Gravity Waves - An Update

 This is an update to my October 2017 post on the subject.  Here's the link:  Sigma 5: Gravity Waves.  It's been more than three years.  Surely, something has changed.  Indeed it has.  But before proceeding both backward and forward, let me review the results I reported in my earlier post.

These results were produced by a "gravity wave observatory" called LIGO.  For more than a decade LIGO had nothing to report.  The reason was simple.  It's detector was just not sensitive enough.  But it went through several generations of upgrades.  That last one (Advanced LIGO) did the trick.  The data collection run, tagged "O1" for "Observation run 1", ran from from September of 2015 to January of 2016. It produced three events.  Each event was caused by two large black holes spiraling together till they merged.

The observatory was then shut down for minor upgrades.  At completion, the O2 run took place.  It ran from December 2016 to August 2017.  O1 and O2 together eventually resulted in 11 events being detected.  When I wrote my post five of them had been reported on.  Since then, another round of upgrades has been installed.  Upon completion, the O3 run was started.  It had to be shut down in the middle so it was informally broken into the O3a run (April 2019 to September 2019) and the O3b run (November 2019 to March 2020).

All together, 56 detection events have been identified.  And a third observatory (LIGO is actually two observatories, one located in Washington State, and the other in Louisiana) has been brought online.  VIRGO is slightly smaller than the two observatories that combine to make up LIGO, but is sensitive enough to detect many of the same events that LIGO can.  With three observatories measuring the same event, its location can be narrowed down to a much smaller slice of the sky.  And, in general, more information about the event can be collected.

LIGO is currently shut down so that still more updates can be installed.  The O4 run is currently slated to start in June of 2022.  And VIRGO is also getting upgraded.  And other facilities will be coming online soon.  They are scattered all over the globe.  There are even plans for "LIGO in space", a LIGO-like instrument that would be bigger than it is practical to go with an earth based observatory.  Once those first observations proved that it was possible to detect gravity waves funding has ramped up dramatically.

But that's enough of the present and the future for the moment.  Let's go to the past.  And let's do it by asking a simple question:  what's the speed of light?

It has been possible to make observations spanning distances like ten or twenty miles for millennia.  Back then it was obvious that sound traveled at a finite speed.  You could observe an action and then note a delay measured in seconds before the sound associated with that action reached you.  That made it obvious that, if light was not instantaneous, then it at least traveled much faster than sound.

Reasonably accurate measurements of the speed of sound were successfully made hundreds of years ago.  We have had a very accurate estimate of the speed of sound for perhaps two hundred years.  And scientists were able to establish that sound worked by oscillating something.  Normally, this was air.  But it could be water.  And the speed of sound through water was higher than that of air.  And sound couldn't travel through a vacuum at all.  So, scientists have long had a good idea of how sound worked.

And the obvious thing to do was to apply what they knew about sound waves to light waves.  If the analogy held then light should have a propagation speed.  But what was it?  "Fast" just doesn't tell us much.  Efforts to determine its speed date back to at least 1629.  Experiments with cannons and the like determined that it was too fast to measure using standard methods.

That led to an effort based on astronomy.  This effort produced the first measurement that was at least in the ball park.  Romer in 1676 made detailed observations of the orbits of the moons of Jupiter.  He calculated that in order for the observations to make sense it must take about 22 minutes for light to cross from one side of the Earth's orbit around the Sun to the other.  That would have been great if astronomers of the day knew exactly how big that orbit was.  They didn't.  The best guess put the speed of light at about 140,000 miles per second.

What was important about this is it told scientists just how small the time intervals were that they would need to be able to measure.  Say they wanted to measure the propagation time of light over distance of 10 miles. At 140,000 MPS light would take 0.00007 seconds to traverse that distance.  A stop watch just wasn't going to cut it.

An early scheme depended on a rapidly rotating wheel with teeth on it.  Taking light as particles for the moment to make the explanation simple, arrange for particles of light to be shot past the wheel and along to a distant mirror.  There they are reflected back, again past the wheel and on to a detector.  If things are arranged such that the light has to travel through the part of the wheel where the teeth are, then it will be blocked when a tooth is in the way but can pass freely when it hits a gap.

Now, spin the wheel at high speed.  If the wheel is spinning at just the right speed then a particle of light can pass through one gap between teeth on the wheel, bounce off the distant mirror, and then return just in time to pass through the adjacent gap.  This setup allows time to be sliced into very small intervals very accurately.  Simple calculations suffice.  And a different sized disk or a different rotation speed can used to fine tune the interval to whatever is necessary.

