Tag Archives: GR Centennial

Gravity 5: Putting Einstein in the Navigator’s Seat

by Shane L. Larson

When Einstein put general relativity forward in 1915, the world had barely entered into the electrical era. Automobiles were not unheard of, but were not common. The great Russian rocket pioneer, Konstantin Tsiolkovsky, had published the first analysis of rocket flight through space in 1903, but the first successful liquid fueled rocket would not be flown until 1926 by American rocket engineer, Robert H. Goddard, reaching an altitude of just 41 feet. Earth gravity, though weak by the standards of general relativity, was a formidable foe. Of what possible use was general relativity?

The great rocket pioneers  Konstantin Tsiolkovsky (L) and Robert H. Goddard (R). They were actively trying to design machines to escape Earth's weak gravity at a time when Einstein was developing general relativity to understand gravity in more extreme situations.

The great rocket pioneers Konstantin Tsiolkovsky (L) and Robert H. Goddard (R). They were actively trying to design machines to escape Earth’s weak gravity at a time when Einstein was developing general relativity to understand gravity in more extreme situations.

At the time general relativity was first described, it was very much in the form of what is today called “fundamental research.” It described Nature on the deepest levels. It extended the boundaries of human knowledge. It challenged our conceptions about how the Cosmos was put together. But for all practical purposes, it had little impact on the average person. It did not contribute to the Technological Revolution, electrifying the world and changing the face of industrial manufacturing. It did not provide a reliable way to make crossing the Atlantic faster or safer. It did not transform the way steel was made or assembly lines were automated. It did not make the lives of the common worker easier, nor scintillate the conversations around family dinner tables.

Chicago in 1915, when general relativity was first presented. South State Street (L) and Water Street (R). Horses were still common, electricity was just coming to cities, and buildings were short by today's standards. General relativity was "fundamental research" and, at the time, had little direct bearing on everyday life.

Chicago in 1915, when general relativity was first presented. South State Street (L) and Water Street (R). Horses were still common, electricity was just coming to cities, and buildings were short by today’s standards. General relativity was “fundamental research” and, at the time, had little direct bearing on everyday life.

In fact, the implications and predictions of general relatively were not fully understood in those early years. It has taken a full century to come to grips with what it is telling us about the structure of the Universe. Over time, it has slowly become a prominent tool to understand astrophysics and cosmology, but those applications are still the purview of exploratory, fundamental science.  It is only now, after a century of tinkering and deep thinking that the full potential of general relativity is being realized. Today, it impacts the lives of every one of us through the magic devices we carry in our pockets that tag our photos with the locations they were taken and help us navigate to business meetings and ice cream shops. Virtually every phone and handheld electronic device in use today uses global positioning system technology (GPS), which cannot work without a full and deep understanding of general relativity.

How do you navigate around the world? When I was a youngster, I would go to camp in the Rocky Mountains every summer. Those long ago days were filled with all manner of woodland adventures, ranging from ropes courses, to archery, to cliff jumping into swimming holes. My favorite activity, however, was hiking and navigating. We tromped all over the forests and mountainsides of Colorado, and every now and then stopped to pinpoint our location on a paper map of the forest. It was an activity that agreed well with me, instilling a lifelong love of maps.  So how did it work?

Traditional navigation using a compass and map. (L) The direction to multiple known landmarks is measured with a compass. (R) Those directions are transferred to a map, passing through the landmark. The place where the sightlines cross is your location.

Traditional navigation using a compass and map. (L) The direction to multiple known landmarks is measured with a compass. (R) Those directions are transferred to a map, passing through the landmark. The place where the sightlines cross is your location.

The basic notion of navigation on paper is to recognize some landmarks around you — perhaps two distinct mountain peaks in the distance.  Let’s call them “Mount Einstein” and “Mount Newton.”  Using your compass, you determine the direction from your location to each of the mountain peaks. Perhaps Mount Einstein is due northwest, and Mount Newton is north-northwest (a hiking compass is finely graded into 360 degrees, so you could have more precise numerical values for direction; the procedure is the same one I describe here with cardinal directions).

Now, you go to your paper map, and locate the two mountain features you are looking at. When you find Mount Einstein, you draw a line on your map that goes through Mount Einstein, pointing due northwest. If you are standing anywhere along that line, you will see Mount Einstein due northwest.  Now you do the same thing with Mount Newton, drawing a line that points due north-northwest. If you are standing anywhere along this line, then you will see Mount Newton due north-northwest.  If you extend your two lines as far as you can, you will see they cross at one place and one place only. This is the only place a person can stand and see these two landmarks in the directions indicated — it happens to be exactly where you are standing!

In the modern era, many of us navigate using GPS technology, built directly into our smartphones.

In the modern era, many of us navigate using GPS technology, built directly into our smartphones.

