![maximizes the proper time between two events, is called a geodesic, and I will state the following ‘geodesic
principle’: An object with no forces acting on it will always follow a geodesic, which is the longest proper time
path between two points in spacetime.” [Emphasis added]
All objects travel through spacetime at a constant speed equal to the speed of light. If an object moves
at a certain speed through space, its speed through time is reduced due to the minus signs in the
equation. This leads to some counter-intuitive results. Consider the example depicted below.
Alice and Bob live on an orbiting space station, depicted by the dashed red circle around the Earth.
Since they are in a state of free fall and experience no external forces, both of them follow the longest
possible paths in spacetime. Bob is assigned to go on a ten-year mission to the outer reaches of the
solar system, so he must accelerate during his trip, requiring external forces that reduce his path length
through spacetime. He departs the space station from Point A in spacetime on January 1, 2024 and
rejoins Alice at Point B ten years later.
Meanwhile Alice orbits the Earth 58,400 times during the ten years. Since her path through spacetime
is a geodesic, she doesn’t travel through space at all in her frame of reference, as shown as a red
horizontal line along the time axis. Bob’s path travels through both time and space with respect to
Alice. Although Bob’s path appears longer than Alice’s in Euclidean space, it is actually a shorter path
due to the bizarre hyperbolic geometry of spacetime. In effect, Bob took a shortcut through spacetime.
A shorter spacetime path is equivalent to less proper time, so when Bob and Alice are reunited, his
clock has lost time during his journey compared to Alice’s. Furthermore, even though both Bob and
Alice traveled through spacetime at the same speed, c, and Alice took a longer path, they still managed
to come together again at Point B! This is completely counter-intuitive to those of us whose brains are
wired for living in the everyday world of Euclidean geometry.
The yellow arrow in the above illustration is the path of a light ray radiating from Point A. Light
travels the maximum possible distance in space, Δx, in a time interval Δt: Δx = c Δt. From the
Minkowski formula, we find that the spacetime distance light traveled, Δs, is zero! In other words,
light is motionless in spacetime, and it doesn’t experience the passage of proper time, Δτ. The above
example covers the gist of relativity without taking gravity into account.
Part III – The Physics of Spacetime with Gravity
Einstein’s genius was his ability to visualize complicated situations using simple thought experiments.
During the “miracle year” 1905, he published four milestone papers, one on special relativity, one on
the photoelectric effect, one on energy-mass equivalency, and another on Brownian motion. Following
that, he turned his attention to the general case of relativity with acceleration and gravity. Part II,
- 5 -](https://image.slidesharecdn.com/gravity-240317143211-c1b24973/85/Gravity-Superstrings-or-Entropy-A-Modest-Proffer-from-an-Amateur-Scientist-7-320.jpg)
Gravity: Superstrings or Entropy? A Modest Proffer from an Amateur Scientist
- 1. Gravity: Superstrings or Entropy? A Modest Proffer from an Amateur Scientist by John Winders
- 2. Note to my readers: You can access and download this essay and my other essays directly from the Amateur Scientist Essays website using this link: https://sites.google.com/site/amateurscientistessays/ You are free to download and share all of my essays without any restrictions, although it would be very nice to credit my work when quoting directly from them. If you would like to leave comments via email, you can send them using this link: mailto:Amateurscientist@hotmail.com
- 3. Part I – String Theory I wanted to learn as much as I could about string theory, which is currently the hottest topic in the theoretical physics community. There are a lot of string theory experts I could turn to, but their writings a videos seem to proselytize instead of inform.1 Peter Woit has physics degrees from Harvard University and a PhD in particle physics from Princeton. He currently holds a position in the mathematics department at Columbia University and is skeptical about string theory in general. He seemed to be someone who has a lot to say about the topic without trying to sugar coat it and promote it, so I picked up a copy of his book, Not Even Wrong – The Failure of String Theory and the Search for Unity in Physical Law. I must admit his book was rough going. In the introduction to the book, Woit said he was going to explain the topics of quantum field theory, gauge symmetry theory, and the standard model of particle physics without using math, which was a hopeless task in my case. After struggling through the first ten chapters without understanding most of it, I finally got to the part about string theory. The motivation behind string theory was the Theory of Everything, which is sometimes referred to as the Grand Unified Theory. Most physicists believe (mistakenly) that quantum mechanics and general relativity are “incompatible.”2 Ever since the quantum revolution occurred in the 1920s, The Holy Grail of Physics has been finding a way to quantize gravity.3 The genesis of string theory was in 1968 when a graduate student, Gabriele Veneziano was investigating the properties of something called the S-Matrix. The S-Matrix is a quantum operator that maps the states of elementary particles before they interact to particle states after the interaction. The “S” stands for scattering, which is one of the ways particles can interact when they collide. It seems Veneziano stumbled on a formula using Euler’s beta function that seemed to fit the bill, and he published a paper on his discovery. In 1970, Leonard Susskind and Holger Nielsen proposed a simple physical interpretation of Veneziano’s formula by replacing sub-atomic particles with even tinier strings that inexplicably obey classical mechanics. Elementary particles being replaced by tiny strings obeying classical mechanics? Based on what I’ve learned, the whole point of quantum mechanics is that tiny particles do not obey classical mechanics, which is why it’s called quantum mechanics instead. This is a case of abductive reasoning, which often leads to scientific dead ends. According to Merriam Webster, abductive reasoning is: “A syllogism in which the major premise is evident but the minor premise and therefor the conclusion only probable. Basically, it involves forming a conclusion from the information that is known.” An example of forming a conclusion from the information that is known is when somebody fits a polynomial curve to a set of data points and then assumes the curve they discovered is some kind of fundamental law.4 Another way of putting it is, “If it walks like a duck and quacks like a duck, it must be a duck.” Susskind and Nielsen assumed if tiny classical strings agreed with Veneziano’s formula, then elementary particles must be tiny classical strings. The current standard model of elementary particles has a number of unexplained “free parameters” whose values can only be determined through measurements and cannot be derived from theory. Originally, it was hoped that string theory could provide the theoretical basis for those values, but it soon went far beyond that modest goal. 1 I tried to compile a list of the top promoters of string theory. Edward Witten is referred to as “string theory’s leading thinker” and Leonard Susskind is called the “godfather of string theory.” What I learned, however, is that almost everybody in the physics community is now promoting it, with a few exceptions like Woit. 2 The two theories are different because they apply to very different scales, but they most assuredly are not incompatible. To find out why, read my essay “Relativity and Quantum Mechanics are Not Incompatible”. 3 This was Albert Einsteins dream. He wasted 40 years of his career pursuing it from 1915 until his death in 1955. 4 This kind of curve fitting resulted in the Raleigh-Jeans catastrophe, where a black body would radiate an infinite amount of heat. In 1900, Max Planck’s energy quanta resolved it and started the quantum revolution. - 1 -
- 4. In order for the mathematics of string theory to “make sense,” the original version requires a 10- dimensional spacetime background – one time dimension and nine space dimensions, six of which are so tightly “curled up” they are unobservable on everyday scales. But by expanding the spacetime background to 11 dimensions, gravity seems to magically emerge from the mathematics. Now physicists believe that the Holy Grail of unifying gravity with the “other three forces of nature” seems to be within reach. Woit explains that the electromagnetic, weak and strong forces are already sort of unified in the sense that at very high energies all three seem to have the same strength. Gravity, on the other hand seems to be the weak sister of the other three. Here, something called “supersymmetry” comes into play, converting the plain-vanilla string theory into superstring theory, where all four forces of nature come together at super-high energies.5 Unfortunately, superstring theory comes with baggage, referred to as a landscape, where an implausibly large number of possible different kinds of universes could emerge from within it. That number is estimated to be in the range of 10500 . This is kind of disappointing because unless you know which of these universes we live in, you cannot use string theory to make any kind of falsifiable predictions about our universe, making it a non-scientific theory.6 Not to be discouraged, Susskind and others turned that particular lemon into lemonade. There are a plethora of constants in