This admittedly inaccurate explanation gives you the idea.  The point is that with the proper equipment built along these lines things can be arranged so that the light passes through one slit going and a different slit coming.  A wide range of time delays can be accommodated.  It is simply a matter of dialing the setup in.  Once the right combination of rotation speed and disk/tooth size is found, it is a small step to translate the settings into the speed of light they represent.

And, as a result, Foucault was able to come up with a speed of 298,000 KM/s in 1862.  This is very close to the modern value of just under 300,000 KM/s.  Others improved his setup and came up with similar values.  By 1887 Michelson and Morley were confident that they could measure the speed of light very accurately.

Measuring the speed of light very accurately was of secondary importance to them.  Their primary interest was in learning was how fast and in what direction the Earth was moving.  To do that they needed to very accurately measure the speed of light.

In normal circumstances sound travels through air.  Air is the "medium of transmission", the thing that sound vibrates in order to move.  But what was the medium of transmission of light?  It had to be something, didn't it?

And there were all those things that weren't the medium of transmission.  After all, unlike sound, light can easily travel through a vacuum.  And since a vacuum is, by definition nothing, all the usual suspects get immediately eliminated.  So, scientists posited the existence of something called the "luminiferous aether".

Assuming something exists just because its existence is convenient is not good enough for scientists.  They need actual proof that it does exist.  And a good place to start is by trying to measure its properties.  And the fundamental property that aether had was its ability to transmit light.

And it was assumed that, everything else being equal, the propagation speed of light in aether was constant.  That was true for sound and air.

If you kept air moving at a constant speed and kept its temperature constant, and so on, then the speed of sound through it was constant.  Conversely, you could determine some of the attributes of air by measuring the speed of sound through it.  For instance, fast moving air would result in a different measured speed of sound than slow moving air.

So, assuming the parallel held, the speed and direction the aether was moving could be inferred from a careful measurement of the speed of light in various directions and at various times.  And it was assumed that the aether didn't move.  The Earth moved through the aether.  So, differences in the speed of light led to different speeds for the aether, which in turn led to a measurement of the speed of the Earth through the aether.

All this was speculation piled upon speculation and scientists knew it.  But the measurements were expected to turn up something, even if it wasn't exactly what "aether theory" predicted.  And repeated measurements should lead to some ideas about aether theory being discarded and other ideas being confirmed.  That was all par for the course.  But what everybody agreed on was that careful measurements would turn up differences in the measured speed of light.

After all, it was known that the Earth traveled around the Sun at relatively high speed.  And the direction of travel changed with the season.  That amount of change alone should have been enough to change the speed of light by a measurable amount.  The apparatus had been designed to easily and unambiguously detect changes of this magnitude.  If other changes turned up as the measurement process progressed, that would just be a bonus.

The problem is that the Michaelson-Morley experiment turned up the result that no one expected.  And this "unexpected result" phenomenon pops up in Science all the time.  It is a normal part of science.  Scientists expect it to happen regularly.  They just don't know when it will happen and when it won't.  And what this means is that all those "scientists reject my belief, not because it is wrong, but because it doesn't fit in with what they already believe" arguments are nonsense.

If someone provides hard evidence that current scientific thinking is wrong then scientists change their thinking.  That's what scientists were forced to do because Michaelson and Morley got the result they did.  No scientist liked the result they got.  But other scientists were able to reproduce the result in well conducted experiments.  So, scientists had to find a way to live with the result, which they eventually did.  Scientists reject "unscientific" beliefs, not because they are unscientific, but because they are not backed by solid evidence.

Scientists have been forced by the results of experiment to reject all kinds of sensible ideas. They have been forced to accept ideas that were far more weird and unnatural and unbelievable than anything an outsiders has thrown at them.  Why?  Because some well done experiment or observation has forced them to.  And the Michaelson-Morley result was one of many instances of this.

The Michaelson-Morley result eventually led Einstein to publish his Special Relativity theory in 1905.  Their result had dealt a near-fatal blow to the idea that the luminiferous aether existed.  But it wasn't until Special Relativity that scientists bailed completely on it.  The theory worked.  It also did away with the need for aether to exist at all.   The real kick in the pants, however, didn't come until ten years later.  Einstein introduced General Relativity in 1915.  That's when things got really weird.

