come into being.

In 1939, Einstein himself published a paper, “On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses,” attempting to use General Relativity to prove that black holes were a physical impossibility. Within months, Robert Oppenheimer published “On Continued Gravitational Contraction,” using general relativity to prove that they should exist. It would take until 1972 for the effects of a presumed black hole (Cygnus X-1) to be observed, thus proving that the mathematical slight-of-hand, in fact, corresponded to a physical thing.

To me, it is quite remarkable that this “imagine if…” turned out to be something that is real and not particularly uncommon. That, combined with the easy math, goes a long way towards explaining why a book like General Relativity expends so much of itself on Black Holes rather than just General Relativity in general.

So, contrary to my title, we learn that Einstein was initially wrong about Black Holes. In “general” (oh I slay myself), though, his is a story of being right when those around him thought otherwise. When his peers felt that his descriptions of the physical universe were just too strange to be true.

This brings me around to that frequent subject of this site and my ponderings upon the true nature of gravity. If my feel for the state of the science, as rough as that is, is correct – this remains even today a subject for debate.

Specifically, I am thinking about Einstein’s assertion that there is no difference between an object in free fall within a gravitational field and an object outside of any effects of gravitation. In this understanding, it must be the shape of time and space that causes the apparent application of forces rather than the balance between gravity, acceleration, and momentum that makes the Newtonian understanding of gravity.

The alternative to Einstein’s explanation is that gravity is, truly, a “force field” similar to an electromagnetic field. That when an object in orbit which sees its forces balance out to zero, as Newtonian physics would explain, that this doesn’t mean those forces don’t exists. It is instead a convenient mathematical outcome. It’s a fine distinction but, in this this conception, we must consider Einstein’s version something other than the fundamental nature of the universe, as he proposed it. Instead, that fundamental nature has something to do with gravitons or unified fields – things we don’t yet fully understand – and Einstein’s thought experiments are just a nice way to get our minds around this deeper reality.

Dr. Susskind is in this second camp. Similar to my own line of thinking (if I may be so bold as to put myself in a league with a Stanford theoretical physics professor), he has gone further and proposed a thought experiment that would allow one to distinguish between these two concepts. His experiment involves a 2,000 mile tall man.

Imagine, he suggests, an astronaut who is 2,000 miles in height. Our big fella decides to don a spacesuit and engage in some outer space sky-diving, dropping back towards home from an orbital height. Imitating an atmosphere-limited jumper, his body is aligned flat with the earth’s surface, perpendicular to the direction in which he is falling. Encountering a sudden sense of dislocation combined with a momentary blindness, he finds himself unsure as to whether he is floating in the vastness of space or freefalling towards his home planet – two states which Einstein says are identical. But for our very tall spacefarer, they are not. If at some orbital height near the earth, he will be able to determine that “down” has different directions for his head versus his toes. More technically speaking, he will sense some compression as the two ends of his massive body move towards the earth following converging trajectories. If he were all alone in deep space, by contrast, he would feel no such sensation.

While the 2,000 miles of manflesh is necessary to make the differential significant, one might imagine that ability to detect the phenomenon exists (Planck’s insights notwithstanding) for a more reasonably sized body. It should be possible to design an experiment to measure the effect and, proof in hand, show that the nature of gravity is a force field and that Einstein’s thought experiments fail when one strays outside their applicability.

For my own peace of mind, I can’t fully accept Dr. Susskind’s reasoning and I am further comforted that there are others, far better educated than I, who feel similarly. My gut tells me that his 2,000 mile mind experiment glosses over certain realities. When he imagines his man in orbit, isn’t he, at least in some ways, imagining inhabiting the world of Newtonian physics? Might he be neglecting some reality of space-time curvature? Might he be ignoring the difference between a point mass and a distributed mass and the limitations of communication across distances. Can a 2,000 mile tall man even evaluate simultaneity properly so as to determine that he is both in free fall and under a gravitational-induced stress? I’ll even offer (without details) how this thought experiment has some common features with “ladder paradox” in special relativity. Might have similar resolution?