This navigational process is called triangulation and it is the most basic form of locating your position. But when was the last time you navigated around the city with a paper map and a compass? This is the future, and if you are in downtown Chicago and want to get from the ice cream shop to the Adler Planetarium, you whip out your smartphone and ask your favorite Maps program to give you some navigational instruction!

How does your phone know where you are? Your phone has a microchip inside it that uses a network of satellites to locate your position on Earth by figuring out where you are with respect to each satellite. In essence, it is kind of like the triangulation method we just discussed.

Third generation GPS satellite (GPS IIIa).

Third generation GPS satellite (GPS IIIa).

The Global Positioning System satellite network is a constellation of 32 satellites orbiting at an altitude of approximately 20,200 km (12,600 mi, almost 50x higher than the International Space Station). Each of the satellites carries on board an accurate atomic clock that is synchronized to all the other satellites. They sit in orbit, and transmit the current time on their clock.  Those signals spread outward from the satellites, and can be detected on the ground by a GPS receiver, like the one in your smartphone.

Each satellite transmits the same signal at the same time. If you are the same distance from two satellites, you get the same signal from both satellites at the same time.  But suppose you are closer to one satellite — then the time you get from one satellite is ahead of the other! The time you receive from each satellite tells you the distance to the satellite (for aficionados: distance is the speed of light multiplied by the time difference between the received satellite time and your clock, if you ignore relativity!) . The exact position of the satellites in their orbits is known, just like the position of Mount Einstein and Mount Newton were known in the map example above. You can triangulate your position from the satellites by simply drawing a big circle around each satellite as big as the separation you figured out from the timing — you are standing where those big circles cross. GPS allows you to exactly pinpoint your location on the surface of the Earth!

GPS satellites broadcast their own time signals which your phone receives on the ground. Above, the "310" time signal from the red satellite is reaching you at the same time as the "309" signal from the blue satellite. This tells your phone is is closer to the red satellite than the blue satellite. The position of the satellites is known, so your phone uses this information to compute the distance to each of the satellites, and triangulates its position.

GPS satellites broadcast their own time signals which your phone receives on the ground. Above, the “310” time signal from the red satellite is reaching you at the same time as the “309” signal from the blue satellite. This tells your phone it is closer to the red satellite than the blue satellite. The position of the satellites is known, so your phone uses this information to compute the distance to each of the satellites, and triangulates its position.

So what does this have to do with general relativity? One of the predictions of general relativity is that massive objects (like the Earth) warp space and time. The warpage of time means that clocks down here on the surface of the Earth (deep down in the gravitational well), tick slower than clocks carried on satellites high above the Earth.

General relativity tells us time moves more slowly deep down in the gravitational well. If you are going to navigate using clock signals from satellites (GPS) you have to account for this!

General relativity tells us time moves more slowly deep down in the gravitational well. If you are going to navigate using clock signals from satellites (GPS) you have to account for this!

Being appropriately skeptical, you should immediately ask “Okay, how much slower?” and once you hear the answer ask “Does that make a difference?” The military commanders in charge of developing GPS in the 1970s famously asked exactly these questions, uncertain that we had to go to all the effort to think about general relativity for navigation by satellite.

The GPS time correction calculation is well understood, and only takes a couple of pages to work out.

The GPS time correction calculation is well understood, and only takes a couple of pages to work out.

The time difference between a clock on the ground and a clock in a GPS satellite due to general relativity warping time is about 1 nanosecond for every two seconds that passes.  What’s a nanosecond? It is one billionth of a second. What kind of error does a nanosecond make? GPS navigation is based on how long it takes radio signals (a form of light) to get from a GPS satellite to you. Light travels about 12 inches in a nanosecond (watch the indefatigable Admiral Grace Hopper explain what a nanosecond is), so for every nanosecond your timing is off, your navigation is off by about 1 foot.  The accumulated error is about 1000 nanoseconds every 30 minutes, amounting to a difference of 1000 feet. This is a substantial difference when you are trying to accurately navigate!

Every satellite in the GPS constellation is constantly in motion, orbiting the Earth once every 12 hours.

Every satellite in the GPS constellation is constantly in motion, orbiting the Earth once every 12 hours.

This is not the only correction that has to be accounted for. The GPS satellites are also moving along their orbits, so there is a speed difference between you and then. One of Einstein’s early discoveries was special relativity which said that moving clocks run slower than clocks that are standing still. So while the warpage of spacetime is making your clock on the ground tick slower than the satellite’s, the satellite’s motion makes its clock tick slower than yours!  These two effects compete against one another, and both must be accounted for. Special relativity means the satellite clock ticks about 0.1 nanoseconds (1 ten-billionth of a second) slower for every second that passes compared to your clock on the ground. On a 30 minute walk then, this produces an error in location of almost 200 feet.

einsteinPocketBoth special and general relativity were discovered in an era where they had little application to everyday life. None-the-less, as the years have worn on clever and industrious scientists and engineers have discovered that they both have important and profound applications. Both special and general relativity have grown into important tools in modern science and technology, with applications in the most unexpected places in our lives. Usually, it is hidden from me and you under the slick veil of marketing and glossy industrial design, but they are there none-the-less.  Just remember this the next time you’re out walking around, using your phone to navigate: there is a whole lot of Einstein in your pocket.