In 1905 Einstein had built Special Relativity around the idea that the speed of light is constant.  That's pretty weird.  In order to make things work the theory demanded that all these other not-light things must stretch and shrink.  There were still lots of things that remained unchanging.  But still, some things that we had thought were unchanging, changed in these predictable ways in circumstances that Einstein laid out.

Okay.  That's lot to buy, but the Michaelson-Morley result demanded some kind of weirdness.  And Special relativity weirdness was pretty much the minimum amount of weirdness that would get the job done.  The problem is that General Relativity took weirdness to a whole new level.  We're now talking bat-shit-crazy weird.

You see, General Relativity requires space itself to stretch and shrink.  Space, to put it another way, is the luminiferous aether.  And it behaves in many ways like the air that sound travels through.  It's what vibrates to transmit gravity.

Newton said "objects in motion tend to continue in that motion".  Gravity works by literally warping space.   So an object thinks it is continuing to travel in a straight line.  But gravity causes space to warp and that caused the "straight line" course of the object to bend, not because the object has changed direction, but because "straight" is no longer straight.  Like I said, bat-shit-crazy.

And this "space is wiggly" business means that there are such a thing as "gravity waves", instances of space wiggling because, you know, gravity.  And I think you can now understand why I, for one, was not having any of it.  I was not convinced that gravity waves even existed even though lots of smart people whom I deeply respected believed that they did.  But the LIGO results did me in.  They were right and I was wrong.

And I have to admit that I am actually happy that I turned out to be wrong.  Because, as I observed three years ago, "[e]very time something previously invisible has become visible, tremendous discoveries have been made".  And it is important to understand that the first tremendous discovery has already been made.  We now know with absolute certainty that gravity waves exist.  That's a tremendous discovery if there ever was one.

Beyond that, we know that General Relativity computations about the characteristics of gravity waves work pretty well.  For instance, they get their strength about right.  Why not 100% right?  Maybe.  But we know so little about the events behind the observations at this point that we can't say with certainty.  All we know about many events comes from running LIGO data through General Relativity.  That results in, for instance an estimated mass.  Is the estimate correct?  At this point we have no way of knowing for sure.

But even if the calculations are off by some they still tell us things.  Remember that first estimate for the sped of light.  It was close enough to tell us where the decimal point went.  And that was valuable information.

And we now have 56 events to go on.  The first event was scary.  Was it some kind of screw up?  Was it some kind of unusual event or was it pretty typical?  With one event it's hard to judge.  With 56 events patterns emerge.  A lot of the events are two black holes spiraling together to merge into one.  We now have some idea of how common that event is.  One of the early events was a neutron star merging with a black hole.  Scientist got very excited about that one.

There have been some events that fall outside the accepted theories for how these kinds of events are supposed to progress.  The details are complex and I don't really understand them.  But the scientists are very excited by what they are seeing.  It would be nice if everyone else was too.  But they are not.

Something the general public doesn't understand is that scientists are actually happier when the data doesn't conform to current theory.  It's just more fun and interesting to be on the hunt for a new theory to replace an old broken one.  That's as good as it gets.  It's what made Einstein famous.

Next best is to come up with a modification to an old theory.  Sometimes you don't have to throw the whole thing away.  Maybe you change parts of a theory but leave the rest alone.  If the result is that the revised version now fits all the experimental data then that is a very good result.

It's progress but not the best outcome if you can change a theory and the new theory is a better but not a perfect fit for the experimental data.  That's an improvement, but it also is evidence that more work is needed.  Unlike many, scientists expect their theories to agree with all the data, not just most of it.

The scientists that feed off of the LIGO data have gone from not excited to very excited.  Before 2016 they had no data to work with.  That's not very exciting.  Now that they have data, and lots of it, to work with they are very excited.

Unfortunately, things have gone in the other direction in terms of general interest.  There was a flurry of press coverage back in 2016.  Although the first event LIGO observed happened in 2015 it wasn't announced until then.  For reasons I go into in the previous post the first event was suspicious.  No one wanted to make an announcement they would later have to take back.  So there was a long delay while things were checked and rechecked.

Fortunately, the second event came along pretty quickly.  That's when I and many others relaxed.  It was real.  And a third event followed shortly thereafter.  And VIRGO came on line.  This was enough to maintain interest until about the summer of 2017.  I wrote my post in October of 2017.  It turns out that interest by the press and by the public was already waning by then.  Press coverage since has been almost nonexistent.

