Tag: relativity

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GENERAL & SPECIAL RELATIVITY

Since Einstein said that E=mc2 , why does a massless photon have energy?

Someone asked a similar question on Quora. My answer garnered nearly a million views and many dozens of comments. It gave me an opportunity to gather thoughts on a subject that has puzzled folks for decades.

Of course, I’m a pontificator, not a scientist. I got advice from working physicists and incorporated what they taught me.

One thing I learned from science writer Jim Baggott is that Einstein first published his famous equation in this form: 

M = \frac{E }{ C^2}

When written this way, it becomes clear that anyone who knows the total energy of anything can calculate in principle its total mass.

Einstein knew nothing at all about the Higgs field but today physicists agree that the mass it creates is less than 5% of what mass they have discovered. 

In fact, nearly 99% of the mass of a single proton is derived from the energy of “massless” gluons that constrain its two up-quarks and one down-quark. Gluons are bosons which don’t interact with the Higgs field; quarks, which are fermions, do.  

In the end, it’s all about energy, which it turns out is equivalent to mass, which according to Baggott is what quantum fields do. Quantum fields like the Higgs field make mass. Perhaps the electromagnetic field — which makes photons — does the same. 

Here is Einstein’s equation for energy:

E^{2}=\left ( mc^{2} \right )^{2} +\left (pc \right )^{2}

Since

p=\frac{hf}{c} 

and

m=\frac{hf}{c^2}

it follows that it might be reasonable to imagine that photons have both internal mass and inertial mass, which causes Einstein’s equation for energy to give the following result:

E=\sqrt2 ({hf})

All that is left is to divide by c2 to get mass, right? 

m=\sqrt2 (\frac{hf}{c^2})

Most folks think the internal mass of a photon is zero. Period. End of story. They use the two mass and momentum terms in Einstein’s equation to calculate total energy of massive objects, yes, but photons, they insist, lack internal mass. They lack the internal fermionic structures associated with all massive particles.

Photons do have inertial energy proportional to their critical frequency though, which suggests that they possess perhaps equivalent inertial mass, which drives the photoelectric effect.

When physicists take the energy measure of photons, they drop the mass term in Einstein’s equation. They set mass to zero and cancel out the first term, mc2.  It leaves the second term — pc — which for photons simplifies to hf, inertial energy correlated to frequency, right? Energy can be measured in eVs, electron-volts, which are also units of mass. 

If photons have internal energy, their total energy in the universe is undervalued by 1.414 (the square root of 2). Accounting for this added mass reduces the Cosmic energy deficit to near zero. 

PHOTON MASS

I should add that overestimating mass and disrupting popular models of the Cosmos is something most scientists think is a bad idea. 

The gluon is the only other massless particle currently in the standard model, but it has never been observed as a free particle. All gluons are buried inside hadrons. It is their binding energy in quarks that makes as much as 99% of the measured mass of protons and neutrons. 

So, there is precedent to possibly reevaluate mass equivalence of photons. 

Some readers might wonder about the massless graviton. This particle is theorized to exist, yes, but has not been observed or added to the Standard Model.  The same is true for dark matter and dark energy — no physical evidence; not added to the Model.

It doesn’t mean dark energy and matter don’t exist. Cosmologists see way too much gravity everywhere they look. The problem is they can’t explain exactly what is causing it. 

As for my answer to the original question published on Quora, it was as accurate as my limited experience could make at the time, but the subject is controversial and several issues are not yet settled, even by experts. Some disputes might never be settled.

Who knows? 

Not me. I’m a pontificator, right? 

What follows is a version of the answer:

You might be mistaken about energy.

According to the complete statement of Einstein’s most well-known equation, energy content is a combination of a particle’s mass and its momentum. The equation you cite is abbreviated. It is a simplified version that is missing a term.

Einstein’s complete equation is strangely analogous to Pythagoras’s geometry of right-triangles. When anyone thinks about it though, aren’t the frequencies of light at right angles to its propagation? Light waves are transverse, right? 

Here is a more complete version of Einstein’s equation:

E^{2}=+\left ( mc^{2} \right )^{2} + \left (pc \right )^{2}

—where m is internal mass and ρ is momentum. Internal mass is often referred to as “rest mass” because it is invariant in all reference frames and unchanged by velocity or acceleration. Momentum is inertial energy measured in equivalent mass units called electron volts (eVs). 

Massless particles like photons have momentum that is correlated to their wavelengths (or frequencies). It’s their frequencies that give massless particles like photons their energy content. So without (rest) internal mass the equation becomes:

E=ρc

—where  p=\frac{hf}{c} for massless photons.

So, E = hf 

[“h” is Planck’s constant. “f” is frequency. “c” is light speed.]

Of course, in classical Newtonian physics ρ = mc. The mass term is critical.

Screen shot from Khan Academy showing derivation of photon momentum. Typed mark-ups by me show mass equivalence when ρ is set equal to mc. When mc = hf/c, then m = hf/c*c, right? Click the pic for a better view in a new window. 

On the other hand, in quantum mechanics the total mass of photons cannot be zero either—photon internal mass is set equal to zero and eliminated. Inertial energy based on the photon’s critical frequency (the 2nd term in Einstein’s equation) becomes its equivalent mass. I’m not sure everyone agrees. 

The beauty created by setting photon rest-mass (internal energy) to zero is it transforms the maths of relativity and quantum mechanics into structures that seem to be consistent and complete — able, one hopes, to meld into theories of everything; TOEs, if you like. The problem, of course, is that the convention of setting to zero leaves thrashing in its wake 95% of the mass and energy which “other” stories claim is hidden unseen “out there” within and around galaxies to move them faster than they ought. 

The Abraham-Minkowski controversy seems to touch the argument.  Click the link and scroll to the end of the article to learn how many things are disputed, not known, or unexplained. The science is not settled, although several physicists claim that the controversy is resolved by postulating an interaction inside dielectrics (like glass) of photons with electron-generated polaritons.  