nature, like G, h, c, kB, for which there is no apparent reason for the values they have, and it is assumed that those values were just arbitrarily set at the moment when our universe was created (the Big Bang). However, these constants must be “just so” for the universe to be hospitable to intelligent life. In other words, a universe such as ours must be very “unlikely,” meaning that ours must be just one out of an enormous number of possible universes. Susskind et al latched on to the enormous superstring landscape with a potential of 10500 universes as a way of supporting something they call the anthropic principle, which proposes the reason why our universe is so hospitable to intelligent life is because we’re here!7 This conclusion is based on a common fallacy that is somewhat similar to abductive reasoning, i.e., deriving a conclusion from information that is known. The reasoning goes like this. If A is true, then B is almost certain to occur; thus, if B occurs, then A is almost certain to be true. In this case, A represents a very large number of possible universes which can emerge from the huge landscape of string theory, and each of them have different random values of free parameters in their standard models. If B represents a universe that is hospitable to intelligent life and if A is true, then it is almost certain that a Type B universe like ours will emerge somewhere in the landscape. Since B has in fact occurred (our universe is hospitable to life after all), therefore A must be true. Symbolically, this can be written as follows. P(B|A) ≈ 1 B→P(A) ≈ 1 P(B|A) is the probability that B is true, given that A is true. P(A) is the probability that A is true. So if B is true, then A is true. Unfortunately, this argument is entirely false. The conditional probability P(B|A) has no bearing at all on what the probability P(A) actually is. If it did, you could substitute almost anything for A and claim A is true.8 The anthropic principle relieves theorists of explaining why the so-called constants of nature have the values they do. Using string theory involves something called a perturbation expansion. It is similar to a Stirling power series expansion used to calculate the value of a mathematical function. Take the exponential 5 Don’t ask me what any of this means, because I have no idea. 6 Karl Popper was one outspoken critic who said the lack of falsification rendered string theory as pseudoscience. Susskind disparages those critics by calling them “Popperazi.” 7 According to Susskind, everyone at Stanford University now believes in this tautology. 8 Theists would say that A stands for “God Created It” and nobody could argue that He didn’t. - 2 -
- 5. function y = ex for example. The power series for y = ex is y = 1 + x + x2 /2 + x3 /6 + x4 /24 + … + xn /n! It turns out that the terms for large values of n vanish, making the Stirling series converge to a finite value ex . String theory is solved using a similar perturbative process by combining a series of expressions. Unfortunately, the higher-order terms don’t get smaller, so the combinations equal infinity. Quantum electrodynamics (QED), invented by Paul Dirac in 1928, uses a similar perturbative process that also generates infinities. Murray Gell-Mann and Francis Low figured out a way to eliminate those infinities using a dodgy mathematical technique called renormalization, which uses measured quantities to subtract away the infinities.9 But according to Woit, renormalization doesn’t work with string theory. In simple terms, you can’t use string theory to calculate anything. The fact that renormalization doesn’t work did not discourage the proponents of string theory. By 1985 there were five different string theories. In that year, Edward Witten (aka the Leading Thinker) made a startling statement at a string theory conference at the University of Southern California. He said that all five theories were actually part of a larger theory he called M-Theory, although he wouldn’t reveal what M stood for. Witten promised that when physicists figured out what M-Theory actually is, it would solve all the shortcomings of the existing theories, but as of 2024 nobody has figured it out. One thing that keeps string theory alive, besides the hope that someday it will work, is that the mathematics behind it are so “beautiful.” In Woit’s book, David Gross is quoted as saying that string theory could not be wrong because its incredibly beautiful mathematics could not be accidental. The idea that string theory reflects reality is that beauty begets beauty. In other words, since the physical universe is so beautiful, it must be based on a beautiful theory, and string theory is the most beautiful theory there is. First of all, this line of thinking is referred to in logic as a category error, where a quality is assigned to something that can only apply to a different category.10 I think the main reason why string theorists consider the mathematics as “beautiful” is because certain things appear to pop out unexpectedly, like gravity. Unfortunately, other things that don’t exist also appear to pop out, like a plethora of nonexistent massless particles and particles having mass that should be – but aren’t – detected by experiments using current particle accelerators. I would judge such a theory as being ugly instead of beautiful. Beyond all of this, it seems that working with string theory is an excruciatingly difficult task, described as a form of mental torture by some practitioners of the craft. Einstein warned about relying solely on complicated mathematics to describe reality. Here are some of his quotes. “Fundamental ideas play the most essential role in forming a physical theory. Books on physics are full of complicated mathematical formulae. But thought and ideas, not formulae, are the beginning of every physical theory. The ideas must later take the mathematical form of quantitative theory, to make possible the comparison with experiment.” Evolution of Physics, 1938 “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” Lecture on Geometry and Experience, 1921. “It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.” Lecture on the Method of Theoretical Physics, 1933. John A. Wheeler had this to say about the Theory of Everything. “Behind it all is surely an idea so simple, so beautiful, that when we grasp it - in a decade, a century, or a millennium - we will all say to each other, how could it have been otherwise? How could we have been so stupid?” Final lecture on Quantum Measurement, 1979. 9 Feynman wrote in 1985, “The shell game that we play is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process!” Fortunately, the shell game works in the case of QED. 10 As an example, wine tasters often assign “notes” (as in music) to fermented grape juice. - 3 -
- 6. String theory may be beautiful, although beauty is in the eye of the beholder. But it’s clear to me and to most people that it is anything but “simple,” because if it really were simple, it would be taught in high school science classes. In conclusion, I am highly skeptical that string theory (or M-Theory) will lead to anything scientifically useful for the following reasons. • It is based on the false belief that quantum mechanics is incompatible with general relativity.11 • It assumes gravitation is a “force” that needs to be “unified” with the electromagnetic, weak, and strong forces. This requires a force carrier particle, called the graviton, which does not exist in the standard model of particle physics. • It started out as a misguided attempt to replace sub-atomic particles with sub-sub-sub-atomic particles called strings that inexplicably obey the deterministic laws of classical mechanics instead of quantum physics; a version of Einstein’s idea of “hidden variables” that was finally disproved through a series of experiments beginning in 1980 that violated Bell’s inequality. • It elevates a conjecture incapable of making falsifiable predictions into a scientific theory. Even worse, the theory suggests the existence of elementary particles that do not exist. • It leads to the anthropic principle, which is based on the false premise that P(B|A) = 1 somehow proves A is true. • It employs abductive reasoning, which often leads to scientific errors. • It makes a number of category errors, such as comparing its “beauty” to that of physical reality. So if string theory is a dead end, what can be said about the phenomenon of gravity? The following sections will explore the principles of general relativity with a surprising connection to thermodynamics. Part II – The Physics of Spacetime without Gravity Shortly after Einstein’s paper on special relativity was published in 1905, Hermann Minkowski pointed out that all of the equations in that paper could be derived from a fundamental principle that space and time were parts of a four-dimensional manifold called spacetime, as expressed in the Minkowski equation below. c2 Δs2 = c2 Δt2 – Δx2 – Δy2 – Δz2 Here, c is the speed of light, Δt is a change in the temporal dimension of spacetime and Δx, Δy, and Δz are changes in the three spatial dimensions. The symbol Δs represents a distance along some path through spacetime. The minus signs in this equation indicate that the geometry of spacetime is exotic; it is based hyperbolic geometry instead of a plain-vanilla Euclidean geometry based on plus signs. What this means is that if a body travels at some velocity through space, its velocity through time is reduced. This leads to some counter-intuitive results as we shall see shortly. Sometimes Δτ2 is substituted for Δs2 , where Δτ represents “proper time” as measured by a clock. The following statement is taken from a course in general relativity at Wake Forest University.12 “Now, in the absence of forces, we know that objects move at a steady pace in a straight line. We normally think of a straight line as the shortest distance between two points, but in relativity, it turns out that a constant velocity motion along a straight line is actually the longest proper time between two events in space time. This path, which 11 The so-called incompatibility stems from the fact that quantum mechanics is inherently acausal and stochastic. Quantum indeterminacy is absolutely necessary in order for causality and determinism to exist at classical scales, as documented in my essays “It from Bit” and "Quantum Mechanics and Relativity Are Not Incompatible". 12 Refer to the class notes “General Relativity”. - 4 -