But the data keeps pouring out.  The upgrade from the O1 setup to the O2 setup was modest.  But it was enough to increase the rate of event detection.  The modest upgrade that was sandwiched between the O2 and the O3 runs has also increased the rate of event detection.  LIGO will be down for a long time between the O3 and the O4 runs.  The currently scheduled starting date for the O4 run envisions a 27 month gap.  The gap is so long because the upgrade will be much more extensive.

Each upgrade increases the sensitivity.  That means that events similar in size to currently detectable events can be detected further out.  Since a "cube" law is involved, a 10% increase in sensitivity translates into a 33% increase in the volume covered.  Also, smaller events that happen within the old volume can now be detected.  The difference is not as dramatic, but it should result in still more events being detected.

So LIGO started out as what appeared to be a boondoggle.  For a long time it ate lots of money while producing no science.  But the project did a one-eighty in 2016 when that spectacular discovery of the first event was announced.  The discovery of the second event didn't strike the public as nearly as spectacular.  But in many ways it was more important.  It proved that the first event wasn't a one-off.

Unfortunately, the public saw not much difference between the first event and the second, so they started tuning out.  And the public was treating each new event as routine a long time before LIGO got to the 56th one.  And routine is not newsworthy.  So, the press has been checked out since 2018.  It is possible but unlikely that O4 will produce a result that is spectacular enough to put LIGO on the front page again.

That is sad.  The quality of the science is increasing by leaps and bounds.  A big reason for this is the large pool of events, the very thing that makes the whole enterprise boring to the public.  And the O4 run should make things worse at generating buzz by producing data much more quickly than any previous run.

But more data is good for science.  Many more events means that comparisons can be made and patterns can be confirmed or disproven.  There is lots more data to use to test theories against.  Most theories will be found wanting but that's okay.  It's how science works.

The practical effect of something as exotic as gravitational waves can not be predicted.  No one knew that the time of its development that an obscure and insanely difficult physics theory called Quantum Mechanics would prove to be the foundation upon which all the integrated circuits that power all of our modern electronic devices are built.

Some theoretical work, and at this point LIGO is all about the theoretical, never seems to lead to anything practical.  But time after time, something wildly theoretical and of no apparent practical use, ends up allowing us to go from "why are people we don't care about and who live in an obscure corner in China getting sick?" to "we are now making life saving vaccines out of something that the public has never heard of called 'mRNA'."  And these mRNA vaccines are so powerful that they can stop a deadly world wide pandemic in its tracks.  And only a year separates these two events.

60 Years of Sceince - Part 23

This post is the last in a series that dates back several years.  In fact, it's been going on for long enough that several posts ago I decided to upgrade from "50 Years of Science" to "60 Years of Science".  And, if we group them together, this is the twenty-third main entry in the series.  You can go to Sigma 5: 50 Years of Science - Links for a post that contains links for all the entries in the series.  I will update that post to include a link to this entry as soon as I have posted it.

I take Isaac Asimov's book The Intelligent Man's Guide to the Physical Sciences as my baseline for the state of science when he wrote the book (1959 - 60).  In this post I will review the section titled  "Fusion Power".  This is the last section of the last chapter in the book.  But there is an appendix.  So I will finish up by taking a look at what's in it. Since there is no more to the book, there is no reason to continue the series.  To work.

"Fusion Power" addresses a potential that has remained unfulfilled to this day.  Nuclear fission potentially provides access to such large amounts of power as to be almost unimaginable.  This potential has been turned into reality in the form of fission powered electric power plants that supply a substantial portion of the electricity we consume.

For all their problems, and in spite of the fact that they have not lived up to the potential Asimov and many others saw back in 1960, facilities of this type actually exist.  And they actually produce large quantities of electric power on a routine basis.

The same can not be said for fusion based electric power production.  But before we go into how this sorry state of affairs has come to be, let's review what Asimov had to say on the subject.  He starts out by noting that at the time of the book, physicists had been dreaming of harnessing nuclear fusion for twenty years.  Why the interest?    Because fusion is the process that powers our Sun.

The Sun is at roughly the midpoint of the time it will spend as a type of star called a Yellow Dwarf.  At some time in the future it will go through a series of metamorphoses that will turn it into a type of star called a White Dwarf.  That doesn't sound so bad, but for us it is.  A White Dwarf is tiny.  And it only puts out an infinitesimal amount of the heat and light that a Yellow Dwarf star like out Sun produces.