NOTE BY EDITORS: On 18 April 2021 a writer massively abbreviated and modified the article in Wikipedia on the A-M controversy. The writer deleted the entire list of disputed claims. Please click the link in this sentence to review a list of unsolved problems in modern physics. Photon mass inside dielectrics isn’t on the list. 

The permittivity of “empty“ space (called the electric constant) qualifies as a dielectric, does it not? Isn’t space itself—with its Maxwell-assigned permeability (the magnetic constant) and permittivity (electric constant)—a dielectric?

Arthur Eddington wrote in chapter 6 of his book Space Time and Gravitation (read pages 107-109) that the dielectrics of space around the Sun increase proportionally with the intensity of the gravitational field. Light waves closest to the sun slow down more, which pulls the wavefront that lies farther out to deflect still more to catch up. Like glass, gravity refracts light.

Light falls into the Sun like any solid rock, but refraction adds to light’s “Newtonian” deflection to give Einstein’s predicted result. Unlike slow rocks, light travels fast enough to avoid capture by the sun. 

It’s not clear to me how many physicists agree with Eddington, but then again, it’s not obvious whether humanoids are able to visualize reality. It’s one thing to write equations and symbolic algorithms that match well with observations. It’s quite another to acquire a natural intuition for what might be true. 

Empty space isn’t empty, right?

As for the Abraham-Minkowski dispute: how important might it be to decisively resolve ambiguities concerning photon mass?

Perhaps the dispute is swept under a rug because disagreements about something as fundamental as photon mass mean that physicists might know less than they let on. The controversy seems to me at least to have the potential to crash the tidy physics of light and mass built by hard work and much history.

Isn’t it better to pretend everything is just fine until physicists finally agree that everything really is?

Maybe the subject involves some aspect of national security which requires obfuscation. It wouldn’t be the first time. 

What I think can be safely said is that momentum and mass of quantum objects seem to have no meaning until they are brought into existence by measurements. The math looks like nothing we know; sometimes physicists use the results as mathematical operators that don’t commute the way some might think they should.

PHOTONS AND GRAVITY

I reviewed the math.  I saw the term that makes the deflection difference (it’s really there) but did not understand enough at the time to tease out a satisfying reason why photons seem to bend nearly twice more in a gravitational field than early acolytes of Newton conjectured. I guess I like Eddington’s explanation best. 

According to Wikipedia, Einstein’s theory approximates the deflection to be:

\frac{4GM}{(c^2)b}

“b” is the distance of a photon’s closest approach to a gravitational object like our Sun.

Here’s some guesses I made before reading Eddington:

Maybe light deeply buried in a gravity field near a star like the Sun will experience the flow of time more slowly—it’s an effect common to all objects in a gravity field; it affects all objects the same way and is unaffected by their mass or lack of it.

It might have something to do with Schwarzchild geodesics. The geodesics of spacetime paths are longer and more curved in a gravity field than what anyone might expect from a simple application of Newton’s force law, which is oblivious to the spacetime metrics of Einstein. 

Schwarzchild metrics help to explain the “gravitational lensing” of faraway objects when their light approaches Earth from behind massive gravitational structures in the far reaches of space. Light careens around the structures so that astronomers can see what would otherwise remain forever hidden from them. 

Here is another guess:

It might be that light spends more time in a gravitational field than it should due to special-relativity-induced time dilations so that photons have more time to fall toward the star than they otherwise would. This guess is certainly wrong because the time differential would be governed by a Lorentz transformation.

Photons of light don’t undergo Lorentz transformations because, unlike massive objects that travel near the speed of light, they don’t have inertial frames of reference. Any line of reasoning that ties Lorentz transformations to photons leads folks into rabbit holes that contradict the current consensus about the nature of light. Light speed is a constant in all reference frames. Space and time expand and shrink to accommodate it. 

Electron-like muons (which have rest masses 205 times that of electrons) are short-lived, but their relativistic speeds increase their lifetimes so that some of those that get their start in the upper atmosphere are able to reach Earth’s surface where they can be observed. Their increased lifespan is described by a Lorentz transformation. It’s tempting to apply this transform to photons, but theorists say, no. It doesn’t work that way.

Time contractions and dilations are Special Relativity effects that apply to objects with inertial mass that move in some specified reference frame at velocities less than the speed of light, yes, but never at the speed of light, right?

Nearly every physicist will insist that photons have no internal mass; they travel in vacuum at exactly the speed of light—from the point of view of all observers in every reference frame. Photons don’t have inertial reference frames in the same way as muons or electrons.

Changes in time and position caused by a photon’s location in a gravity field are completely different; they are described by a vastly more complicated theory of Einstein’s called General Relativity.

Here is one way to write his formula:

The terms in this expression are tensors, most of them. Click the link, anyone who doesn’t think tensors are difficult to write and manipulate. 

Here is another way to think about photon energy and behavior:

Light follows the geodesics of spacetime near a massive object—like the sun. Gravity is the geodesic.

The difference for massive objects traveling at relativistic speeds is that their momentum and inertia enable them to skip off the geodesic tracks, so to speak.

Because massive objects always travel at speeds less than light, their “clocks” slow down through an additional dynamic (a Lorentz transformation) that works at cross-purposes to gravity. Massive objects lock onto the gravity geodesics for a shorter period of time. They undergo less gravitational time dilation than does light because they spend less time constrained on its geodesics. They jump the geodesic tracks to become constrained by the dynamics of the Lorentz transformations. 

The result is that massive objects traveling at relativistic velocities less than light deflect less toward the star (Sun) than does light.

What makes General Relativity unique is it’s view that gravity and acceleration are equivalent. Acceleration is a change in the velocity and/or the direction of motion. Massive bodies such as stars curve and elongate the pathways that shape the space and time around them.