- 7. maximizes the proper time between two events, is called a geodesic, and I will state the following ‘geodesic principle’: An object with no forces acting on it will always follow a geodesic, which is the longest proper time path between two points in spacetime.” [Emphasis added] All objects travel through spacetime at a constant speed equal to the speed of light. If an object moves at a certain speed through space, its speed through time is reduced due to the minus signs in the equation. This leads to some counter-intuitive results. Consider the example depicted below. Alice and Bob live on an orbiting space station, depicted by the dashed red circle around the Earth. Since they are in a state of free fall and experience no external forces, both of them follow the longest possible paths in spacetime. Bob is assigned to go on a ten-year mission to the outer reaches of the solar system, so he must accelerate during his trip, requiring external forces that reduce his path length through spacetime. He departs the space station from Point A in spacetime on January 1, 2024 and rejoins Alice at Point B ten years later. Meanwhile Alice orbits the Earth 58,400 times during the ten years. Since her path through spacetime is a geodesic, she doesn’t travel through space at all in her frame of reference, as shown as a red horizontal line along the time axis. Bob’s path travels through both time and space with respect to Alice. Although Bob’s path appears longer than Alice’s in Euclidean space, it is actually a shorter path due to the bizarre hyperbolic geometry of spacetime. In effect, Bob took a shortcut through spacetime. A shorter spacetime path is equivalent to less proper time, so when Bob and Alice are reunited, his clock has lost time during his journey compared to Alice’s. Furthermore, even though both Bob and Alice traveled through spacetime at the same speed, c, and Alice took a longer path, they still managed to come together again at Point B! This is completely counter-intuitive to those of us whose brains are wired for living in the everyday world of Euclidean geometry. The yellow arrow in the above illustration is the path of a light ray radiating from Point A. Light travels the maximum possible distance in space, Δx, in a time interval Δt: Δx = c Δt. From the Minkowski formula, we find that the spacetime distance light traveled, Δs, is zero! In other words, light is motionless in spacetime, and it doesn’t experience the passage of proper time, Δτ. The above example covers the gist of relativity without taking gravity into account. Part III – The Physics of Spacetime with Gravity Einstein’s genius was his ability to visualize complicated situations using simple thought experiments. During the “miracle year” 1905, he published four milestone papers, one on special relativity, one on the photoelectric effect, one on energy-mass equivalency, and another on Brownian motion. Following that, he turned his attention to the general case of relativity with acceleration and gravity. Part II, - 5 -
- 8. above, introduced acceleration as a curved trajectory on a time versus space plot. Einstein’s inspiration for attacking the general case came from his realization that a man falling off a ladder under the influence of gravity does not experience any force of acceleration while he is falling.13 Furthermore, he realized that any physics experiment done in a 1-g accelerating frame of reference in deep space will produce exactly the same results when it is performed on Earth’s stationary 1-g surface. In other words, gravity is equivalent to acceleration.14 From the example shown on the previous page, Bob’s acceleration is indicated by curvatures in his path plotted on a graph of time versus space. But as far as Bob is concerned, his path is a straight line. So an alternative way of looking at this situation is that Bob’s path is indeed straight, and spacetime itself is curved around him while he accelerates. Since gravity is equivalent to acceleration, this means that gravity has the same effect as acceleration: Spacetime curves in the region surrounding a gravitating body. Einstein spent ten grueling years figuring out precisely how spacetime was stretched, bent and twisted under the influence of gravity, and his achievement culminated in two papers published in 1915, which made a famous prediction about the bending of light near the Sun. Arthur Eddington validated that prediction during a total eclipse of the Sun on May 29, 1919. If asked “why” gravitation curved spacetime, Einstein would simply answer that gravity is equivalent to acceleration, and acceleration is equivalent to curving spacetime; end of story. But that’s not really an answer. It’s the same kind of answer he gave when someone asked him what time is. He said, “Time is what is measured by a clock.” Really? Does a clock really measure “time”? Or does it count things, like grains of sand falling through an hour glass, or the number of rotations of the Earth on its axis, or the number of revolutions of the Moon around the Earth, or the number of swings of a pendulum, etc., etc. And what do counting those things have to do with “time”? There must be a more fundamental reason behind gravitational curvature of spacetime. I believe the answer is a quantity known as entropy. Part IV – The Physics of Spacetime Based on Entropy Thanu Padmanabhan, a theoretical physicist from India, showed that the exterior Schwarzschild equation can be rearranged in a way that is identical to the thermodynamic equation T dS = P dV + dE at the boundary of spacetime.15 This is hugely significant because it indicates that spacetime is more than just a mathematical abstraction; in many respects it behaves like a four-dimensional physical material that can be bent, twisted and stretched and possesses thermodynamic properties of temperature (T), pressure (P), energy (E) and entropy (S). Most theoretical physicists already accept the idea that the vacuum has pressure and energy. The idea of spacetime temperature and entropy may be a bit alien to them, but I believe it can be validated. Thermodynamic entropy, S, as defined by the Gibbs equation is equivalent to information, H, as defined by Claude Shannon because the same identical expression is embedded in them. S = – kB Σ pj Loge(pj) ↔ H = – Σ pj Log2 (pj) , where pj is the probability of a particular internal state, j. Thus, H = S (1.44 / kB), expressed in bits. 13 Einstein called this his “happiest thought.” 14 This is the reason why I insist that gravity is not a force. There is no force on a body as long as it is only under the influence of gravity. The only force that would be present is a force that causes a body to deviate from its geodesic path, causing an acceleration and curvature in its spacetime path. Since gravity is not a force, there is no need for the long sought-after (and non-existent) force-carrier particle called the graviton. 15 See Padmanabhan’s paper, “Thermodynamical Aspects of Gravity: New Insights”. - 6 -
- 9. My essay “It from Bit” derived a conceptual model of the physical universe from the Minkowski equation and the Bekenstein-Hawking equation applied to a curved two-dimensional temporal boundary enclosing an expanding three-dimensional spherical hyperbolic space. The salient features of this model are summarized in the following bullets. • The temporal boundary represents the Now moment. It is a two-dimensional curved surface with a uniform radius of negative curvature; i.e., it is the surface of a hyperbolic sphere.16 • The radius of curvature increases at the speed of light: R = c t • All events begin and end on the temporal surface. The three-dimensional interior of the hyperbolic sphere is a residue of the expanding Now moment, representing the “past.” • The expanding Now surface can be compared to a temporal tsunami traveling at the speed of light, and objects possessing mass are swept along by the tsunami. This is the underlying reason why all objects “travel” through spacetime at the speed of light. • Observers on the Now surface perceive the universe as three-dimensional space surrounding them, with three degrees of spatial freedom, x, y, and z. However, no matter which directions observers point into space, they always point along the direction of the radius of curvature, R, back through time toward a point called the temporal Beginning. Objects visible in the interior space are projections of prior Now moments that represent the “past.” • There is a 1:1 correspondence between physical properties on the expanding two- dimensional Now moment surface and the three-dimensional interior of the “past.” • The three-dimensional space filling the interior of the hyperbolic sphere is comprised of entropy. In the absence of gravity, each bit of entropy is spread evenly throughout the entire hyperbolic sphere and each of those bits of entropy is also present on the Now surface (see the correspondence principle in bullet No. 6 above). • The number of bits in the interior increases with volume and the cube of the radius: H R3 • The density of entropy present on the Now surface, expressed in bits/meter2 , increases in proportion to the radius: dH/dA R = c t. • This increase in density