Even so, thanks to fusion, the Sun has been able to continuously produce massive quantities of energy for billions of years.  And it will continue to be able to do so for several billion years more.  Is there any better argument for the potential represented by fusion power?

Asimov correctly concludes that "[i]f somehow we could reproduce such reactions on the earth under control, all our energy problems would be solved."  The "under control" part is important.  At that time we already knew how to build a large "H" bomb.  It used fusion to create an amount of energy that was measured in megatons.  That's far too much of a good thing.

It's not the inefficiency of fossil fuel burning that is the problem. It is the side effects, the greenhouse gasses, etc.  Other non-nuclear options have problems that I have listed elsewhere.  Fission, the other "nuclear" option, has turned out to have problems that I have also addressed elsewhere.  But, assuming it could be controlled, and assuming little or no radioactivity would be generated, a reasonable assumption, then fusion based power generation would be a wonderful thing.

Asimov opines that fusion power would produce no radioactive waste.  This is actually an open question.  Some designs produce no radioactive waste.  Others do.  But even the designs that do produce radioactive waste look like they would produce far less radioactive waste than a fission based power plant.  He also notes that pound-for-pound fusion produces 5-10 times more power than fission.  So what's the hold-up?

He postulates the development of a fusion reactor based on Deuterium.  It is far rarer than regular Hydrogen.  But, as he notes, traces can be found in regular ocean water.  If efficient extraction processes can be found or developed then the fuel supply becomes effectively unlimited.

Deuterium has long been a subject of interest to nuclear physicists.  It is much easier to induce it to fuse.  The easy way to think of the problem is in terms of temperature.  Deuterium requires super-high temperatures to induce it to fuse.  But regular Hydrogen requires ultra-high temperatures, temperatures far higher than Deuterium requires.  Both regular Hydrogen and Deuterium are non-radioactive.  Putting it all together, Deuterium seems like the smart way to go.

Asimov then goes on to practical considerations.  With fission, physicists already had a starting point when it came to figuring out how to control it.  The "nuclear pile" they had built while figuring out how to build an "A" (atomic - fission) bomb provided a working example of a small, controlled, fission environment.  The problem is that there is no pile-equivalent that was developed along the way to the creation of a successful "H" (Hydrogen - fusion) bomb.

All "H" bomb designs use an "A" bomb as the mechanism necessary to initiate the fusion reaction.  No one ever figured out a half-measure way to get the job done.  So the developers of a fusion based power plant had to start from scratch.

Asimov whined about a lack of effort when it came to fusion reactor design.  This might have been true at the time.  But the problem has since received a large and persistent amount of attention.  Asimov lays out the two big problems.

The first problem is achieving super-high temperatures.  He estimated that reaching 350 million degrees would be necessary.  That's no problem in the vicinity of an exploding "A" bomb.  But we want to be able to do it in a relatively normal building that is situated relatively close to homes and businesses.  His estimate turned out to be way to high.  But millions of degrees are certainly necessary.

The second problem is holding everything together long enough to extract the power and turn it into electricity.  An "H" bomb literally blows itself apart in a small fraction of a second.  A practical fusion power plant must be able to produce power steadily for minutes, hours, days, weeks, even years.  Asimov tackles the first problem first.

His suggestion is a magnetic bottle.  A donut shaped cavity is evacuated.  Deuterium is inserted and an extremely strong magnetic field is applied.  For reasons I am not going to get into, this mechanism can heat the Deuterium to extremely high temperatures.  This causes it to turn into a plasma.  I am also going to skip most of the differences between gases and plasmas.

I am also going to mostly skip over the fact that we took a vacuum and then added a gas.  Doesn't that ruin the vacuum?  It doesn't if only a small amount of gas is inserted.  And it turns out that a plasma acts effectively as if it is a series of wires.  So we can use magnetic fields to "pump" lots of energy into it from a short distance (a few feet) away.  That heats it up.

The fact that we are doing this in a vacuum means that, if we ae clever enough, we can keep the super-hot plasma from ever touching the cold walls of the donut.  This means that the plasma can stay hot as it not transferring any energy from itself to the cold walls.  Conversely, the walls can be kept at something like normal temperatures because they don't make contact with the super-hot plasma.

And it turns out that all of this works.  But only to a certain extent.  No one has been able to get a device to heat a plasma to a sufficiently high temperature and then keep it there for any length of time.  One unexpected problem turns out to be plasma instability.  The plasma starts forming waves.  And those waves keep getting bigger and bigger.  They quickly get big enough to mess everything up.  The temperature crashes, or something else goes wrong, and the whole thing quickly "quenches".