Photons traveling on these longer spacetime paths accelerate by their change in direction, but their velocity doesn’t change in any reference frame. Something has to give. What gives, what changes is the expected value of deflection. The light from distant stars bends more than it should.

SOME HISTORY

No one who lived before 1900 could know that the geodesics of space-time elongate (or curve) in the presence of mass and energy, which are equivalent, correct? No one in bygone eras could have known that time slows down for massive objects that approach light-speed, either.

A man named Joann Georg Soldner did a calculation to show how much a Newtonian “corpuscle” of light would bend in the Sun’s gravity, which he published in 1804. He assumed that photons had mass and fell toward the Sun like any other object.

When Arthur Eddington’s observations showed that starlight deflected more than Soldner had calculated, Einstein’s theories of relativity got a boost in credibility that lives on into modern times.

I should add that Eddington knew about Einstein’s predictions when he made his experimental observations in 1919 because Einstein had already published his general theory.

EXPLANATIONS

I would very much like to read a coherent, verbal (non-mathematical) explanation of exactly why and how Einstein’s general theory can lead to an accurate and reasonable prediction at odds with Newton about the angle of deflection of photons near a star.

Here is a synopsis of an explanation that I heard from a working physicist:

Soldner used Newton’s view to calculate deflection using only the time the photon spent in the gravitational field. Einstein did the same but then modified his calculation to account for the bending of space in the gravitational field. The space component nearly doubled the expected deflection.

The theorist’s explanation satisfied me. It sounded right.

Notice the speed of the hands on the clocks and how they vary in space-time. Clocks slow down when they are accelerated or when they are immersed in the gravity of a massive object like the star at the center of this GIF. Stronger gravity makes clocks run slower. Under General Relativity, gravity and acceleration do the same thing. Click on the pic for a better view.

On the other hand, I believe (secretly and in agreement with Newton’s acolytes) that photons must have a mass equivalence that for some reason is being discounted, but no one I’ve read believes the idea makes sense beneath the shadow of a relativity theory that has the reputation for being fundamental, flawless, and complete.

After all, the mass of any object in a gravitational field is irrelevant to its trajectory because the mathematics cancels it, right?

F=ma=\frac{GMm}{r^2}

Little “m” appears on both sides of the equation so it can be divided away.

The problem is that the equations for gravity—especially over cosmological distances—are not necessarily settled. These are serious anomalies that are not yet resolved to everyone’s satisfaction. Some have direct consequences on the ability of organizations like NASA to conduct accurate landings on Mars and the Moon. Click the link in this paragraph to review six of the biggest puzzles followed by seventeen alternative theories designed to bring the discrepancies to account. 

Anyway, mass-energy equivalence of photons might permit Lorentz transforms on light to help to resolve certain problems in cosmology and the transmission of light through medias where gravity is not a factor. It might also simplify understanding of annoying Shapiro effects, which slow down communications with explorer craft inside our solar system.

ANOTHER EXPLANATION

Since I haven’t yet found a good explanation—and with a promise to avoid nonsensical personal predispositions—here is my attempt to explain:

In GPS (Global Positioning Systems), dilations of time—in both the velocity of satellites in one frame and their acceleration in another frame (gravity)—must add to provide accurate information to vehicles located in another frame.

These time dilations can work at cross-purposes. It requires expensive infrastructure on the ground to coordinate the information so that drivers of vehicles don’t get lost.

A massless object moving at the speed of light is going to follow the geodesics of the gravity field. This field is a distortion of space and time induced by the presence of the mass of something big like the Sun.

If massless energy does not obey the laws of Special Relativity (like GPS satellites do), then its velocity must necessarily have no influence whatever in the deflection of light near a star. It might seem like all the deflection comes from the distortion of spacetime, which is gravity.

Photons ride gravity geodesics like cars on a roller coaster. According to appendix III in Einstein’s 3rd edition of his book, Relativity, the Special and General Theory—published in English by Henry Holt & Company in 1921—it’s only half the story.

The other half of the measured deflection comes from the Newtonian gravitational “field”, which accelerates all objects in the same way. This field further deflects light across the spacetime geodesics toward the sun to double the expected angle.

I’m not entirely convinced that modern 21st century physicists believe it’s quite that way or quite that simple.

CONCLUSION

The theory of general relativity helps theorists to describe the distortion of metrics in spacetime near massive bodies to predict the deflection angle of passing photons of light. What we know is that predictions based on the theory don’t fail.

It’s like the theory of quantum mechanics. It never fails. It’s foundational. No one has yet been able to explain why.

Somebody, please, tell me I’m wrong.

Here is a link that addresses the math concerning the deflection disparity between Newton and Einstein.

Billy Lee

Link to comments on Quora

Readers interested in this subject will learn things from the material in these comments, I promise. 

Billy Lee 

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25 ANSWERS

Two months ago, I discovered QUORA. It’s been around since 2009.

Since 2010, Quora has enabled people to ask experts questions about topics they like; even to answer questions on subjects they claim to know something about.

Quora is a site for geeks and nerds, and so far I  like it. The people who hang out in the areas I hang out tend to be polite, kind, and smart. If they like someone, they follow them and are notified when they post. So far, ten people have signed on to follow me. It’s a start. I think most are from India.

During the first six weeks, 150 or so of my answers were viewed 35,000 times; I got nearly 175 “upvotes”, which enabled many of the answers to move to the head of the line. I wrote most answers in the wee hours between 2 AM and 7 AM when I couldn’t sleep. Insomnia inspired me.

What follows are 25 of the most popular answers I posted to the first 150 or so questions that caught my interest. They are sequenced by popularity — the most read first .

Why not read a few? How many questions can anyone answer? Not many, I’m thinking.

Who knows what you might learn?

What? 

Someone thinks they know better than a pontificator with no bonafides?

I don’t think so.