on the Now surface is sensed as the “flow of entropy” and is expressed as proper time measured by clocks. Thus, for non-accelerated bodies, proper time Δτ = Δt. • Since the area of the Now surface increases with the square of the radius, A R2 , and the density of entropy present on the surface increases with the radius, dH/dA R, the total entropy on the surface increases with the cube of the radius, H R3 , which equals the total entropy in the interior (see the correspondence principle in bullet No. 6, above). • Cosmological temperature, TC, is proportional to the temporal curvature, which is inversely proportional to the radius of curvature, R: TC R-1 • Spacetime energy is proportional to entropy times temperature: dE = kBTC dH. Thus, the energy present on the temporal Now surface is proportional to R2 and is equal to the total energy in the hyperbolic sphere’s interior (see the correspondence principle in the bullet No. 6, above). • Newton’s universal gravitational parameter, G, is not a constant. Instead, it is proportional to cosmological temperature and inversely proportional to the radius: G TC R -1 16 One cannot visualize what a sphere in hyperbolic space actually looks like. Every point on the surface of the sphere curves away from the center even though the all points on the surface have the same radius. The surface area of a hyperbolic sphere is equal to 4π R2 , the same as a sphere in Euclidean space. However, the volume of a hyperbolic sphere is equal to ⅔π R3 , or half the volume of a Euclidean sphere with the same radius. - 7 -
- 10. I admit the last bullet item surprised me, yet I convinced myself this is an unavoidable consequence of entropic gravity in an expanding universe. Although it flies in the face of conventional scientific wisdom, it provides a testable and falsifiable prediction, which string theory sorely lacks. I was able to flesh out approximate present-day numerical values of some of the parameters of the model, replacing the proportionality symbol, with an equals sign. I did this by using the current measured value of G with the assumption that R ≈ 13.8 × 109 light years at the present time.17 Entropy density of space: dH/dV ≈ 1044 bits/m3 = Constant Cosmological temperature: TC ≈ 10-30 K Energy density of space: dE/dV ≈ 10-10 J/m3 Notes: dH/dV is constant over time. Cosmological temperature, TC, is not related to the so-called cosmic microwave background temperature, TCMB = 2.7 K. The energy density of space, dE/dV, is proportional to TC, so it decreases over time in proportion to t-1 . The present value is within an order of magnitude of the energy density 10-9 J/m3 based on the cosmological constant. I consider this agreement very encouraging.18 The nice feature of my proposed model is that all relationships are are based on basic fundamental principles, and the numerical values of the parameters that arise from the model are based on current experimental measurements of G along with the fundamental constants kB, ħ, and c. Part VI – The Physics of Entropic Gravity So far, this essay has ignored gravity, except for its equivalence to acceleration. We saw earlier that Bob’s travel to the outer reaches of the solar system involved acceleration, which curved his trajectory on the graph of time vs space. Alternatively, his acceleration could be modeled as a straight path through curved spacetime. We also learned that geodesics are paths that maximize distances through spacetime, so gravity should alter spacetime in a way that continues to steer objects along paths that maximize those distances. Furthermore, we know that spacetime distances are equivalent to proper time as measured by clocks. We are finally at the point where we will learn why proper time measured by clocks is related to entropy. An article in “Quantum Information Theory” entitled “The New Thermodynamic Understanding of Clocks” discusses recent experiments, which prompted Gerard Milburn of the University of Queensland in Australia to state, “A clock is a flow meter for entropy.” We learned in Part IV that the density of entropy on the expanding two-dimensional temporal Now surface increases in proportion to time. This is because the total entropy on the surface matches the entropy within it according to the correspondence principle (bullet No. 6), so both must increase in proportion to R3 . I propose that the linear increase in the entropy density across the Now surface corresponds to the “flow of entropy” Milburn described. Since expansion is linear with time, as long as entropy is uniformly distributed over spacetime, intervals in proper time, Δτ, will be the same as Δt, which is the fastest rate a clock can record. We now can see how nicely the pieces of the reality puzzle fit together. 17 I used a value of R based on the currently-accepted cosmological model, where the Big Bang occurred 13.8 billion years ago. Part VIII will show the Big Bang value is actually too small. 18 On the other hand, the vacuum energy based from QED is higher by a whopping 122 orders of magnitude. Even still, the standard cosmological model has flaws, which are discussed in Part VIII, below. For one thing, it assumes that the total mass-energy of the universe was constant over 13.8 billion years. In the entropic model, the total mass-energy began as almost zero and increases in proportion to t2 . In 1937, Nobel laureate Paul Dirac reached this same conclusion from his large number hypothesis, and he also found that G t-1 , just like the entropic model. - 8 -
- 11. Clocks measure physical changes, like the motion of a pendulum, the rotation of the Earth, the orbiting of the Moon, etc. I’m convinced that all physical changes are facilitated by the increase, or flow, of spacetime entropy in the Now moment. So what happens to spacetime surrounding a large, massive body like the Sun? The most reasonable answer is that mass causes a displacement of entropy in the region of spacetime where the body is located. Since a given volume of space is equivalent to a set number of bits of entropy, a redistribution of entropy would alter volume and distort space. Since clocks measure entropy flow, such a redistribution of entropy would distort time as well. There is a basic ironclad principle that governs how this distortion must occur: Any redistribution of entropy in space must be coordinated with changes to the time dimension such that no clock traveling along any possible path through curved spacetime can run faster than a clock traveling through empty, flat spacetime. In short, Δτ ≤ Δt for all paths through spacetime. This rule, like so many other rules in physics, assures that causality is preserved. There are other examples of rules that are in place to preserve causality, such as: 1) no massive body can attain the speed of light with respect to any other massive body, 2) there are no hidden variables involved in quantum measurements, 3) quantum influences must travel instantaneously across space, 4) quantum entanglement can involve only one pair of quantum states, and 5) quantum states cannot be cloned. Einstein was engaged in an intense effort over ten years trying to figure out exactly how mass distorts spacetime. This resulted in the general theory of relativity with field equations employing difficult tensor algebra and nonlinear differential equations. Einstein couldn’t figure this out all alone, so David Hilbert provided help with the higher math. To this day, exact solutions to Einstein’s field equations exist only for a few special, highly-symmetrical cases. I don’t pretend to have anywhere near the mathematical skills of Einstein or Hilbert, although I do believe the underlying principles of entropic gravity are valid, and when they are applied, the mathematics would be in good agreement with general relativity. Incidentally, I’m not alone in my belief that entropy underlies space, time, mass, energy and gravity. Erik Verlinde is a Dutch physicist who is developing an entropic approach to gravity and momentum. His work is in progress, but it looks like he will make a major breakthrough in understanding how gravity works. One of his early papers on this topic, "On the Origin of Gravity and the Laws of Newton", is a good introduction to his work. Verlinde has extended this idea quite a bit since then, and he has since applied it to the mystery of “dark matter” and the anomalous rotations of spiral galaxies. Part VII – A Falsifiable Prediction of Entropic Gravity One of the things that prevents the string conjecture from being a true scientific theory is a lack of falsifiable predictions. The entropy-based spacetime I propose makes one prediction that has not been tested experimentally in the past, and it is highlighted on Page 10 of this essay and could easily be tested experimentally. It is well established that there are variances in measurements of Newton’s gravitational “constant” using torsion balances that are well beyond the noise and known measurement errors of those instruments. The variations seem to go through repeated cycles of approximately 5.9 years. Interestingly, variations in the length of day (LOD) measurements on Earth over the same cycle coincide with the variations in G-measurements.19 These changes also seem correlated with sunspot activity, which could correspond to changes in the nuclear-fusion energy output in the interior of the Sun caused by changes in gravitational pressure. It is my belief that changes in measurements using torsion balances are not due to instrument errors, but reflect actual changes in the value of G. 19 The source of the data can be found in “Measurements of Newton's Gravitational Constant and the Length of Day” by John Anderson, Gerald Schubert, Virginia Trimble, and Michael Feldman. - 9 -