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This post is part of an ongoing series written for the General Relativity Centennial, celebrating 100 years of gravity (1915-2015).  You can find the first post in the series, with links to the successive posts in this series here: http://wp.me/p19G0g-ru.

Gravity 04: Testing the New Gravity

by Shane L. Larson

In the world of artistic painting, connoisseurs have a word: pentimento. It is the revelation of something the artist hid from us.  There are many reasons why changes to a composition may come to light. Sometimes it is because as paint ages, it becomes more translucent, revealing a previous facial expression or position of a hand. Sometimes, close and careful study reveals that a slight alteration was made to disguise a mistake or a shift in ideas about the composition. And still sometimes technology can be used to see through the painting to what lies beneath — the artist’s original sketch or painting that was altered in the final production

nakedCookieJarThe word pentimento is an Italian word, meaning “repentance.” Its use in the context of art is an implication that the artist has been caught red-handed, changing their mind about a particular composition! The idea of repentance and being caught red-handed carries a certain amount of emotional baggage in our culture; I suspect it is ingrained in us at an early age, when our parents catch us doing something we’d rather them not know — like stealing cookies from the cookie jar, or seeing what we look like if we cut our eyelashes off, or getting caught reading Scientific American under the covers with a flashlight (I just made all of those things up — my parents never caught me doing any of those!).

But science is different. Part of the game is about being wrong and getting caught. There is no shame in changing your mind, no repentance for previous incorrect speculations about the nature of the Cosmos. You make up cool ideas, that you present to the world not as art, but as proposed mathematical explanations for how the Cosmos works. Any crazy idea is fair game, with one requirement: you have to also suggest a way for us to do an experiment to test if your crazy idea is right! If it’s right, we go think of new experiments; if it is wrong, then we look at your crazy idea and figure out which crazy bits of it aren’t quite right. We make some changes, turning it into a new crazy idea, and then go conduct another test.

Science is always these two parts — the first part, describing the world, is called “theory”; the second part, testing your ideas, is called “experiment.”

The leading header of the paper where Einstein introduced general relativity, his writeup of the presentation he made to the Prussian Academy of Sciences in November, 1915.

The leading header of the paper where Einstein introduced general relativity, his writeup of the presentation he made to the Prussian Academy of Sciences in November, 1915.

Albert Einstein introduced General Relativity to the world in a presentation to the Prussian Academy of Sciences in November of 1915; his written summary of that presentation may be read (in German) online: Die Feldgleichungen der Gravitation (“The field equations of gravitation”).  If you look at that paper, you will see it looks a lot like your introductory physics textbook looked — lots of mathematical symbols and equations. What does it all mean?  This is the “theory” side of gravity, where we imagine the Laws of Nature that describe gravity — in this case, the idea that gravity can be explained as the curvature of spacetime.

General relativity was a tremendous step forward in our understanding of gravity — it was consistent with special relativity and extended our understanding into physical regimes that Newtonian gravity could not address. But at the same time, especially early on, very little was actually known about GR. It was mathematically difficult to work with (in the lingo of physics, it is “non-linear”), and in 1915 there were no known astrophysical systems that absolutely required general relativity to describe them. Einstein knew it was fine to write down new and possibly crazy ideas about the Universe, but the real task was to decide if there were ways to test those ideas. Were there some observations that could be made and compared against the predictions of general relativity? Observations that confirmed the predictions of general relativity would demonstrate its viability as a description of gravity. There have been many tests of general relativity discovered over the course of the past 100 years, but Einstein himself set the stage for testing his ideas by proposing three immediate tests that scientists could put their efforts into.

The first test was one that Einstein used to convince himself that general relativity was going in the right direction. In 1859 Urbain Le Verrier had noticed something odd about Mercury’s orbit.  Like all planets, Mercury has an elliptical orbit — it is a slightly squashed circle, closer to the Sun on one end than on the other. The point where the orbit comes closest to the sun is called perihelion, and it lies in a particular direction. Over time, the direction to perihelion changes — the orbit of Mercury pivots slowly, in a dance that astronomers call precession.

The point of closest approach to the Sun is called perihelion, and occurs in a certain direction (green dashed lines). Over time, an orbit pivots slowly -- it precesses -- changing the direction to perihelion.

The point of closest approach to the Sun is called perihelion, and occurs in a certain direction (green dashed lines). Over time, an orbit pivots slowly — it precesses — changing the direction to perihelion.