No way!   😉

1)   What are some of the most popular “mathematically impossible questions“?

Freeman Dyson — one of the longest-lived and most influential physicists and mathematicians of all time — argued that it is impossible to find a whole (or exact) number that is a power of two where someone can reverse its digits to create a whole number that becomes a power of 5.

In other words, 2^{11} = 2048 , right? Reversing the digits to make 8402 does not result in an exact number that is a power of 5.

In this case,  8402^{1/5} = 6.09363  plus a lot more decimals. It’s not a whole (or exact) number. Not only that, no matter how many decimal places anyone rounds-off 6.09363… , the rounded number raised to the power of 5 will never return 8402 exactly.

Dyson claimed that this conjecture must be true, but there is nothing in mathematics that enables anyone to write a proof. He claimed that there must be an infinite number of similar statements—-each one true, none provable.

Click the link below to learn more.

TRUTH

The Snowplow Problem is another “impossible” problem. My differential equations professor assigned it with the promise that anyone who solved it would receive a 4.0 grade, regardless of their performance on tests. I was the only student he ever taught who actually managed it.

The problem goes like this: It is snowing at a constant rate. A snowplow starts plowing snow at noon. By one o’clock the plow has traveled one mile. By two o’clock the plow travels an additional half mile. At what time did it start snowing?

It took me 3 days and two pages of calculations, but I got my 4.0.

Note from the Editorial Board: Over 50 people on Quora submitted answers to Billy Lee’s Snow Plow problem. One person had the right answer, but would not produce his proof. He did point out an unusual feature of the solution that Billy Lee had not noticed before. Billy Lee characterized the feature as ”very surprising.” When pressed Billy Lee refused to reveal the secret. 

2)   How much force is one Newton?

A newton is the force that an average sized apple makes on your hand when you hold it. No matter where in the universe you are; no matter on what planet you stand or how strong the gravitational field, a newton of force always feels the same.

A newton is one kilogram of mass that is accelerating at one meter per second per second. Gravity on Earth accelerates everything at nearly 10 meters per second per second. A kilogram of mass feels like 2.2 pounds on earth. One tenth of 2.2 pounds is 0.22 pounds or 3.5 ounces, which is the weight of a typical apple. The weight is the force that you feel against your hand. It is one newton.

On the moon, an object with the mass of a large brick would feel as light as an apple on earth due to the moon’s lower gravity. The force of the brick in your hand would feel like one newton.

3)   x + y = 4 .  and  . x^x + y^y = 64 .   What are x and y?

The simplest way to solve is to make y = (4-x) and create an equation in terms of x.

An easy version to create and solve is

{x^x + (4-x)^{4-x} = 64}

You can solve it by hand using iteration or throw it into an app like Wolfram Alpha and let them solve it in a few seconds.

Either way, one value for x is .606098…. The other is 3.393901… , which you can assign to y. The two numbers add to 4.000… and when substituted into both initial equations return the right results.

4)   If I had 1,000,000,000,000,000 times 1,000,000,000,000,000 hamsters floating in space in close proximity, would gravity turn them into a hamster planet?

Assuming the question is serious, it deserves a serious answer.

A typically fat hamster weighs around one ounce, which is about 0.03 kilograms of mass. The number of hamsters in your question is 10E30.

Multiplying the mass of a single hamster by this large number gives the result of 3E28 kilograms.

To compare, the mass of planet Earth is 6E24 kilograms. The mass of the proposed population of hamsters is 5,000 times the mass of the earth.

The sun contains 67 times more mass than the hamster population. If the hamsters are close enough together to hold paws, a hamster planet is almost certain. I haven’t worked out how long the process to congeal would take, but I can estimate that the hamsters would probably die of starvation before the inexorable forces of gravity completed their work.

The hamster planet would be formed mostly from three elements: hydrogen (64%), oxygen (33%), and carbon (10%). 3% would be trace elements like calcium and maybe lithium.

The most likely outcome, given enough time, is a planet-like object. The hamsters have only one-fifth of the mass to make the smallest of the smallest suns — red dwarfs, which populate 67 to 80 percent of the Milky Way Galaxy.

There are way too many hamsters to make a reasonably sized moon.

Their mass (3E28 kg) happens to fall on the border between the range of masses that are required to form celestial objects known as brown dwarfs and the less massive sub-brown dwarfs — sometimes referred to as free-floating planets.

Brown dwarfs don’t have enough mass to ignite like a star, but they do produce heat and can accept small orbiting planets. The chemistry of brown dwarfs is not well-understood and is a bit controversial.

It’s a toss-up, but my vote goes to the notion that the hamsters will eventually form a very large but ordinary planet — a free-floating planet — which I referred to earlier as a sub-brown dwarf. This hamster planet might wander through space for millions (or even billions) of years before being captured by a massive-enough star to begin to orbit.

Because the elements of hydrogen and oxygen are likely to become the constituents of frozen moisture (or water ice), there is the risk that the ice might melt into oceans and perhaps boil away if the hamster planet approaches too close to a star (or sun). In the case where the planet loses its water, a carbon planet with 50 times the mass of earth would form.

Otherwise, should the planet find itself in a far-distant future orbiting in the “goldilocks” zone around a sufficiently massive star, the water would not evaporate. Life could arise in the planet’s oceans. It’s possible.

Life-forms might very well crawl up out of the water and onto land someday where — over the eons and under  ideal conditions — they will evolve into hamsters.

5)   Why is evolution a valid scientific theory despite the fact that it can’t be conclusively proven due to the impossibility of simulating the millions-of-years processes that it entails?

Evolution is a fact that is thoroughly established by observations made in many disciplines of science. Changes in species happen fast or slow; in the lab and in the field.

The mystery is how one-celled life got established so quickly — it was solidly established within one billion years of earth’s formation. It’s taken 3.5 billion years to go from one-celled life to what we have now.