- 12. So this is my prediction: According to my proposed entropy model, changes over time to spacetime temperatures and curvatures at the temporal boundary (the Now) would produce corresponding changes to G since G TC. Assuming this is true, those changes will cause corresponding changes throughout the entire solar system over a 5.9-year cycle. Therefore, measurements of the LOD of all the planets in the solar system will be in in sync with the variations in LOD of the Earth. Verlinde’s paper, cited on the previous page, showed that inertial mass and gravitational mass are both proportional to cosmological temperature, TC. The “constant” G simply relates the force required to move mutually attracting bodies apart, being proportional to the product of their masses divided by the square of the distance between them. Thus, any change to spacetime temperature will produce changes to the masses which are reflected in changes to G and the moment of inertia, I, of a rotating sphere. ΔTC Δm ΔG ΔI ΔLOD The fact that the data show the percent change ΔG to be greater than the percent change ΔLOD should not be a problem. The Earth is not a solid sphere, but a squishy one instead. Any tendency to increase the speed of rotation will result in increased centripetal acceleration and cause the bulge at the Earth’s equator to increase, particularly since much of the Earth’s surface is water. An increase in the bulge increases the moment of inertia, so since angular momentum is conserved, an increased bulge will oppose any increase in the Earth’s speed of rotation. In other words, ΔLOD for a squishy sphere like the Earth would be smaller than the ΔLOD of a more rigid sphere like Mars. If the ΔLOD of Mars is larger than the ΔLOD of Earth over the 5.9 year period, that would further support the prediction. Part VIII – The Trouble with Hubble The standard cosmological model (Big Bang Theory) is partly based on Einstein’s general relativity field equations and partly based on some astronomical observations and a lot of unsubstantiated assumptions. The entropic gravity model was derived from first principles and is discussed in much detail in my essay "The Universe on a Tee Shirt". The table below compares the two models. Standard Cosmological Model Entropic Gravity Model The universe began as a singularity of infinite density and infinite curvature. The total mass- energy of the universe has remained constant throughout time per the conservation law. The universe began with a single bit of entropy having around 24 times the volume and 10% the mass of a proton. The total mass-energy of the universe increases in proportion to t 2 . In the beginning, the expansion of the universe began at a rate greater than the speed of light and has slowed down over time. Lately, the rate of expansion has increased due to “dark energy.” Since the beginning, the radius of the universe has increased at the speed of light and it will continue to increase at that rate forever. The density of entropy is constant throughout space. The vacuum energy density is constant over time based on the universal cosmological constant, Λ, and Newton’s gravitational constant, G. The vacuum energy density is proportional to the constant entropy density times the cosmological temperature, TC , which is proportional to t -1 . Distance measurements are based the apparent brightness of stars with known intrinsic luminosities and applying the inverse square law. Red shift measures closeness of an object to the Beginning where time is t=0, and Now is receding from the Beginning at the speed of light. Based on red shifts versus distances, the age of universe is approximately 13.8 billion years. Based on the vacuum energy density, the age of the universe is greater than 26 billion years. - 10 -
- 13. The last row of the table under Entropic Gravity Model deserves some explanation. In “The Universe on a Tee Shirt,” the following formula was derived for the mass-energy density of space. ρ = 1 / (8π G tU 2 ) kg/m3 Here, tU is the age of the universe based on the value of G that decreases over time: G = KG / tU, where KG would be a universal constant. Solving for tU in the above equation, gives the following result. tU = √1/(ρ 8πG) Note that since both ρ and G are inversely proportional to time, the above expression has units of seconds. Based on cosmological-constant arguments, there is reason to believe that ρ ≤ 10-9 J/m3 at the present time. Using the present measured value of G and converting energy units into mass units and seconds into years, we find that tU ≥ 26.5 billion years, or about double the conventional scientific guesstimate of 13.8 billion years. There are others who challenge the current estimate of 13.8 billion years. Rajendra Gupta, an adjunct professor at the University of Ottowa, has come up with a very similar figure of tU = 26.7 billion years. He used a “tired light” model resulting in “constants of nature” that change over time, like Dirac did with the large number hypothesis. Needless to say, Gupta’s views were not well-received among physicists and cosmologist, some of whom have ridiculed his work.20 However, recent surveys using the JWST telescope shows the number of galaxies in the universe is about 10 times larger than was previously thought. Hmm … a volume 10 times larger has a radius a little over 2 times larger. The figure below shows red shifts versus distance of both models plotted on the same distance scale. The curved Hubble line shows stars of three different frequency shifts (yellow, orange, red) as being much nearer to us than those same stars that lie along the straight line of the entropic gravity model. This makes sense if G decreases over time. Intrinsic brightness of stars increase with surface gravity, so if stars in the “past” are assumed to have certain distances based solely on intrinsic brightness, they would actually be much brighter and thus farther away since G was greater in the past. 20 This would also explain why Gupta is an underpaid and overworked adjunct professor instead of a tenured full professor without teaching duties. Scientists who run counter to orthodox beliefs can expect this level of respect. In case you are interested, you can access his paper "JWST early Universe observations and ΛCDM cosmology". - 11 -
- 14. Part IX – A Superstring Promise Fulfilled by Entropy Instead One of the unsolved “mysteries” of physics is why the gravitational “force” is so much weaker than the other three forces in nature. I always felt comparing gravity to the electromagnetic force was kind of a dumb comparison – and a clear example of a category error. What does a comparison like that even mean? The mass ratio of the Sun to Jupiter is about the same as the mass ratio of a proton to an electron, so I guess it means that if the Sun were a proton and Jupiter were an electron, the ratio of electrostatic to gravitational attractions between them would be many powers of ten. String theory promised that gravity, electromagnetism, the strong nuclear force and the weak nuclear forces would become “unified” at high energies. String theorist and Harvard professor Lisa Randall even states the reason why gravity is so currently “weak” is that it somehow “leaks away from the gravity brane” into other hidden dimensions, whatever that’s supposed to mean.21 Dirac believed the ratio of the radius of the universe divided by the radius of a proton was the same order of magnitude as that gravity/electrostatic “force” ratio. He concluded that since the universe is expanding, gravity was much stronger in the past – the exact conclusion I reached with the entropic gravity model. This gave me an idea. Since gravity is proportional to the curvature of the temporal dimension at the spacetime boundary (the Now), then a new set of ratios below could be established. G0 / Gt = Rt / R0 = (Vt / V0)1/3 = (Ht / H0)1/3 , where G0, R0, V0 and H0 are the values of G, the radius, the volume and the total entropy of the universe at the Beginning, and Gt, Rt, Vt and Ht are those values on the present Now surface. We found in Part VIII that tU = 26.5 billion years, so Rt = 26.5 billion light years. Vt = ⅔ π R3 because of hyperbolic geometry. Converting light years into meters, Vt = 1.77 × 1075 m3 . Recalling from Part V that a cubic meter is equivalent to 2 × 1043 bits of entropy, Ht = 3.54 × 10118 bits. We will assume that at the Beginning, the total entropy of the universe, H0, was a single bit.22 Thus, G0 / Gt = (35.4 ×10117 )1/3 = 3.27 × 1039 Dirac’s ratio of universe / proton radius works out to be around 1039 , which is right in the ballpark of the ratio G0 / Gt above. I don’t know if multiplying present-day’s tiny G by that ratio would satisfy string theorists who are troubled that gravity is too “weak” compared to the other forces, but it sure seems to me that the “four forces of nature” do come together at very high energies in the Beginning. Speaking of high energy, the cosmological temperature at the Beginning, T0, would be 7.94 × 1040 times greater than today’s temperature. Using a revised time tU = 26.5 billion years instead of 13.8 billion years the cosmological temperature in the Beginning is as follows. T0 = 7.94 × 1040 × 7.4 × 10-30 K = 588× 109 K In terms of infinite Big Bang temperatures, 588 billion K may not seem all that impressive, and it’s a lot cooler than the 9.9 trillion K world-record temperature set by the folks at the Large Hadron Collider near Geneva. But remember, we are talking about cosmological temperatures, and not the 2.7 K sky temperature Penzias and Wilson measured with their microwave antenna in 1964. The bottom line is this: Superstring theorists like Randall have promised they would “unify” gravity with the “other three forces of nature” based on a theory of strings and branes vibrating in parallel supersymmetric dimensions. As of 2024, string theory failed to deliver on that promise; instead, it seems that a complete theory of entropic gravity may do the trick. 21 She is quoted as saying that in this article from the Harvard Crimson. 22 H0 = 1 bit was a completely arbitrary choice on my part, but it seems reasonable to me. H0 = 1 nat would also suffice. - 12 -