Some precession is expected, because the Sun is not a perfect sphere (it is actually a bit squashed, fatter at the equator because it is spinning), but Le Verrier had looked at 150 years of observations of Mercury’s orbit and discovered the perihelion was shifting by an anomalously large amount — 43 arcseconds every century! That is to say, the angle of Mercury’s orbit was pivoting by an extra angle over the course of 100 years, equivalent to 43 arcseconds. How big is 43 arcseconds? Take a quarter and put it 382 feet away (a bit longer than a standard US football field) — 43 arcseconds is the angle between pointing from one side of the quarter to the other side of the quarter. It is a VERY small angle! But astronomers had detected this small change in Mercury’s behaviour through diligent and careful observations of the Cosmos.

43 arcseconds is about the apparent visual size of a US quarter when viewed from a distance of 382 feet (116.4 meters), slightly farther than the length of a regulation NFL football field.

43 arcseconds is about the apparent visual size of a US quarter when viewed from a distance of 382 feet (116.4 meters), slightly farther than the length of a regulation NFL football field.

When he was developing his new way of thinking about gravity, Einstein realized that the anomalous precession might be described by general relativity. He calculated that general relativity predicts an extra 43 arcseconds in perihelion precession for Mercury, the exact amount observed by astronomers. This not only resolved a 50 year old mystery in astronomy, but firmly convinced Einstein he was on the right track.

The second test is arguably one of the most famous tests in the history of gravity, and its success catapulted Einstein into the public eye, making him a world-wide celebrity. One of the central features of general relativity is that everything experiences gravity — everything “falls.” This is certainly true for things like rocks and slurpees, but Einstein also realized it should apply to light. Light, like all freely moving objects, wants to travel in a straight line, and generally it does so. This is one of the features that makes it such a useful messenger in astronomy: if you receive some light on Earth, and look back along the direction the light came from, you should be looking at the object that generated the light!

[A] When viewed alone in the sky, two stars (yellow and red) have a well defined separation, defined by the direction you have to point to look directly at them. [B] They appear separated, and that amount can be measured. [C] During a total solar eclipse, light from the yellow star passes near the Sun and is bent. Looking back along the line of sight, the yellow star appears to be closer to the red star than it was when the Sun was not in the way. [D] The deflection of starlight is the amount the position of the yellow star appears to move on the sky.

[A] When viewed alone in the sky, two stars (yellow and red) have a well defined separation, defined by the direction you have to point to look directly at them. [B] They appear separated, and that amount can be measured. [C] During a total solar eclipse, light from the yellow star passes near the Sun and is bent. Looking back along the line of sight, the yellow star appears to be closer to the red star than it was when the Sun was not in the way. [D] The deflection of starlight is the amount the position of the yellow star appears to move on the sky.

If on its long journey through the Cosmos a little bit of starlight (called a “photon”) passes near an object with strong gravity, the gravity will bend the path the light travels on. This is exactly what would happen to any massive object. If an asteroid is flying through deep space, it will travel in a straight line. If it strays too close to an object with strong gravity, like Jupiter or the Sun, the gravity deflects that asteroid and it ends up travelling in a different direction.

One of Eddington's images recorded during the 1919 Eclipse. The small horizontal hash lines mark the stars that would be measured.

One of Eddington’s images recorded during the 1919 Eclipse. The small horizontal hash lines mark the stars that would be measured.

So how can you measure the bending of light predicted by general relativity? The trick with light is if you want to see the deflection, it has to pass through a strong gravitational field. Einstein suggested you could look for the deflection of starlight during a total solar eclipse. The gravity of the Sun is strong enough to bend the path of light by a measurable amount; during an eclipse when the Moon blocks most of the light from the Sun, the stars near the edge of the Sun’s disk should be visible.

The first realization of this test was organized in the United Kingdom by the Astronomer Royal, Frank Watson Dyson, and Arthur Stanley Eddington. Eddington led an expedition to the island of Príncipe, off the west coast of Africa, to observe the total solar eclipse on 29 May 1919. Eddington imaged several stars around the eclipse, and confirmed general relativity’s predictions. These measurements are difficult to make, and their accuracy has often been debated, but the experiment has been repeated during many eclipses since then, continuing to confirm the predictions of general relativity.

The leading header of the paper summarizing Eddington's measurements to confirm the deflection of starlight.

The leading header of the paper summarizing Eddington’s measurements to confirm the deflection of starlight.

The last proposed experiment is called the gravitational redshift. Think about tossing a rock up in the air. What happens? When the rock leaves your hand, it has some initial amount of energy that physicists call “kinetic energy” — energy associated with motion. As it climbs, it slows down. It looses kinetic energy, expending it to fight upward against gravity. Einstein argued based on the Equivalence Principle that a photon must also expend energy to climb upward against a gravitational field.

Photons travelling upward in a gravitational field lose energy, becoming redder. Photons travelling down gain energy, becoming bluer.

Photons travelling upward in a gravitational field lose energy, becoming redder. Photons travelling down gain energy, becoming bluer.