Part of me wants to believe that there it is possible to gain an intuitive understanding of General Relativity; one that does not need to depend on advanced math. General Relativity: The Theoretical Minimum gets me a little bit closer to bringing it all together in a way that, as a mildly committed amateur, enables me to comprehend the nature of the universe. If I found any issues with Dr. Susskind’s presentation, these were minor relative to the value that this book provided to me.

As to Einstein, I previously highlighted the date on which he published his paper on General Relativity. Yet it was on THIS day, November 18th, that he made his discovery. He was looking at a problem with Mercury’s orbit and the fact, since 1859, its perihelion (the point where the planet is closest to the sun) advanced 43 inches2 per century more than physics predicted. While physical explanations were considered (e.g. an invisible moon around Mercury), an alternative was that Newton’s law of gravitation was wrong.

Since 1911, Einstein has realized that empirical validation of this theories would have to come through astronomical observations. In the case of Mercury’s orbit and Newton’s inverse square law, this ε, the discrepancy or error factor, provided an opportunity to put his theories to a test. Applying his theory of gravitation to Mercury’s orbit produced, exactly, the measured orbital advance of 43 inches per century with no additional, unknown factors required. The computations were finalized on November 18th, 1915 and this discovery informed the paper that he submitted on the 25th.

  1. It is a spheres whose dimensions are described by the Schwarzschild radius. The book explains what that means. I will not. ↩︎
  2. The calculation in 1859 was an error of 38″ per century. By 1882, the number was corrected to 42″ per century. ↩︎

Attractive Theory?

I can’t believe that it has been more than a year since I’ve written anything here. I’d blame the lingering effects of the Great Pandemic, except that doesn’t make any sense at all.

Even today, all I have to offer you is something partial and inconclusive. It is just a question, really. But to ask it I need to think about some other questions and so set the stage for my final, wild conjecture.

Great Questions

What is the nature of gravity? What is it that causes massive objects, with no physical interconnection, to exert forces upon each other over such vast distances?

I’m no expert, and as such I have no problem mixing current and viable theories together with ones that make no sense. So what might be the mechanism through which gravity exerts its pull? Is there a gravitational ether through which an object’s mass can influence other bodies? How about gravitons, cycling back and forth between (astronomically speaking) nearby entities? Are we simply missing something important in a unified field theory that would explain the nature of a gravitational field? Is there even any such thing as gravity in the first place?

Einstein’s General Relativity explains that gravity does not, in fact, exist. The acceleration that we feel, standing here on Earth every day, is not due to the Earth’s downward pull upon us, but rather due to the fact that we are standing in an accelerating frame of reference. His demonstration of the truth of this is – if we were in, instead, an accelerating reference frame in empty space, we would experience exactly the same environment. This is called the “equivalence principle.”

To put it in simplified terms, the reality is that the earth and everything on it is accelerating upwards at (from where we sit) 9.8 m/s2. So then why isn’t the earth and everything on it exploding outward into a million pieces? If that’s what you are imagining, you are trying to think of that acceleration as only a spatial phenomenon. Einstein’s explanation requires considering the integrated space-time in which the universe exists.

Simple Conceptualization

The short (very short) answer is that our non-inertial frame is accelerating in space-time even as the spatial coordinates, as centered on the earth, remain fixed. In terms of the governing equation, the spatial acceleration term is exactly balanced out by the space-time curvature term. We are accelerating in space so as to PREVENT us from accelerating* in space-time.

This conveniently solves the problem of the mechanism by which gravity affects objects through vast distances of spatial vacuum. If there is no gravity, there is no “force” holding an object within its stable orbit… it’s simply an object in motion that shall remain as such, as Newton might have said. It does so, however, by creating a new and equally-perplexing problem.

If an orbit is simply a straight line in space-time, this suggests that the presence of a large mass has “distorted” space-time itself. In this understanding, a large mass (or a small one for that matter) bends space all around it so as to create a “gravitational well.” But by what mechanism can a lump of matter distort the very nature of time and space and for vast distances all around it? Isn’t this an even harder “reality” to conceptualize than determining the physics of a gravitation force (which, absence of a unifying theory notwithstanding, doesn’t look all that different from an electromagnetic field)?

Which Came First?