- 15. Appendix A – What Is Entropy Anyway? This essay is filled with references to a mysterious substance called entropy having two equivalent forms: Thermodynamic (Bolzmann) and Informational (Shannon). Entropy originated with thermodynamics in the Age of Steam. Scientists and engineers discovered that only a fraction of useful work can be extracted from heat energy. Rudolph Clausius noted that the amount of “non-useful” energy exiting a steam engine reduces useful work, and he named this reduction en- “energy” -tropy “change.” It was noted that heat always flows from hot to cold regions, smoothing out temperatures and reducing the amount useful energy. This always increases entropy of a system; thus, the second law of thermodynamic was born: Entropy of an isolated system can never decrease. Ludwig Boltzmann defined entropy based on a statistical foundation, using the formula below.23 S = kB loge (W) , where S is entropy, W stands for Wahrscheinlichkeit, the German word meaning probability, and kB is the Boltzmann constant, a fundamental constant of nature that relates changes in energy to changes in entropy at a constant temperature T: Δe = kB T ΔS. In fact, temperature can be defined in terms of the relationship between Δe and ΔS: T ≡ Δe / (kB ΔS). The term W isn’t really a probability with values between 0 and 1. It is actually the total number of internal states of a system if they all have the same probability, p, for a given measurable external state of pressure, temperature, energy, and so on. The more internal states there are, the lower the probability of being in a particular state, so W = p-1 . Thus, another way to express entropy is S = – kB loge (p). Some years later, Josiah Gibbs generalized this formula for cases where the internal states of a system have different probabilities by expressing S as a weighted sum of the logarithms of those probabilities. S = – kB Σ pj loge (pj), summed over all N possible states j = 1, 2, … , N Unfortunately, almost everyone (including some physicists) say entropy equals “disorder,” but the equation above contains nothing that remotely resembles “disorder.” Disorder is just a subjective judgment about things without any mathematical meaning. The confusion arises from the fact that “disorderly” states outnumber “orderly” ones, and so as the number of internal states increases, the “disorderly” states tend to vastly outnumber the “orderly” ones. However, “disorder” reveals nothing about what entropy really is. Entropy is only a function of probabilities and nothing else. Claude Shannon was the genius who introduced the world to information science. He realized information content is inversely proportional to the likelihood of that information, referring to it as a “surprise factor.” Based on “surprise factor,” he developed the following equation for information.24 H = – Σ pj log2 (pj), summed over all N possible symbols j = 1, 2, … , N Dropping kB from Gibbs’ formula and using the base-2 logarithm instead of the base-e logarithm makes Shannon’s equation identical to Gibbs’ equation with H expressed as a dimensionless number, the “bit.” The bottom line is entropy is equivalent to information.25 23 That formula is carved into Boltzmann's tomb. 24 For example, weather reports for the middle of the Sahara Desert would likely be the same every day; i.e., hot and dry. Since probability of hot and dry weather in the middle of the Sahara high, the surprise factor of that information is low. But if a flood warning were issued, the surprise factor of that information would be high because floods rarely occur in the middle of the Sahara. If p HD is the probability of “hot and dry” weather and p FW is the probability of “flood warning,” in general weather reports for the middle of the Sahara contain HWS = – pHD log2 (pHD) – pFW log2 (pFW) bits of information. Since pHD 1 and pFW 0, then HWS 0 bits in general for weather reports in the Sahara. 25 Shannon often referred to information as entropy in order to emphasize this point. - 13 -
- 16. Appendix B – TOE on a T-Shirt Physicist and astronomer Adam Frank had this to say about the Theory of Everything (TOE): “Once you have this TOE you are done. You know everything there is to know in principle. It will be so simple and elegant that it should be expressible via an equation that fits on a T-shirt.” I seriously doubt the equation of string theory will ever fit on a T-shirt, but I know of an equation that does fit, and I believe it captures everything there is to know in principle about the universe based on the entropic gravity model. The equation on the T-shirt above is another version of a formula derived in a previous essay. EU = π kB T c3 tU 2 / ħ G In the original version, EU was expressed in kg, so EU is converted to Joules by multiplying the above expression by c2 , altering it to the final form below. EU = π kB T c5 tU 2 / ħ G Because the term T/G is a constant, it is easy to see that the energy of the universe (both on the Now surface and in the interior space) is proportional to tU 2 = R2 /c2 . By dividing EU by kBT, we get an expression for Shannon entropy H (expressed as nats instead of bits) on the Now boundary, and this equals the Shannon entropy of the interior space from the correspondence principle. H = EU / kB T = π c5 tU 2 / ħ G = π c3 R2 / ħ G Since G varies over time and is proportional to R – 1 , then H is proportional to R3 and is proportional to the volume of the interior space. Therefore, the density of H with respect to volume, dH/dV is constant over time. The density of H with respect to the Now surface area, dH/dA, is proportional to R, and the increase of dH/dA over time ΔtU is equivalent to the “flow of entropy” or proper time registered on a clock, Δτ. By working with the TOE equation based on entropic gravity, the state of the universe can be determined as a function of tU for all Now moments back to the Beginning. Note that even the hyperbolic geometry of spacetime is embedded in the TOE formula. - 14 -
- 17. Appendix C – The Gravity of the Situation So you might still wonder what does all this talk about entropy have to do with gravity, such as apples falling from a tree in Newton’s back yard. Whereas the cosmological temperature, TC, and the mass- energy density, ρ, are proportional to tU – 1 , the entropy density dH/dV is constant. Then one could argue that 3-dimensional space is assembled from entropy. I stated previously that a body possessing mass- energy has the property of displacing entropy in the space surrounding it, thus distorting space.26 The diagram below shows such a body embedded in space with the surrounding entropy density shown in shades of red – the lighter shades corresponding to a smaller density dH/dV. The grid is scaled such that the area of each box corresponds to volume of 3-dimensional space. The left-hand portion of the figure shows the local frame near the surface of a massive body. Suppose the entropy density in the center box is ⅛ the universal constant density H ≈ 2 × 1043 bits/m3 far away from the center. Thus the box in the center has to be eight times larger than a far-away box in order to contain the same quantity of entropy. Material objects located inside a box each take up certain percentages of the total volume of the box. From the thermodynamic equation ΔE = kB T ΔH, the mass- energy density, ρ, in the center box is also smaller than ρ in the distant frame while T remains constant. In other words, a mass-energy body displaces both entropy and energy surrounding it. As seen from a distant frame, the center box has ⅛ of its volume measured in the local frame, along with everything in the box. Lengths, widths and breadths of objects in the center box are ½ of those dimensions in the local frame. This agrees with general relativity, which also shows that space and everything in it shrinks near the surface of a large gravitating body, relative to a distant frame. When two mass-energy bodies approach each other, they displace binding energy, decreasing the effective mass-energy of the two bodies by the amount of binding energy displaced. Pulling the two bodies apart requires force = energy/distance in order to restore positive binding energy. This force is called the “force of gravity,” but it’s really a force resulting from the displacement of entropy and energy in the region of space around the bodies (see next page). Einstein showed that gravity is equivalent to acceleration; therefore, the force of acceleration, F = ma, also can be shown to be the result of displacing entropy in the local frame of an accelerating mass- energy body toward the direction of acceleration (see next page).27 26 Don’t ask me why, because I have no idea. But bear in mind that physicists unanimously accept Einstein’s GR field equations, even though none of them knows why mass-energy is able to stretch, bend or twist spacetime. 27 The so-called Unruh acceleration temperature could be a related phenomenon, where a body undergoing an acceleration, a, “experiences” a temperature T = ha / c kB. However, it should be noted that temperature alone cannot produce the force of acceleration, F = m a. Also, even a very modest acceleration of 1 m/sec2 produces an Unruh temperature 4 × 10–21 K, or 1,000,000,000 times higher than the current cosmological temperature, TC. - 15 -