But photons — all photons — propagate at the speed of light! The notion of “kinetic energy” as it applies to objects like rocks is hard to extend to photons. But the Equivalence Principle demands that a photon climbing up through a gravitational field must give up energy. How? It can change its color. Photon energy is directly related to its color — blue light is more energetic than green light which is more energetic than red light.  A photon can give up energy as it climbs upward against gravity by changing its color, shifting from bluer light toward redder light.

Measuring the change in color of light is easy to do, but notoriously difficult to attribute to general relativity because all kinds of things change the color of light! But in 1959, Robert Pound and Glen Rebka successfully measured the gravitational redshift at the Jefferson Laboratory at Harvard.

Pound and Rebka’s triumphant measurement concluded a more than 40 year effort to complete the three classical tests proposed by Einstein in 1915. Since those early days, many other tests of general relativity have been suggested, and measured. To date, no experiment has uncovered any chinks or holes in the theory. If there had been, then general relativity would have been relegated to the trash bin of Cool Ideas that Failed, and we would have moved onward to look for a new understanding of gravity. Instead we find ourselves in that happy frame of mind where we use general relativity to describe the Cosmos with swagger and aplomb. There may yet be another revolution in our understanding of gravity, but if there is, I am confident that it will have to successfully include both general relativity and Newtonian gravity as parts of its core infrastructure.

In the end, there is a bit of pentimento in the game of science, but it is not on our part — it is Nature’s. General relativity is the latest in a series of tools that we have developed and used to peer closely at Nature. Slowly — ever so slowly — we are seeing through the paint Nature has clothed herself in. The secrets of the Cosmos are becoming slowly transparent, revealing the clockwork wonder of the Universe that hides beneath.

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This post is part of an ongoing series written for the General Relativity Centennial, celebrating 100 years of gravity (1915-2015).  You can find the first post in the series, with links to the successive posts in this series here: http://wp.me/p19G0g-ru.

Gravity 03: Curvature & the Landscape of the Cosmos

by Shane L. Larson

Albert Einstein is one of the most easily recognized figures in our culture, so much so that he is recognized in imaginary fantasies, like this one of Albert being a master of the electric guitar in my band ("MC Squared and the Relatives"). In reality, his colleague Robert Oppenheimer noted that Einstein was "almost wholly without sophistication and wholly without worldliness ... There was always with him a wonderful purity at once childlike and profoundly stubborn."

Had he been alive when Queen formed in 1970, perhaps Einstein might have jammed with them.

“Is this the real life? Is this just fantasy?”  So opens the classic rock song by Queen, Bohemian Rhapsody. Trying to understand modern gravity often puts one in the frame of mind that the mental machinations we go through are somehow not connected to reality. Albert Einstein’s genius was that he persevered through those uncomfortable feelings. He willfully ignored traditional ways of thinking about the real world, and imagined new and inventive ways to describe how Nature behaves.

One of those inventive ways of thinking was to noodle about unusual situations, like the Equivalence Principle. What a happy little thought — an idle daydream to imagine an elevator on the Earth, or flying in a rocket, and asking what would happen if you did something as simple as drop an apple? Dropping an apple is an act of everyday life, but the conclusion seemed almost a fantasy. Einstein’s thought experiment had discerned that there was no way for a person to distinguish if they are in a rocket controlled elevator, or under the influence of the gravity from a planet. At face value, the conclusions would seem to be this: since I can’t tell the difference between gravity and a rocket, maybe gravity isn’t real at all.

Any normal person might throw up their hands in exasperation and decide this discussion of the Equivalence Principle is nonsense and go work on something simpler (like brain neurochemistry). But Einstein was a persistent fellow, and pushed a little harder.  He asked, “Is this the real life? Is this just fantasy?” Could it really be the case that there is no way to tell if you are in a gravitational field? What can we experience — what can we observe — that convinces us that we are caught in the grip of gravity?

We understand the world through experiments; they are the medium by which we observe. All experiments — thought experiments in particular — are recipes. Experiments produce results (knowledge about the Cosmos) the same way a recipe produces a cookie. We look at that result, and we ask ourselves a few fundamental questions — they are the same for physics experiments as cookies. Why is the result this way? Can I change what went into the result? Will I get a different result if I change what I put in?

Science and baking are both built around experiments that seek to discover what small changes reveal about the thing you are looking at. In cookies, a relatively small change can change a chocolate cookie into an almost identical vanilla cookie (TOP). A more substantial but not completely different approach might make a peanut butter cookie (BOTTOM). But getting a chocolate chip cookie (RIGHT) takes a completely different approach. [Photo by S. Larson; I don't know what happened to these cookies after the #science was done; sorry.]

Science and baking are both built around experiments that seek to discover what small changes reveal about the world. In cookies, a relatively small change can change a chocolate cookie into an almost identical vanilla cookie (TOP). A more substantial but not completely different approach might make a peanut butter cookie (BOTTOM). But getting a chocolate chip cookie (RIGHT) takes a completely different approach. [Photo by S. Larson; I don’t know what happened to these cookies after the #science was done; sorry.]