My question, one that it has now taken me over 4,000 words to formulate, is quite simple. What if we have the cause and effect backward? Rather than matter distorting space and time, maybe it is the ripples in space and time, caused by the vast energies involved in the formation of the universe, that results in mass being exactly where and when it is.

To give an example, maybe our sun is at the exact shape, size, and position that it is because there happens to be, right there, a sun-sized distortion in the shape of the surrounding space. Sounds crazy, right? But how can you tell the difference? Did the gravitation hole that the sun sits in gather up the sun? Or did the sun make the hole? Chicken or egg? If it’s the former, though, no weird theories are needed to explain how the sun “bends” space… the space was already bent.

Throwing DARTS

For myself, it is interesting that, just as I was trying to articulate this concept and get it written down, I read of exactly the sort of experiment that might help dis/prove my point. NASA recently launched a spacecraft so as to collide with an asteroid. The experiment, called the Double Asteroid Redirection Test (DART), was designed to help evaluate our ability to divert astral objects from their course. What I read about DART had to do with that double asteroid thing and the measurement of experimental results.

NASA’s spacecraft struck the smaller of a pair of asteroids; the smaller being in orbit around the larger. The mission’s success was measured through the deviation of the orbit of that smaller, struck asteroid around the larger of the pair. In the first couple weeks of October, calculations confirmed an orbital shift similar to what was expected.

The fun part, as far as this post is concerned, came near the end of the article. It mentioned that we earthlings have more in our arsenal besides diverting a trajectory through a collision. These include, said the DART team, “shooting asteroids with ion beams” or using “a so-called gravity tractor.” This last is defined as “a spacecraft that looms near an asteroid and exerts gravitational pull on the space rock for an extended time.”

If there is no gravity, how does this work? It must require an altering of the “shape” of space/time in a non-trivial way for what, frankly, is a fairly modest expenditure of energy. Is there a way to tease out some of the reality of an answer to my “which came first” question? Maybe I can come back here (in less than a year, this time) with some thoughts about thought experiments.

*The straight line, inertial frame in which we wish to remain is “free fall.” Or to put it another way, an orbit. Let’s save the implications of that for another post.

Related

In the fall of 1915, after ten years of analysis, Albert Einstein presented his gravitational field equations of general relativity in a series of lectures at the Royal Prussian Academy of Sciences. The final lecture was delivered on November 25th, 104 years ago.

Yet it wasn’t until a month or so ago that I got a bug up my butt about general relativity. I was focused on some of the paradox-like results of the special theory of relativity and was given to understand, without actually understanding, that the general theory of relativity would solve them. Not to dwell in detail on my own psychological shortcomings, but I was starting to obsess about the matter a bit.

Merciful it was that I came across The Perfect Theory: A Century Of Geniuses And The Battle Over General Relativity when I did. In its prologue, author Pedro G. Ferreira explains how he himself (and others he knows in the field) can get bitten by the Einstein bug and how one can feel compelled to spend the remainder of one’s life investigating and exploring general relativity. His book explains the allure and the promise of ongoing research into the fundamental nature of the universe.

The Perfect Theory tells its story through the personalities who formulated, defended, and/or opposed the various theories, starting with Einstein’s work on general relativity. Einstein’s conception of special relativity came, for the most part, while sitting at his desk during his day job and performing thought experiments. He was dismissive of mathematics, colorfully explaining “[O]nce you start calculating you shit yourself up before you know it” and more eloquently dubbing the math “superfluous erudition.” His special relativity was incomplete in that it excluded the effects of gravity and acceleration. Groundbreaking though his formulation of special relativity was, he felt there had to be more to it. Further thought experiments told him that the gravity and acceleration were related (perhaps even identical) but his intuition failed to close the gap between what he felt had to be true and what worked. The solution came from representing space and time as a non-Euclidean continuum, a very complex mathematical proposition. The equations are a thing of beauty but also are beyond the mathematical capabilities of most of us. They have also been incredibly capable of predicting physical phenomena that even Albert Einstein himself didn’t think were possible.

From Einstein, the book walks us through the ensuing century looking at the greatest minds who worked with the implications of Einstein’s field equations. The Perfect Theory reads much like a techno-thriller as it sets up and then resolves conflicts within the scientific world. The science and math themselves obviously play a role and Ferreira has a gift of explaining concepts at an elementary level without trivializing them.