- 18. The figure above illustrates a local frame with two masses near each other. Each of the boxes contains an equal quantity of entropy, H. The masses displace entropy around them and their combined effect is to push entropy out of the space between them, decreasing mass-energy density in that region by releasing gravitational binding energy. Keeping the masses from moving toward each other requires a restraining force, F, operating on each of the masses. Note that the force is not really a “gravitational force” but it is a restraining force to prevent the release of binding energy. If the distance between the masses increases by Δr, the energy increases by ΔE = F Δr. This force is equal to the change in energy with respect to distance, which is proportional to the change in entropy expressed by Newton’s law. F = ΔE / Δr = kB T ΔH/Δr = Gm1 m2 /r2 The force of acceleration is another entropic phenomenon. An accelerating mass-energy body creates a linear entropy gradient toward the direction of acceleration in the local frame as depicted below, which is equivalent to an entropy gradient produced by gravity. The mass-energy density in the region ahead of the accelerating body increases while the mass-energy density in the region behind decreases. The natural tendency is for the mass to follow a geodesic path through spacetime, which would eliminate the entropy gradient. Energy must be added to the body to maintain the acceleration, requiring an external force. As soon as the force pushing on the accelerating body is removed, the acceleration instantly stops and the entropy gradient disappears. A crude analogy is a speedboat pushing a bow wave in front of it and creating a wake behind. After the speedboat’s motor is shut off, the bow wave and the wake disappear, and the water surrounding the boat returns to its original undisturbed state. - 16 - F = kB T ΔH/Δx = m a
- 19. Appendix D – The Entropic Gravity Model Illustrated The illustration above is a 3-dimensional representation of entropic gravity model, bearing in mind that spacetime is 4-dimensional. Here, spacetime is separated into a 3-dimensional space with a radius, R, expanding at the speed of light and a curved 2-dimensional surface; the temporal boundary surrounding space, the “Now” moment where everything happens. The tiny yellow dot in the center of the model represents the temporal “Beginning” of space.28 In order for the pieces to fit together properly, the 2-dimensional surface must have negative curvature, which is a hyperbolic sphere, also known as a pseudosphere. The volume of a pseudosphere is only half as large as an ordinary 3-dimensional sphere in Euclidean space, or V = ⅔ π R3 . Negative curvature means the surface curves away from the center at all points instead of curving around it. This is depicted as the blue surface curving away from the center, a small portion of the Now surface.29 The surface area of a pseudosphere is the same as an ordinary 3-dimensional sphere, A = 4 π R2 . One quantum nat of entropy is depicted as the orange object within the spatial volume with an area that projects onto the temporal surface. The volume of the entropy quantum is a constant. The number of entropy quanta must therefore increase in proportion to the volume of the pseudosphere. H = Volume of the Pseudosphere / Volume of a Quantum Nat = π R3 c3 / G ħ R Note that the value of H above agrees with H computed on Page 14, above, using the TOE on a T-shirt equation. Also note that GħR/c3 is a constant, equal to 3/2 times the volume of one nat of entropy in m3 . This constant combines Newton’s gravitational parameter G, the reduced Planck constant ħ, the radius of the universe R, and the speed of light c, disposing of the popular scientific myth that general relativity and quantum mechanics are somehow “incompatible.” I consider this to be a true universal constant of nature, so I named it β, the first letter of βαρύτητα, the Greek word for gravity. A quantum nat projecting on the “Now” surface has an area cross-section equal to 4β/R, so as the “Now” surface expands, the entropy density on the surface increases linearly with R and tU. This flow of entropy is proportional to the time measured by a clock traveling along a geodesic path. 28 An additional note: The space+time model depicted above can be thought of as a 3-dimensional surface that surrounds 4-dimensional hyperbolic Minkowski spacetime within it. 29 It’s impossible to show a true picture of a pseudosphere; however, a 3-D model having a surface of uniform negative curvature is shown in this article from Wolfram Math World. - 17 -
- 20. Appendix E – No Pressure No Worries One major problem with the current cosmological model is maintaining the universe in a more or less orderly state. This requires a term added to the general relativity field equations, Λ, called the cosmological constant. This prevents the universe from collapsing under its own weight (the “Big Crunch”), but it could also cause the universe to expand exponentially (the “Big Rip”). The “measured” value of Λ is on the order of 10-52 m-2 , but cosmologists worry a lot about this number because it had to be set with incredible precision at the moment of the Big Bang. The problem has been compared to balancing a pencil on its point.30 Fortunately, the entropic gravity model takes care of this problem automatically. Recall in Part IV of this essay that at the boundary of spacetime (the temporal Now surface) the Schwarzschild equation is equivalent to the thermodynamic equation: T dS = P dV + dEU The P in the above equation is pressure at the boundary, and it works as follows. Since external pressure is always zero, positive internal P will speed up the expansion, negative internal P will slow down the expansion, and zero internal P will maintain the expansion at the current rate.31 By rearranging the thermodynamic equation, internal pressure is solved below.32 P = TC dS/dV – dEU / dV We already know how to compute all of the terms above from the entropic gravity model: • dEU / dV = c2 / (8π G tU 2 ) the mass-energy density of the vacuum expressed in Joules • TC dS/dV = kB TC (dH/dA)(dA/dV) • dH/dA = one nat divided by the area of one nat at the boundary = 1/ (4 G ħ /c3 ) = c3 / 4G ħ • dA/dV = 2/R since A = 4 π R2 and V = ⅔ π R3 for a pseudosphere • TC = ħ / (4 π tU kB) • Substituting the above for dH/dA, dA/dV, and TC, P = c3 / (8π tU GR) – c2 / (8π GtU 2 ) At this point we could replace R=ctU, but in this case we’ll allow the expansion to have an arbitrary rate of speed, R= vtU, instead. P = (c/v – 1)c2 /(8πG tU 2 ) It’s easy to see that when v > c the internal pressure is negative, reducing the rate of expansion; when v < c the internal pressure is positive, increasing the rate of expansion; and when v=c the internal pressure is zero, maintaining the rate of expansion at R=ctU. In other words, the universe has a built-in speed governor as a requirement of the Minkowski equation to maintain an expansion rate, c. Although c cannot be altered in the relationship e =mc2 , the expansion rate could vary, as is the case for the standard cosmological model where the expansion rate supposedly varies over time; however, in the entropic gravity model the expansion regulates itself to maintain v =c, thus preventing collapse or runaway expansion without needing any ad hoc free parameters. 30 Even worse, achieving the cosmological balancing act with Λ requires G to be constant, which Einstein and everyone else assumed is true. Unfortunately, it isn’t. 31 This is opposite from the standard cosmological model, where negative pressure inexplicably speeds up expansion. 32 The positive and negative terms of P correspond to “dark energy” and “dark matter” in the standard model. The difference is that the entropic model produces these terms automatically instead of having to enter ad hoc values into the standard model by hand (and two fewer free parameters to invent). - 18 -
- 21. Appendix F – One-Bit Universe in the Beginning This appendix is a more detailed determination of the conditions at the Beginning. At this point, I thought it would be worthwhile to display the various parameters of the entropic gravity model on a graph. The main parameter the TOE equation is tU, or the age of the universe. Singularities appear in certain of the parameters when tU = 0, so I’ll define the Beginning as a single bit of information, H, which is the entropy of a coin toss with the same head/tales probabilities. One bit of information equals 1.44 nat. I’m going to use a revised age of the universe tU =26 billion years. The total volume of the universe is 5×10122 times the volume of one bit, corresponding to a radius ratio of 7.94×1040 , or almost a 41 orders-of-magnitude span of time and distance from the Beginning to the Now. The figure below shows a graph of the parameters of the TOE equation displayed using dimensionless logarithmic scales. The lines trace orders of magnitude of the Beginning values over time, which all converge at the magenta sphere on the left, representing the Beginning. It’s interesting to compare volume, radius, mass-energy and density of the one-bit universe to the same properties of a proton: Property One-Bit Universe Proton Volume - m3 6.2 ×10 -44 2.5 ×10 -45 Effective radius - m 3.1 ×10 -15 8.4 ×10 -16 Mass equivalence - kg 1.3 ×10 -28 1.7 ×10 -27 Density - kg/m3 2.1 ×1015 6.8 ×1017 Temperature can be inferred from the mass of a single bit: T=Mc2 loge 2/kB =588×109 K. The one-bit universe was roughly 24 times the volume of a proton and 10% as massive. But although these conditions seem pretty extreme, they’re a far cry from the “singularity” conditions that were supposed to precede the Big Bang in the standard model. Interestingly, a photon with a wavelength twice the effective radius, 6×10-15 m, would have a mass equivalence of h/λc=5.5×10-28 kg. The entropic gravity model doesn’t yet answer the question of exactly how matter particles in the universe were produced in a high energy, high density and high temperature vacuum state, so the model needs a bit more refinement to address the formation of ordinary matter. Stay tuned!! - 19 -