Science is a game of tearing apart ideas and seeing what makes them tick. Changing the assumptions, the recipe of the experiment, could change the outcome. So once again, Einstein returned to our thought experiment with the elevators, and imagined something new. There is a big assumption hidden in our thought experiment — the rooms you and I were confined to were “small.”

Why should that matter? Let’s imagine that our rooms were larger — much larger — and consider each of them in turn.

First, think about your room, on a spaceship. This is a BIG spaceship, of the sort that only the Galactic Empire has the metal and economic resources to build. The entire bottom of the spaceship is covered by rockets, all of them pushing with the same force to make you go.  Now conduct the apple dropping experiment again, first at one end of the spaceship, and then at the other end, very far away.  Both apple drops show the same thing — the apple falls directly down, parallel to the walls of your spaceship.

No matter how large you make a room, if it is being propelled uniformly by rockets, apples all over the room fall straight to the floor, along paths that are everywhere parallel to one another.

No matter how large you make a room, if it is being propelled uniformly by rockets, apples all over the room fall straight to the floor, along paths that are everywhere parallel to one another.

Now think about my room, on the planet Earth. This is a BIG room, far larger than any room ever built as it is large enough that if my floors are flat, the curvature of the Earth falls away from under my room at both ends (though I am not aware of this — no windows, right?). If I drop my apple at either end of my huge room I make an astonishing discovery — my apple does not fall parallel to the wall! It lands farther from the wall than it started. Given the outcome of my experiment, I could imagine all sorts of plausible explanations.

Perhaps the walls of your gigantic room are repulsive.

Perhaps the walls of my gigantic room are repulsive.

Maybe the walls are repulsive!

That’s an interesting idea; maybe it’s true, maybe it’s not. Can I test it?

Sure!  I build a few new walls at different places in the big room and drop many apples many times. What I find is this: if a wall is closer to the center of the room, a dropped apple falls closer to straight down. At the exact center of the room, two apples fall straight down and land the same distance apart as when they were released. Two apples dropped at opposite ends of the room are closer together when they land on the floor! Physicists get grandiloquent about this and call it “tidal deviation.”

What is going on? The walls clearly aren’t repulsive — a wall in the center of the room doesn’t push apples away from it at all.  We have talked about the lines of force that show the gravitational field.  The gravitational field always points to the center of the source of gravity. What this experiment seems to show is that if my room is big enough, I can detect the shape of the gravitational field!

(TOP) When my apples are dropped, their paths are not parallel; we say there is a tidal deviation between the paths. This is a key experimental signature of gravity. (BOTTOM) We can understand the tidal deviation of the apple paths if we imagine they are following the lines of force in the gravitational field (this is how Newton would have explained it). But this is not the only way to explain gravity!

(TOP) When my apples are dropped, their paths are not parallel; we say there is a tidal deviation between the paths. This is a key experimental signature of gravity. (BOTTOM) We can understand the tidal deviation of the apple paths if we imagine they are following the lines of force in the gravitational field (this is how Newton would have explained it). But this is not the only way to explain gravity!

This idea of the shape of the gravitational field, and its relation to the motion of falling objects, would be a key part of Einstein’s mathematical development of general relativity: it led him to the thought that motion and geometry could be connected.

That may seem like an odd thought, but the fundamental building blocks of elementary geometry are exactly the elements of motion that we discovered in our Giant Room Apple Dropping Experiments: lines can be parallel or not parallel. Einstein recognized that was important, so he explored it. We can too! Let’s think about a flat table top.

If I have two Matchbox cars on my table, and give them a push, they travel in a straight line and never stop (in the absence of friction — every little kid’s dream!).  If I take those two cars and set them in motion  exactly parallel to one another, what happens? The two cars speed off across the table and their paths never cross, no matter how far they go. In many ways, this example is like our two apples on opposite ends of the Gigantic Rocket-Propelled Room — the apples both started falling on straight lines, parallel to each other, and they ended up hitting the floor falling on straight lines that were still parallel to each other.

On a flat surface, two lines that begin parallel stay parallel, no matter how far you extend the lines across the surface.

On a flat surface, two lines that begin parallel stay parallel, no matter how far you extend the lines across the surface.

So this leads to the inevitable question: is there a way in geometry to make the Matchbox cars start out along parallel paths, but ultimately draw closer together? This would be analogous to the Gigantic Room the size of Earth, where apples dropped on opposite ends of the room landed closer together.  As it turns out, the answer to this question is YES.

Imagine a sphere, like a playground ball or a desk globe.  The surface of the globe is two dimensional, just like the table top — there are only two directions you can go: front-back, or left-right. Suppose I take my two cars and set them on the equator in different spots, but both are initially travelling due north — the paths are parallel! What happens? Eventually, the paths of the two cars get closer together, and if we wait long enough, they cross.