Why so fast to get life started; why so slow to get to human intelligence and civilization?

People have a lot of ideas, but no one is sure. Some life forms have orders of magnitude more DNA than humans. Only 2% of human DNA is used to make the proteins that shape us.

So, yes, there are lots of questions.

NO CODE

6)   Why do cosmologists think a multiverse might exist?

Many high-level, theoretical physicists have written about the obvious problem our universe seems to have, which is that it has too many arbitrary constants that are too tightly constrained to be explained by any theory so far. No natural cause has been found for so many constants, so it’s fertile ground for theorists.

Stephen Hawking, among others, has said that the odds of one universe having the physics that ours has is 1E500 against. He is joking in his English way, because such a large number is essentially an infinity. It’s not possible to constrain a universe like ours by chance unless there are an infinity of choices, and we happen to be in the one that supports intelligent, conscious life.

Two ways of getting to infinity are the concepts of multi-verse and the new one proposed by Paul J. Steinhardt of Princeton University in 2013, which is based on data supplied by the Planck Satellite launched in 2003. Paul is the Einstein Professor of Science at Princeton, so his opinion holds a lot of weight.

Steinhardt has proposed that the universe is ekpyrotic, or cyclic. He has asserted that the universe beats like a heart, expanding and contracting in cycles, with each cycle lasting perhaps a trillion years and repeating, on and on, forever. Each cycle produces conditions — some which are ideal for life. This heart has been beating forever and will continue to do so, forever.

Conscious Life

7)   How will we visit distant galaxies if we cannot travel faster than light? 

We will never visit distant galaxies, because they are too far away; most are moving away from us faster than our current technologies can overtake. At huge distances space itself is expanding, which adds to our problems.

The expansion of space is gradually accelerating. Any increase in performance by space vehicles over the next few thousand years is certain to be overwhelmed by the accelerating expansion of the universe.

As time goes on the amount of objects that are reachable (or even viewable) by earthlings will shrink.

On the happy side, our own solar system has at least 165 interesting places to visit that should keep folks fascinated for many thousands of years. A huge cavern has been discovered on Mars, for example, that might make a safe habitat against some forms of radiation dangers; it seems to be a place where a colony of humans might be able to live, work, and survive — perhaps even flourish.

Elon Musk is planning a mission to Mars soon.

8)   What is the mathematical proof for a+a = 2a ?

Some things that are true can’t be proved. All math systems are based on axioms, which are statements believed to be true but which, in themselves, are not provable.

This link provides a list of axioms for addition: https://sites.math.washington.edu/~hart/m524/realprop.pdf

A lot of interesting philosophical and mathematical work has been done on conjectures that are believed to be true, but can’t be proved.

TRUTH

9)   Can you explain renormalization in physics in simple words?

There is a problem in physics that has to do with the huge variation in scales between the very large and the very small. This problem of scales involves not only the size and mass of things, but also forces and interactions.

Philosopher Robert Pirsig believed that the number of possible explanations that scientists could invent for phenomenon were, in actual fact, unlimited.

Despite all the math and all the convolutions of math, Pirsig believed that something mysterious and intangible like quality or morality guided our explanations of the world. It drove him insane, at least in the years before he wrote his classic book, Zen and the Art of Motorcycle Maintenance.

Anyway, the newest generation of scientists aren’t embarrassed by anomalies. They have taught themselves to “shut up and calculate.” The digital somersaults they must perform to validate their work are impossible for average people to understand, much less perform. Researchers determine scales, introduce “cut-offs“, and extract the appropriate physics to make suitable matches to their experimental results.

The tricks used by physicists to zero in on pieces of a problem where sensible answers can be found have many names, but renormalization is one of the best known.

When physicists renormalize an equation, they cut away infinities and other annoying problems (like dividing by zero). They focus the range of their attention to smaller spaces where the vast differences in scales and forces don’t blow up their formulas and disrupt putative pairings of their carefully crafted mathematics to the world of actual observations.

It’s possible that the brains of humans, which use language and mathematics to ponder and explain the world, are insufficiently structured to model the complexities of the universe. We aren’t hard wired with enough power to create the algorithms for ultimate understanding.

RENORMALIZATION

10)   If a propeller rotates at the speed of light at half of its length, what happens to the outer parts?

Only the ends of the propeller can rotate at near light speed (in theory). At half lengths the speed of the propellers will be half the speed of their ends, because the circumference of a circle is 2πr. (There is no squared term.)

So the question is: will the propellers deform according to the rules of the Lorentz transformation along their lengths due to the difference in velocity along their lengths?

The answer is, yes.

As you move outward along the propeller, it will become thinner in the direction of rotation, and it will get more massive. A watch will tick more slowly at the end than at the middle.

I am not sure how it would look to an outside observer. Maybe such a propeller would resemble in some ways the spiral galaxies, which don’t rotate the way we think they should. Dark matter and energy are the usual postulates for their anomalous rotations. Maybe their shape and motion is related to relativity in some way. I really don’t know.

In reality, no propeller can be constructed that would survive the experiment you describe. But in theory (and ignoring the physical limitations of materials) there would be consequences.

However, no part of the propeller will move at light speed or higher. Such speeds for objects with mass are impossible.

11)   What is the fundamental concept behind logarithms?

The first thing that anyone might try to understand is that the word logarithm means exponent.

Example 1:

log 100 = 2 . What does this expression say? It says that the exponent that makes 100 is 2. What confuses people is this: exponent acting on what number?

The exponent acts on a number called the base. Unfortunately, the base is not written down in the two most common logarithm systems, which are log and ln.

The base for the log system is 10. In the example above, the exponent 2 acts on the base 10, which is not shown. In other words,  10^2 = 100 , right? The exponent that makes 100 from the base 10 (not shown) is (equals) 2.