- 22. Appendix G – Turning Gravity into Matter Appendix F, above, ended with a question of how ordinary matter (protons, electrons, neutrons) were created following the Beginning. Since this essay is all about gravity, I’m going out on a limb by stating that ordinary matter is created from gravitational energy, or more exactly the reduction in negative gravitational energy. Quantum Electrodynamics (QED) postulates that the vacuum ground state is filled with virtual particles of all types.33 Because they are virtual, they haven’t yet turned in to real particles so they shouldn’t make their presence felt in the world at large.34 Pairs of particles can make their presence felt as real particles, albeit very briefly, by “borrowing” energy from the vacuum with a very short payback time period, which is inversely proportional to the amount of energy borrowed, according to Heisenberg’s equation. For example an electron-positron pair borrows energy to create them but they must pay it back (in other words disappear) within about 10-20 seconds. If the vacuum-energy debt can be paid back quickly by a third party, say a photon, then both the real particle and its anti-particle partner can come into being permanently. Unfortunately, the anti-particle will sooner or later encounter another particle identical to its partner, which annihilates both of them and emits a gamma photon, making a net change of zero. The diagram on the left, below illustrates how a gamma photon, labeled γ, supplies enough of an electromagnetic field gradient (energy/time) to literally create a real electron-positron pair from the vacuum. According to QED, there is another way of looking at particle pair creation, as shown in the figure on the right, above. An anti-particle moving forward in time is equivalent to a particle traveling backward in time. This gave me an idea: The gamma photon creates an enormous electrical and magnetic energy gradient in time. Could gravity do the same thing? And if so, when and where could such a thing occur? I believe the answer to the first question is “yes” and the answer to the second question is that it occurs where every else occurs: At the curved Now boundary when G was much larger. In order to approach this question we need to examine the gravitation binding energy of subatomic particles like protons and electrons. Gravitational binding energy is negative, meaning that it reduces the net mass of an object. The classical formula for the binding energy of a solid sphere of mass, m, and radius, r, having uniform density is u=- 3/5 Gm2 /r. Because we have to deal with values of G that are many orders of magnitude greater than today’s value, this requires the use the relativistic version of the binding energy formula instead: u= -mc2 (1-1/√1-rs /r ), where rs is the Schwarzschild radius of m 33 The reason is simple. Virtual particles exist in the vacuum because the vacuum is filled with entropy. 34 However, QED physicists believe the presence of virtual particles are felt, which is why their calculations show the vacuum energy is at least 122 orders of magnitude greater than any plausible cosmological vacuum energy density. This discrepancy is referred to as “the biggest blunder of fundamental physics over the past century.” The source of the implausibly-large QED number comes from measurements of the so-called Casimir effect, named after Hendrik Casimir; however, I think that effect may be a mirage. - 20 -
- 23. When r/rs is very large, the binding energy is almost zero, but as r→rs, u→-mc2 . This is why stars cannot shrink into physical black holes: The binding energy completely cancels the mass of the star as its radius approaches rs causing the physical star to turn into nothing. But that’s a whole other issue. The Schwarzschild radius of a proton in today’s world is rs =2Gmp /c2 =4.5×10-54 m, whereas its physical effective radius is around 10 -15 meters, so the proton’s negative gravitational binding energy is negligible compared to its positive mc2 energy (the same kind of result appears using the classical formula mentioned above). In the early universe near the Beginning, G was many orders of magnitude larger than today’s value, so a proton’s Schwarzschild radius would have swelled to a size comparable to the its physical effective radius, reducing its net mass to zero.35 In this early stage, real protons (and their anti-proton partners) would have emerged effortlessly from the swarm of quantum virtual pairs in the vacuum because their energy payback would have been zero and their payback time infinite. In fact, any object with a sufficiently large m/r ratio could emerge from essentially nothing at this stage. As G decreased, a point was reached when the Schwarzschild radius would shrink beneath the physical radius and the particles would attain positive net mass. For a proton, the crossover point would be reached when 2Gmp /c2 =rp, where mp and rp are the proton’s mc2 mass and its effective radius, or about 10-27 kg and 10-15 m, respectively. Solving for G results in a value around G≈1028 ,which is 1039 times greater than today’s G. If those numbers are correct, that would have occurred very early in the history of the universe, when its R was about ten times larger and its volume was about one hundred times larger than the 1-bit universe. What happens next is most interesting. As the negative gravitational energy dissipates, positive energy is transferred to the particles themselves. The total amount of energy is equal to the mc2 equivalent of the particles themselves, resulting in an incredible temperature rise among those early particles.36 The yellow cone-shaped object above is the timeless spatial component in the entropic gravity model, and the blue negatively-curved surface is the temporal Now, labeled N, going into the future, F, at the speed of light relative to the Beginning, B. Particle pairs are created on the Now; one is swept forward in time and one is left behind in the spatial region. The process of particle creation starts at S and ends at E. The process answers the “missing anti-matter” riddle: All particles created were ordinary matter, either going forward in time or left behind as part of the spatial region’s historical record. The particle pairs are forever entangled quantum-mechanically. The left-behind particles are shown as colored strings tracing world lines of their partners from the Now surface, stored holographically, in the timeless spatial region. There are remaining details that need to be fleshed out in this model, e.g. what kinds of particles were created by gravity and why particles today have the masses they do. 35 It’s easy to turn nothing into tiny particles and increase entropy with help from gravity reduction, whereas according to the second law of thermodynamics, turning huge things like stars into nothing and reducing entropy is impossible. 36 Imagine what it would be like if you were suddenly infused with the energy equivalent of all the mass in your body. - 21 -
- 24. Appendix H – Holography John Wheeler often repeated the statement, “The boundary of a boundary is zero.” What this means in topology is that an N dimensional space is bounded by an N-1 dimensional boundary, but the N dimensional space ceases to have a boundary when it becomes the boundary of an N+1 dimensional space. This is shown below as a set of illustrations below related to the entropic gravity model. Time is usually considered as a 1D line (past and future), as depicted in the first illustration above, bounded by two 0-dimensional points. When time joins itself in two dimensions, it becomes the boundary of a 2D temporal space and it ceases to have its own boundary. The 2D space joins itself in three dimensions, becoming the 2D temporal boundary of timeless 3D space, depicted as a yellow sphere. Bear in mind, that this sphere is hyperbolic, as discussed at length in the entropic gravity model presented in this essay. Taking this a step further, when the 3D entropic gravity model joins itself in four dimensions, it becomes the holographic boundary of 4D Minkowski spacetime. The yellow 3-dimensional space is an expanding light sphere bounded by a 2D temporal hologram. When we look out into space, we are peering into that light sphere. In the picture below, left, an observer is looking into that 3D light sphere at two stars separated by an angle, θ. The 3D light sphere on the left is rendered as the 2D surface of an expanding light cone on the right. Although space appears larger as distance increases, when observers look out into space, they always look through the light sphere toward the Beginning, shown as the tip of the light cone on the right. In this illustration, the observer sits on the Now boundary (reduced to a blue 1D circle instead of a 2D temporal surface). Changing the angle of observation by Δθ in 4D Minkowski spacetime is the same as an observer rotating Δθ around the circular boundary of the light cone; however, the light cone actually rotates Δθ with respect to the observer instead. The triangle to the right of the light cone shows the relationship between Minkowski space, time, and proper time. As previously noted, proper time along any light-like path is zero, meaning that light reaches the observer at exactly the same instant it is emitted from the stars since everything happens in the Now moment. The stars (and everything else in the “past”) are holigraphically-encoded in the timeless, entropy-saturated, spatial light sphere. The 2D/3D time/space boundary projects a holographic image of 4D Minkowski spacetime into the interior space. There is a correspondence principle between them, as there is a correspondence principle between 2D time and 3D space in the entropic gravity model.37 37 This correspondence seems somewhat similar to string theorist Juan Maldacena’s AdS/CFT conjecture. - 22 -