Asimov then moves on to the second problem.  The trick here is that plasmas conduct electricity.  That means that you can steer them with magnetic fields.  This technique is called "magnetic confinement".  Scientists were having some success with magnetic confinement at the time the book was written.  They have since had much more success.  But "plasma instability", the "wave" business I discussed above, has limited the degree of success.  If the plasma instability problem could be fixed then magnetic confinement would work just fine.

Since the time of the book a Russian idea called a Tokamak has become the leading candidate for the best design.  To the untrained eye it looks pretty much like the donut I have discussed above.  But the subtle differences apparently help a lot.  Many design ideas have been tried since the '60s and failed.  The current leading candidate is called ITER.  It is a European led initiative that is based on a Tokamak.

Many billions of dollars have been sunk into ITER.  It is years away from completion so we are many years away from learning how well it works.  And it is a "proof of concept" project.  If it works then a "new and improved" design will be needed.

It will be based on lessons learned from the current ITER.  This follow-on device is supposed to be the first device that can actually produce electricity.  And, if that design works but each device constructed according to that design costs ten or twenty billion dollars to build, then fusion based power production may never pencil out.

There are lots of alternatives to the ITER that some laboratory or another is tinkering with.  Funders have decided to go pretty much all in with ITER. So all these other ideas are starved for cash and operating on a relative shoe string.  So they tend to poke along.  But, if one of them happens to  produces spectacular results then it may displace the ITER/Tokamak design as the front runner.  Don't hold your breath.

In Asimov's time, people were pretty optimistic that fusion power could be pulled off.  But that was sixty years ago.  Since then a lot of designs have come and gone.  And many billions of dollars have gone.  And we are still a very long way from a practical and cost effective device.  Or even one that works at all.

That's where the main part of Asimov's book ends.  So, let's finish up by looking at the appendix.  It is titled "The Mathematics of Science".  It is divided into two sections, "Gravitation", and "Relativity".  Asimov confined himself to a little simple arithmetic for the main part of the book.  Here, he relaxes that restriction somewhat.

You can understand Galileo's take on gravitation by moving on from basic arithmetic to High School algebra.  But before Asimov dives into that he steps back to make a few general observations.

He credits Galileo for the transition from a "qualitative" approach, just describing what's going on in sufficient detail for someone else to be able to recognize it, to a "quantitative" one.  In this latter approach it is important to also be able to measure things with as much precision as can be managed.

That is much easier to do now, than it was then, Asimov notes.  Take time.  There were no clocks capable of more accuracy than a sun dial available to Galileo.  He started out timing things by counting his pulse.  But keeping your pulse even is almost impossible to do.  Galileo knew that so he tried to compensate by devising various water clocks.  I am going to skip over the design details but note that none of his designs was completely successful.

And he had no way to accurately measure very short time periods.  Here, he came up with a trick that worked very well.  Instead of dropping something he rolled it down a ramp.  The shallower the ramp the longer it took the object to roll down the length of the ramp.  This, in effect, slowed things down enough that he could measure things accurately with the tools at hand.

And what he found was that a ball rolling along a flat track maintained a roughly constant speed.  He attributed the minor amount of slowing to friction and decided that, in the absence of friction, the speed would be constant.  This can be represented by the simple algebraic equation "s=k".  "s" is the speed of the object and "k" is some constant that depends on circumstances.  We have now dipped our toes into algebra.

This observation later became the foundation of "Newton's first law of motion".  Newton generalized what Galileo had done, resulting in "v=kt".  Here "v" is a more complicated concept than "s".  "v" (velocity) incorporates the concept of speed but it also incorporates the concept of direction.  So any change in speed, or direction, or both, means that "v", the velocity, has changed.  "k" is our old friend a constant, and "t" is time, that thing Galileo couldn't measure very accurately.

Newton postulated that, when it came to gravity, velocity would change at a constant rate as time passed.  He further postulated that there was something called a "gravitational constant".  So, when applied to velocity in a gravitational field the equation became "v=gt", where "v" and "t" are as before, but "g" is a gravitational constant.  This is still pretty simple algebra, but it is more complex than where we started.