Example 2:

ln 10 = 2.302585… .  What does this expression say? It says that the exponent that makes 10 is 2.302585… . Again, exponent acting on what number?

The base used in the ln system is 2.7182818… ,which is an irrational number that has an infinite number of decimal places. It happens to be a useful number in all branches of science and math including statistics, so mathematicians have decided to represent this difficult-to-write-down number with the letter “e”, which is known as Euler’s number.

The base for the ln system is e . In the example above, the exponent 2.302585… acts on the base e , which is not shown.

In other words,  e^{2.302585...} = 10 , right?

The exponent on e ( which is 2.7182818… and not shown in the original equation above) that makes 10 is (equals) 2.302585… .

All other logarithmic systems express the base as a subscript to the right of the word log.

Example 3:

log_{7}49 = 2

This expression says: The exponent on seven that makes 49 equals 2.

12)   Why do so many spiritual types have mental blocks about science and mathematics?

Everyone has mental blocks about science and math including scientists and mathematicians. Like the lyrics to the old song — people hear what they want to hear and disregard the rest — Einstein, to cite just one example, never accepted most of quantum physics even after it was well established and no longer controversial.

People don’t like the feeling of “cognitive dissonance”. The tension between strongly held beliefs and objective facts can bring unbearable psychological pain to most people. Someone once said that genius is the ability to hold contradictory ideas inside the mind. Most people don’t do that well; they don’t like contradictions.

Here is a link to an essay called Truth that some will find interesting:

TRUTH

13)   Is time infinitely divisible?

Einstein said that time and space (i.e. space-time) depends on mass and energy, which are equivalent. In the absence of mass and energy, space and time are meaningless.

The most recent experiments by NASA have found no evidence that time is anything but continuous. However, the shortest time possible is the length of time it takes light to move the shortest distance possible, which is called Planck time. It is thought to be 5.39E-44 seconds.

Time can be divided into as many smaller increments as anyone wants, but nothing can happen in fewer than the number of intervals that add to 5.39E-44 seconds. Time is a variable that isn’t fundamental. It expands and shrinks in the presence of mass and energy.

Some physicists of the past suggested that the “chronon” might be the shortest interval of time. It is the time light travels past the radius of a classical (at rest) electron — an interval of 6.27E-24 seconds. Its calculation depends only on mass and charge, which can change if a particle other than an “at rest” electron is measured.

It seems to me that time is probably best thought of as being continuous. That said, it doesn’t mean that mass-energy interplay isn’t pixelated — much like a digital camera image. Pixelation is critical to a conjecture concerning the preponderance of matter over anti-matter — a conjecture described in the essay CONSCIOUS LIFE.

14)   Which is bigger:   \frac{3}{5}\;  or  \;\frac{1}{9} ?

Think of fractions as pies, which are all the same size. The bottom number is the total number of pieces into which each pie is cut. The first pie was cut into 5 pieces, which are all the same size. The second pie was cut into 9 pieces, which again are all the same size.

The second pie is cut into smaller pieces than the first pie, because there are more pieces. Right?

Mice come along and eat pieces from both pies. The top number is the number of pieces they left behind; the top number is the number of pieces the mice didn’t eat.

So which pie plate has more pie on it? Is it the 5 piece pie that has 3 pieces left or the 9 piece pie that has 1 piece left?

If you think hard you will figure out that it must be the first plate that has the most pie on it. Right?  

15)   Why is a third of 30 equal to 10 and not 9.999999999, as a third of 10 is 3.33333333? 

You can make three piles of ten objects in each pile. When you count the total, it adds to exactly 30 objects. So the answer of “10” is demonstrably true, right? Three piles of ten adds to thirty.

There is no way to make three piles of any identical objects that adds to 10. Three piles of three adds to nine. Four piles of three objects adds to twelve.

We are required to make three piles of three objects and then add a piece of a fourth object to each pile that is smaller than a whole piece.

It turns out that the fourth object is 1/3 of a whole object. When these three piles of three objects plus 1/3 of an object are added up they equal exactly ten.

The problem in understanding comes from trying to grasp that 1/3 — when written as a decimal — is what mathematicians call a repeating decimal. The rules of arithmetic say that the decimal form of 1/3 is calculated by dividing “1” by “3”.

Following the rules of arithmetic when doing the division forces an answer to the problem that results in a repeating decimal — in this case, 0.333333… .

There is no way around these rules that keeps math working right, consistent, and accurate.

Sorry.

16)   Will we be able to have life extension through cloning? 

Cloning not only doesn’t work, it can’t work.

That said, the idea of cloning is to make a genetic replicant of an existing life-form. Extending life-span would require changes to the genome through other means involving changes to structures called telomeres, probably, which straddle the ends of chromosomes in eukaryotic cells.

Here is a link:  Telomere

A short discussion of cloning is included in the essay at this link:  NO CODE

NO CODE is long (11,000 words), but explains in words, pics, graphics, videos, and links some of the complexities, misunderstandings, and dangers of current genetic-engineering at an undergraduate level. It explains basic cell biology, protein production, and much more.

17)   Why does time slow down when we are on a massive planet or star like Jupiter? 

Gravity is equivalent to acceleration. Accelerating clocks tick slower, according to General Relativity, which has been confirmed by experiments. It has to do with the concept of space-time and the fact that all objects travel through space-time at the same rate.

To understand, it helps to read up on space-time, special relativity, and general relativity. The concepts aren’t easy. The universe is an odd place, but it can be described to a somewhat fair degree by mathematics.

Some of the underlying reasons for why things are the way they are seem to be unknowable.

18)   If the ancients had focused on science instead of religion, could we have become immortal by now? 

Immortality is not possible due to the odds of accidental death, which at the current rate makes death by age 25,000 a virtual certainty for individuals.

Worse: the odds for extinction of the human species as a whole are much higher — it’s a near statistical certainty for annihilation within the next 10,000 years according to experts. It seems counterintuitive, but it’s true.