If two travellers start at the equator travelling due north, their paths are initially parallel. By the time they reach the top of the globe, their paths cross each other --- the paths don't remain parallel because of the curvature of the globe!

If two travellers start at the equator travelling due north, their paths are initially parallel. By the time they reach the top of the globe, their paths cross each other — the paths don’t remain parallel because of the curvature of the globe!

Now, that is no way for parallel lines to behave on a piece of flat two-dimensional paper, but it is perfectly acceptable on a two-dimensional curved surface. THIS is the watershed idea of general relativity — maybe we can describe gravity as curvature. Maybe we can replace the concept of a gravitational force with the idea of particles moving on a curved surface — on flat surfaces, motion along parallel paths stay parallel, but on curved surfaces initially parallel pathways can converge and cross.

Neat idea. But curvature of what?!

Different ways we have devised to measure space or time.

Different ways we have devised to measure space or time.

Einstein brilliantly deduced that since our concern is with the motion of things, it should be curvature of the quantities that we use to describe motion — space and time. Special relativity, which motivated our reconsideration of gravity, was wholly focused on how we measure space and time, and Einstein’s former professor, Hermann Minkowski, had discovered that individually space and time are artificial elements of a single medium — spacetime.  Spacetime is the fabric of the Cosmos, the medium on which all things move. Einstein had become well versed in this notion, and concluded:

Gravity is the curvature of spacetime.

This is the heart of general relativity. So how does it work? General relativity is summarized mathematically by 10 coupled, non-linear, partial differential equations known as the Einstein Field Equations, succinctly written as

efes

Fortunately for us, this mathematics can be captured in a simple, two-line mantra to guide intuition:

Space tells matter how to move.

Matter tells space how to curve.

In geometric gravity — general relativity — you can imagine spacetime like a large, deformable sheet. A particle can move anywhere on that sheet, so long as it stays on the sheet.  In places where the sheet is flat (“flat space”) the particle moves in an absolutely straight line.

But what happens if a particle encounters a large depression on the sheet? The only rule is the particle has to stay in contact with the sheet. It continues to travel in the straightest line it can, but if its path dips down into the depression, the direction the particle is travelling is slightly altered, such that when it emerges on the far side, it is travelling in a new direction that is not parallel to its original course!  Space tells matter how to move, with its shape.

Far from sources of gravity (edges of the sheet) spacetime is flat, and objects travel on straight lines.  Small masses warp spacetime into a gravitational well (left dimple), while larger masses make larger gravitational wells (right dimple). If a particle comes close to a gravitational well, the curvature of spacetime bends its pathway. If a particle gets trapped in a gravitational well, the curvature of spacetime forces it to travel on a closed pathway -- an orbit.

Far from sources of gravity (edges of the sheet) spacetime is flat, and objects travel on straight lines. Small masses warp spacetime into a gravitational well (left dimple), while larger masses make larger gravitational wells (right dimple). If a particle comes close to a gravitational well, the curvature of spacetime bends its pathway. If a particle gets trapped in a gravitational well, the curvature of spacetime forces it to travel on a closed pathway — an orbit.

How do you curve spacetime?  With matter. The large, deformable sheet of spacetime is dimpled wherever there is a large concentration of mass; the larger the mass, the larger the dimple. Matter tells space how to curve, with its mass.  The larger the dimple, the larger deflection a particle passing nearby will feel. This at last, is the long awaited connection to the way we think about Newtonian gravity — the source of gravity is always matter, as we expected.

So we have done away with the concept of a “gravitational force field” and replaced it with the idea of “motion on a curved spacetime.” An astute reader will ask a pertinent question: if general relativity is really the way gravity works, why didn’t we discover it first? Where did Newtonian gravity come from?

Both Newtonian gravity and general relativity make exactly the same predictions when gravity is weak and speeds are slow.  In fact, mathematically, general relativity looks just like Newtonian gravity at slow speeds and in weak gravity. These are precisely the conditions we encounter in the solar system, which is why Newtonian gravity was discovered first, instead of general relativity!

You may have encountered models of spacetime gravitational wells out in the world. This one, in the Milwaukee Airport, captures coins for a museum. [Photo by K. Breivik]

You may have encountered models of spacetime gravitational wells out in the world. This one, in the Milwaukee Airport, captures coins for a museum. [Photo by K. Breivik]

It’s fine, of course, to make up a new idea about gravity. But this is science — fancy theories are only as good as the tests that can be conducted to verify them.  Einstein knew this, and proposed a series of tests for general relativity, which we’ll talk about next time.

PS: I rather enjoyed using Bohemian Rhapsody to start this bit of the General Relativity story. It was written by the great Freddie Mercury for Queen’s 1975 album, “A Night at the Opera.” As we shall see in our next installment, Mercury (the planet) plays an important role in bringing the importance of General Relativity to the attention of the scientific community, just like Mercury (Freddie) helped me explain it here. 🙂

PPS: This week I couldn’t capture all of this in one 3 minute video, so I tried to do it in two.