It turns out that the value of "g" depends on some things.  But in a lot of circumstances "g" is a specific value that doesn't change.  And, it turns out that there is a "G", that really is constant.  You mathematically combine "G" with some other things and you can calculate the value of "g" for a specific circumstance.  Asimov goes into this in some detail, but I am going to skip over it.

I will note that he ends up discussing "sine" (shortened to "sin" in many contexts) a "trigonometry function".  Trigonometry ups the ante when it comes to mathematics by quite a bit.  Trigonometry is normally studied in High School.  But only "math track" students are exposed to it.  Any serious study of the physical sciences involves a knowledge of trigonometry and the ability to use its associated functions.

Digging deeper into Galileo brings us to an equation I don't know how to accurately reproduce in a blog post.  An unusual formulation that is accurate is "d=10tt".  Now "d" is distance" and "tt" just indicates "t" (our usual time) multiplied by itself.  This is usually indicated with a single "t" to which a small superscript "2" is attached.  This indicated that two "t"s should be multiplied together and that value used.  But I don't know how to get the blog software to do the superscript thing.  Anyhow, powers of numbers (multiplying them by themselves multiple times) is another increase in the mathematical degree of difficulty.

Asimov now completely abandons Galileo to focus on Newton.  He starts in territory that requires only High School mathematics.  "A=4 pi r r" (spaces added to improve readability) is such an equation.  Here "A" is area, "pi" is a stand in for the common symbol for the ratio between the circumference of a circle and its diameter.  But I don't know how to get my blog software to spit that symbol out.  And "r", which must be squared, in this case stands for the radius of a sphere.

A more interesting equation associated with Newton is "f=ma".  "f" is force, "m" is mass, and "a" is acceleration.  But why "m" and not "w" for weight?  Because weight is the result of a gravitational field.  As the strength of the field changes, the weight changes.

Newton wanted something that was gravity independent.  If you know the mass and the details of the gravitational field you can calculate weight.  If you know weight and the details of the gravitational field you can calculate mass.  Interestingly, if you know weight and mass, you can calculate the strength of the gravitational filed.

Finally, there are some situations where gravity is not involved but it is useful to know mass.  This led to an interesting question.  We measure weight when we put something on a scale.  What the scale actually measures is force.  Using the formulas discussed above we can translate that into mass.  Specifically, we can calculate the "gravitational mass" of an object in this way.

But the "f" in "f=ma" doesn't need to be a force associated with gravity.  If we know the "f" and the "a" we can calculate the "m".  In many situations not involving gravity, what we are calculating is called "inertial mass".  Einstein asked the question, "is the inertial mass of an object always the same as its gravitational mass"? 

It turns out that there is no effective difference if "im / gm = k".  In other words, if the inertial mass ("im") of an object divided by the gravitational mass ("gm") of the same object always yields the same constant then the two are indistinguishable, so we might as well assume that they are the same thing.

Scientists, including Einstein, have looked for instances where the "im / gm" ratio varies.  So far they haven't found any.  So, until proven otherwise, scientists assume that "im" equals "gm".  If you can find an instance when "im" does not equal "gm", it's a safe bet that there will be a Nobel Prize in your future.

Asimov doesn't move on to the next obvious topic.  High School math is adequate to cover what is called "statics", situations where everything is static, i.e. unchanging.  But what about "dynamics", situations where things are changing?  For that you need calculus.  Newton wanted to study dynamic situations, celestial bodies orbiting other celestial bodies, objects falling in a gravitational field of varying intensity, things like that.

He literally had to invent calculus in order to perform the computations and analysis he was interested in.  The calculus he invented was limited.  As soon as he had developed as much of it as he needed to be able to answer the questions he was interested in, he stopped working on calculus and moved on to other things.

Fortunately for us, a German named Leibnitz developed calculus at the same time.  His version did not suffer from the limitations that Newton's did.  He was a mathematician, so he kept adding improvements and extensions for as long as he could.  In the end his version covered a lot more mathematical territory.

Engineers often use the Newtonian version because it is simpler and well suited to many of the problems they routinely encounter.  Everybody else uses the Leibnitz version.  And it has long since been demonstrated that in the areas where they overlap, they are both completely equivalent.

On to "Relativity".  Relativity consists of "Special Relativity", the version Einstein developed in 1905, and "General Relativity", the more complicated but more all encompassing version he developed in 1915.

But before going here Asimov spends a lot of time on the Michaelson-Morley experiment.  This was an experiment done in 1887 that attempted to measure the direction and speed (i.e. velocity) of the Earth as it travelled though space.  The experiment depended critically on the speed of light being a constant.  At the time no one could imaging things being otherwise.