Stephen Hawking famously was told that every formula he included in A Brief History of Time would cut his sales in half. Hawking compromised by including only Einstein’s most famous formula, E = mc2. Ferreira does Hawking one better, including only the notation, not the full formula, of the Einstein Tensor in an elaboration on Richard Feynman’s story about efforts to find a cab to a Relativity conference as told in Surely You’re Joking, Mr. Feynman. The left side of that equation can be written as, Gμν. This is included, not in an attempt to use the mathematics to explain the theory, but to illustrate Feynman’s punch line. Feynman described fellow relativity-conference goers as people “with heads in the air, not reacting to the environment, and mumbling things like gee-mu-nu gee-mu-nu”. Thus, the world of relativity enthusiasts is succinctly summarized.

The most tantalizing tidbit in The Perfect Theory is offered up in the prologue and then returned to at the end. Ferreira predicts that this century will be the century of general relativity, in the same way the last century was dominated by quantum theory. It is his belief we are on the verge of major new discoveries about the nature of gravity and that some of these discoveries will fundamentally change how we look at and interact with the universe. Some additional enthusiasm shines through in his epilogue where he notes the process of identifying and debunking a measurement of gravitational waves that occurred around the time the book was published.

By the end of the book, his exposition begins to lean toward the personal. Ferreira has an academic interest in modified theories of gravity, a focus that is outside the mainstream. He references, as he has elsewhere in the book, the systematic hostility toward unpopular theories and unpopular researchers. In some cases, this resistance means a non-mainstream researcher will be unable to get published or unable to get funding. In the case of modified gravity, he hints that this niche field potentially threatens the livelihood of physicists who have built their careers on Einstein’s theory of gravity. In fact, it wasn’t so long ago that certain aspects of Einstein’s theory were themselves shunned by academia. As a case in point, the term “Big Bang” was actually coined as a pejorative for an idea that, while mathematically sound, was too absurd to be taken as serious science. Today, we recognize it as a factual and scientific description of the origin of our universe. Ferreira shows us a disturbing facet of the machinery that determines what we, as a society and a culture, understand as fundamental truth. I’m quite sure this bias isn’t restricted to his field. In fact, my guess would be that other, more openly-politicized fields exhibit this trend to an even greater degree.

Ferreira’s optimism is infectious. In my personal opinion, if there is to be an explosion of science it may come from a different direction than that which Ferreira implies. One of his anecdotes involves the decision of the United States to defund the Laser Interferometer Space Antenna (LISA), a multi-billion dollar project to use a trio of satellites to measure gravitational waves. To the LISA advocates, we could be buying a “gravitational telescope,” as revolutionary in terms of current technologies as radiotelescopy was to optical telescopes. The ability to see further away and farther back in time would then produce new insights into the origins of the universe. But will the taxpayer spend billions on such a thing? Should he?

Rather than in the abstract, I’d say the key to the impending relativity revolution is found in Ferreira’s own description of the quantum revolution of the past century. It was the engineering applications of quantum theory, primarily to the development of atomic weapons, that brought to it much of the initial focus of interest and funding. By the end of the century, new and practical applications for quantum technology were well within our grasp. My belief is that a true, um, quantum leap forward in general relativity will come from the promise of practical benefit rather than fundamental research.

In one of the last chapters, Ferreira mentions that he has two textbooks on relativity in his office. In part, he is making a point about a changing-of-the-guard in both relativity science and scientists, but I assume he also keeps them because they are informative. I’ve ordered one and perhaps I can return to my philosophical meanderings once I’m capable of doing some simple math. Before I found The Perfect Theory, I had been searching online for a layman’s tutorial on relativity. Among my various meanderings, I stumbled across a simple assertion; one that seems plausible although I don’t know if it really has any merit. The statement was something to the effect that there is no “gravitational force.” An object whose velocity vector is bent (accelerated) by gravitational effects is, in fact, simply traveling a straight line within the curvature of timespace. If I could smarten myself up to the point where I could determine the legitimacy of such a statement, I think I could call that an accomplishment.