RISK

19)    How do I solve, if the temperature is given by f(x,y,z) =  3x^2 - 5y^2 + 2z^2  and you are located at  (\frac{1}{3} , \frac{1}{5} , \frac{1}{2})  and want to get as cool as possible, in which direction should you set out? 

 You want to establish what the gradient is, establish its direction, then head in the opposite direction, right?

By partial differentiation the gradient is (6x – 10y + 4z), right? You don’t have to take another partial derivative and set it equal to zero to establish a maximum, because all the second derivatives of the variables are equal to one, right? You can drop the variables out and treat them as unit vectors like i, j, & k, correct?

The resulting vector points in the direction of increasing temperature, right?

Changing the signs makes a vector that points in the opposite direction toward cooler temperatures. That vector is (-6, 10, -4).

The polar angle (θ) is 71.068° and the azimuth angle (Φ) is 120.964°. The length (or magnitude) is 12.3288. Right? (We won’t use this information to solve the problem, but I wanted to write it down should I need to refer to it to respond to any comments or to help check my work graphically.)

These directions are from the origin, and you aren’t located at the origin. To determine the direction to travel to get to (-6, 10, -4), you need to subtract your current position. Again, for reference your location is .6333 from the origin at θ = 37.8636° and Φ = 30.9638°. Right?

After subtracting your position vector from the gradient vector, the resulting vector is (-6.333, 9.8, -4.5). Agree?

This vector tells you to travel 12.506 at a polar angle (θ) of 68.9105° and an azimuth angle (Φ) of 122.873° to intersect the gradient vector. At the intersection you must change direction to follow the gradient vector’s direction to move toward cooler temperatures at the fastest rate.

I haven’t graphed out the solution to double-check its accuracy. You might want to do this and let me know if you agree or not.

20)   What is  \sqrt[3]{i} - \sqrt[3]{i}  equal to?

The answer is zero, of course.

But not really. It only seems that way. Each number has three roots.

Depending on which roots are chosen the result can be one of six different sums (as well as zero if both roots are the same). We have to start somewhere so:

What is  i^\frac{1}{3} ?

i =  e^\frac{{i\pi}}{2} .  Right?

Therefore, a third root of i is  e^\frac{{i\pi}}{6} .  Right? It’s not the only root.

It’s the principal root. There are three third roots, which are equally spaced around the unit circle. Right?

It’s clear by inspection that to be equally distributed around the unit circle the other two roots must be  e^\frac{{i5\pi}}{6}  and -i.  Right?

Convert the three roots to rectangular coordinates and do the subtractions.

Here are the roots in rectangular form: (.86603 + .50000 i) , (-.86603 + .50000 i) , and (0.00000 -i).

Here are the six answers (in bold type) to the original question with the subtractions shown to the right:

1.7302 = (.86603 + .50000 i) – (-.86603 + .50000 i)

(.86603 +1.5 i) = (.86603 + .50000 i) – (0.00000 -i)

-1.7302 = (-.86603 + .50000 i) – (.86603 + .50000 i)

(-.86603 + 1.5 i) = (-.86603 + .50000 i) – (0.00000 -i)

(-.86603 – 1.5 i) = (0.00000 -i) – (.86603 + .50000 i)

(.86603 – 1.5 i) = (0.00000 -i) – (-.86603 + .50000 i)

These rectangular coordinates can be converted back to the Euler-form ( e^{i\theta} ).  It’s easy for anyone who knows how to work with complex variables. In Euler-form the angle in radians sits next to i.  The angle directs you to where the result lies on a unit circle. Right?

In fact, the six values lie 60 degrees apart on the circumference of a circle whose radius is the square root of 3. I don’t know what to make of it except to say that the result seems unusual and intriguing, at least to me.

As mentioned earlier, if both roots are chosen to be the same, then in that particular case the result is zero.

21)   What is tensor analysis and how is it used in physics?

Understanding tensors is crucial to understanding Einstein’s General Theory of Relativity.

This question seems to assume that everyone knows what tensors are and how they are represented symbolically. It’s a good bet that some folks reading this question might want some basics to better understand the explanations of how tensors are used for analysis in physics.

If so, here are links to two videos that together will help with the basics:

22)   What is the velocity of an electron?

Electrons can move at any speed less than light depending on the strength of the electro-magnetic field that is acting on them. Inside atoms electrons seem to move around at about one-tenth of the speed of light. You might want to check me on this number. The situation is as complicated as your mind is capable of grasping.

When interacting with photons of light electrons inside atoms seem to jump into higher or lower shells or orbits instantaneously. That said, it is impossible to directly observe electrons inside atoms.

On an electrical conductor like a wire, electrons move very slowly, but they bump into one another like billiard balls or dominoes. The speed of falling dominoes can be very high compared to the speed of an individual domino, right?

So, the answer is: it all depends…

23)   What exactly is space-time? Is it something we can touch? How does it bend and interact with mass? 

Spacetime, according to Einstein, depends on mass and energy for its existence. In the absence of mass and energy (which are equivalent), space-time disappears.

The energy of things like bosons of light — which seem to have no internal (or intrinsic) mass, right? — is proportional to their electric and magnetic fields. Smallest packets of electromagnetic oscillations are called photons.

Many kinds of oscillating fields, like electromagnetic light, permeate (or fill) the universe. In this sense, there is no such thing as nothing anywhere at any scale.

Instruments and tools of science (including mathematics) can give a misleading impression that at very small scales massive particles exist.

According to the late John Wheeler, mass at small scales is an illusion created by interactions with measuring devices and sensors.

Mass is a macroscopic statistical process created by accumulations of whatever it is that exists near the rock bottom of reality where humans have yet to gain access. These accumulations, some of them, are visible to humans; they seem to span 46 billion light years in all directions from the vantage-point of Earth and are displayed for the most part in as many as two-trillion galaxies according to recent satellite data by NASA.