The calculation that would turn the resulting measurements into the velocity of the Earth involve some fairly complex algebra.  But they were well within the capability of a High School student who had completed the "math" track.  I am going to skip over them.  For one thing, they are complicated.  For another thing, we don't need them.  The experiment failed.  The result said that the Earth was not moving through space at all.

We know now and they knew then that "it moves", to quote Galileo on the subject.  If nothing else, it circles the Sun once per year at a distance averaging 93 million miles.  The necessary "orbital velocity" is easily calculated.  That number was far higher than the sensitivity of the experiment.

The "it's not moving" result was shocking.  So lots of people tried unsuccessfully to find a flaw in the experiment's design.  And others reproduced the experiment and got the same result.  Einstein was the first one who was willing to say that "what's going on here is that the speed of light is variable".

In fact, Special Relativity follows directly from the idea that "no matter how you measure the speed of light, and no matter what circumstances you measure it in, as long as there's no acceleration involved, you will always get exactly the same answer".

Fitzgerald had already done some of the work.  He came up with a formula that calculated exactly how much things needed to "contract" to keep the measured speed of light constant.  Fitzgerald's equation included "c" the speed of light.  So the degree of contraction could be related to the number you got when you divided the speed of the object by "c", the speed of light.  Now, as a speed, "c" is very large.  It's conventionally quoted as 186,000 miles per second.

A fast car might go 100 or 200 or even 300 MPH.  That's a tiny fraction of "c".  A commercial airplane flies at just over 500 MPH. A high performance plane might go 2,000 MPH.  Both are only going a tiny fraction of "c".  The speed of pretty much anything we encounter in our day to day experience always amounts to a tiny fraction of "c".  Even a rocket going 20,000 MPH, what we would normally think of as being super-fast, is still crawling along when measured against "c".

Fitzgerald's equation said that for anything going a small percentage of the speed of light, the amount of contraction taking place would be miniscule.  Even if you were going 100,000 miles per second, roughly half the speed of light, the effect would be relatively modest.  You had to be going at 90% or 95% or 99% to see really large effects.  And if you could get to 99.9% or 99.99% then some really strange things would happen.

The fact that the effect was infinitesimal at "normal" speeds was why nobody noticed it, Einstein argued.  Also, note what happens if something is traveling at exactly "c".  Everything either goes to zero or infinity.  This is the basis of the statement that you can't reach the speed of light no matter how hard you try.

There is mathematics that says what might happen if you cold find a way to go faster than "c".  But, if you translate the results into the real world, you get complete nonsense.  And, before you ask, if you succeeded, the things that would happen would instantly render you dead.  (They would also destroy any instrumentation or machinery too, so whatever else you might want to try wouldn't work either.)

Special Relativity works the same as Newtonian Mechanics, in terms of the math required.  You can solve static problems using High School algebra.  You need calculus to solve dynamic problems.  But calculus is all you need.

Asimov does not discuss General Relativity, the version that can handle things when accelerations are involved.  There is a good reason for this from a mathematical perspective.  It took Einstein a decade to go from Special to General Relativity.  He spent several years trying to get anywhere at all.

He finally came up with a couple of key ideas.  But he quickly realized that, if he was going to handle these ideas quantitatively,  he would need to learn a type of mathematics called Tensor Calculus.  He had to devote the best part of two years to doing this.  Fortunately for Einstein, Tensor Calculus had already been invented.  He didn't have to invent it.  He just had to learn how to do it.

All I'm going to say about Tensor Calculus is that it is way harder than regular calculus.  Imagine that you are barely scraping by in High School algebra.  If that's an accurate measure of your mathematical ability then imagine how hard it would be to learn regular calculus.  That comparison gives you a feel for the difference in difficulty that lies between regular calculus and Tensor Calculus.

But the good news is that once Einstein had mastered Tensor Calculus, he succeeded in formulating his ideas in terms of Tensor Calculus, and then using it to compute results.  And, he showed that his results matched reality.  He was able to show that General Relativity provided the solution to several puzzles that had been bedeviling Astronomers.

He even famously made a prediction involving a star appearing to move when the light from that star came close to grazing the surface of the Sun.  The star didn't move, but the "gravitational lensing" caused by the gravitational field surrounding the Sun caused the path that light from the star took to bend on its way to Earth.  And that made it appear that the star had moved.

And with that, we're done.