Mass is thought to interact with everything that can be measured (including everything in the Standard Model) by changing its acceleration (that is, its velocity and/or direction), which is equivalent to changing its momentum.

It is in this sense that mass and energy are equivalent. Spacetime depends on mass and energy. Spacetime does not act on mass and energy; it is its result, its consequence. 

Spacetime is a concept (or model) that for Einstein helped to quantify how mass and energy behave on large scales. It helped explain why his idea that the universe looks and behaves differently to observers in different reference frames might be the way the universe on large scales works.

His mathematical description of spacetime helped him build a geometric explanation for gravity that can be described for any observer by using tensor style matrices; many find his approach compelling but difficult computationally.

WHY SOMETHING, NOT NOTHING?

24)   Hypothetically speaking, if one could travel faster than light, would that mean you would always live in the dark?

The space in which objects in the universe swim does expand faster than light when the expansion is measured over very large distances that are measured in light-years. A light year is six trillion miles.

At distances of billions of light years, the expansion of space between objects becomes dramatic enough that light begins to stretch itself out. This stretching lengthens the distance between the peaks and valleys of the electric and magnetic waves that light is made from, so its frequency appears to drop.

The wave lengths of white light can stretch so dramatically that the light begins to appear red. It’s called red shift.

Measuring the red shift of light is a way to tell how far away an object like a star is. As light stretches over farther distances the ability to see it is lost.

The wavelengths of light stretch toward the longer infra-red lengths (called heat waves) and then at even farther distances stretch to very long waves called radio-waves. Special telescopes must be placed into outer space to see these waves of light, because heat and radio waves radiating from the earth will interfere with instruments placed at the surface.

Eventually the distances across space become so great that the amplitudes (or heights) of the waves flat line. They flat-line because space is expanding faster than light can keep up. Light loses its structure. At this distance the galaxies and stars drop out of the sight of our eyes, sensors, and instruments. It’s a horizon beyond which the universe is not observable.

No one knows how big the universe is, because no one can see to its end. The expansion of space — tiny over short distances — starts to get huge at distances over 10 billion light years or so. The simple, uncomplicated answer is that the lights go out at about 14.3 billion light years.

Because there is no upper limit to how fast the universe can expand, and because the objects we see at 14.3 billion light-years have moved away during the time it has taken for their light to reach Earth, astronomers know that the edge of the universe is at least 46 billion light years away in all directions. Common sense suggests the universe might be much larger. No one has proved it, but it seems likely.

Over the next few billion years the universe that can be seen will get smaller, because the expansion of space is accelerating. The sphere of viewable objects is going to shrink. The expansion of space is speeding up.

The problem will be that the nearby stars that should always be viewable (because they are close) are going to burn out over time, so the night sky is going to get darker.

Most (4 out of 5) stars in the galaxy are red dwarfs that will live pretty much forever, but no one can see them now, so no one will see them billions of years from now, either. Red dwarfs radiate in the infra-red, which can only be seen with special instruments from a vantage point above the atmosphere.

Stars like our sun will live another 4 or 5 billion years and then die. The not-too-distant future of the ageless (it seems) universe is going to fall dark to any species that might survive long enough to witness it.

25)   What does “e” mean in a calculator? 

There are two “e”s on a calculator: little “e” and big “E”.

Little “e” is a number. The number has a lot of decimals places (it has an infinite number of them), so the number is called “e” to make it quick to write down.

The number is 2.71828… . The number is used a lot in mathematics and in every field of science and statistics. One reason it is useful is because derivatives and integrals of functions formed from its powers are easy to compute.

Big “E” is not a number. It stands for the word “exponent”, but it is used to specify how many places to the right to move the decimal point of the number that comes before it.

5E6 is the number 5,000,000, for example. The way to say the number is, “five times ten raised to the sixth power”. It’s basically a form of shorthand that means 5 multiplied by 10^6 .

Sometimes the number after E can be negative. 5E-6 would then specify how many places to the left to move the decimal point. In this case the number is 0.000005, which is 5 multiplied by 10^{-6}.

Bonus Question 1 – What difficulties lie in finding particles smaller than quarks, and in theory, what are possible solutions? 

The Standard Model is complete as far as it goes. Unfortunately, it covers only 5% of the matter and energy believed to exist in the universe.

And humans can only see 10% of the 5% that’s out there. We are blind to 99.5% of the universe. We can’t see energy, and we can’t see most stars, because they radiate in the infra-red, which is invisible to us.

The Standard Model doesn’t explain why anti-matter is missing. It doesn’t explain dark matter and energy, which make up the majority of the material and energy in the universe. It doesn’t explain the accelerating expansion of the universe.

Probing matter smaller than quarks may require CERN-like facilities the size of our solar system, or if we’re unlucky, larger still.

We are approaching the edge of what we can explore experimentally at the limits of the very small. Some theorists hope that mathematics will somehow lead to knowledge that can be confirmed by theory alone, without experimental confirmation.

I’m not so sure.

The link below will direct readers to an essay about the problem of the very small.

ON THE VERY SMALL

Bonus Question 2 – What if science and wisdom reached a point of absolute knowledge of everything in the universe, how would this affect humanity?   

Humanity has reached a tipping point where more knowledge increases dramatically the odds against species survival. Absolute knowledge will result in absolute assurance of self-destruction.

Astronomers have not yet detected advanced civilizations. The chances are excellent that they never will.

Humans are fast approaching an asymptotic limit to knowledge, which when reached will bring catastrophe — as it apparently has to all life that has gone before in other parts of the universe.

Everywhere we look in the universe the tell-tale signatures of advanced civilizations are missing.

RISK

We hope readers enjoyed the answers to these questions. Follow Billy Lee on Quora where you will find answers to thousands of unusual and interesting questions.  The Editorial Board  

Billy Lee

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