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This post is part of an ongoing series written for the General Relativity Centennial, celebrating 100 years of gravity (1915-2015).  You can find the first post in the series, with links to the successive posts in this series here: http://wp.me/p19G0g-ru.

Gravity 2: The Road to General Relativity

by Shane L. Larson

Science is, to some extent, a skill set that can be learned. Like playing piano or solving Rubik’s cubes or cooking Belgian cuisine. Using scientific thinking and applying it to the world is in large part a matter of practice and relentless dedication to getting better. But like all artforms, there is a small element of je ne sais quoi to it as well — a hidden reservoir of intuition and stupendous insight that is unleashed only sometimes.

einsteinAppleApple once had an ad campaign built around the mantra, “Think Different” (grammarians, hold your tongues, and your “ly”s and follow the mantra!). There were images of famous thinkers through the ages who approached the world differently than the rest of us. One of those was Albert Einstein.

Among a community of bright and creative people, it gives me pause to consider those people that we all think of as being remarkable. Albert Einstein is arguably the most famous scientist in history of the world, commanding the respect not just of the general populous, who have grown up immersed in his legend, but also the respect of the scientific community. Why is that? My colleague Rai Weiss, now an emeritus professor at MIT, recently noted that it wasn’t just that Einstein was smart, it was that he exhibited tremendous intuition. His great ability was to look at the same world the rest of us look at every day, and think different.

When Einstein began his quest to refine our understanding of gravity, he knew he was going to have to “think different” — this was, after all, what had led to special relativity in the first place! One of the earliest musings on the road to general relativity was a simple question: how do you know if gravity is pulling on you?

Everything I need to start developing some new ideas about gravity!

Everything I need to start developing some new ideas about gravity!

It’s a seemingly simple question, but it led to an interesting thought experiment. Imagine you and I are each in a small, windowless room with nothing but an apple and our smartphones (so we can text each other the results of the experiment I am about to describe).

Each of us drops our apple, and we see that it accelerates downward — it falls!  The apple starts from rest (at our hands) and speeds up as it falls toward the floor of our small room.  We excitedly text the result to each other and tweet pictures of apples on floors. Should we conclude from these experiments that we are both conducting experiments under the influence of a gravitational field?

[Top] You and I conduct identical experiments (dropping apples) in enclosed spaces and get identical results. [BOTTOM] The reality of what is outside our little rooms may be completely different! A falling apple is equally well explained by the gravity of a planet, or by an accelerating rocket!

[Top] You and I conduct identical experiments (dropping apples) in enclosed spaces and get identical results. [BOTTOM] The reality of what is outside our little rooms may be completely different! A falling apple is equally well explained by the gravity of a planet, or by an accelerating rocket!

Einstein realized the answer to that question should be “No!” There are multiple ways to explain what we saw.  One way is to assume our little rooms are sitting on the surface of planet Earth, where the planet’s gravity pulled the apple down. But another, equally valid way to explain this experiment is to assume the little rooms are really the space capsules of rocket ships, accelerating through empty space (the apple is pressed down to the floor — “falls” — the same way you are pressed back in your seat when a jetliner takes off).  What Einstein realized is that there is no way, based on our experimental results, to tell the difference between these two cases. As far as experiment is concerned, there is no fundamental difference — that is to say, no observational difference — between them. Einstein knew that the laws of physics had to capture this somehow.

[TOP] You and I both find ourselves weightless, floating without feeling forces acting on us. [BOTTOM] The reality external to our rooms could be that we are floating in space, or that we are in a freely falling elevator plummeting to our doom.

[TOP] You and I both find ourselves weightless, floating without feeling forces acting on us. [BOTTOM] The reality external to our rooms could be that we are floating in space, or that we are in a freely falling elevator plummeting to our doom.

What if we consider a slightly different case? Imagine you and I both suddenly found ourselves and our apples drifting weightlessly in the middle of our small rooms. We excitedly text each other that we finally made it to space and tweet messages that we are officially astronauts. Should we conclude that we are both deep in interplanetary space, far from the gravitational influence of a planet? Once again, Einstein realized the answer to that question should be “No!” There is no way to know if we are drifting inside a space capsule in deep space, or if we are merely inhabiting an elevator whose cable has snapped and we are plummeting downward toward our doom!

tweetEquivalence_smallIt is this freefall experiment that really illustrates how we have to learn to “think different” when expanding our understanding of Nature. In Newtonian gravity, we always look at problems with an exterior, omniscient eye toward the problem. A Newtonian approach to the free fall problem says “Of course you are falling under the influence of gravity! I can see the Earth pulling you down from the top of the skyscraper toward your doom at the bottom of the elevator shaft!” But Einstein asked a different question: What does the person in the elevator know? What experiments can they do to detect they are in a gravitational field? The answer is “none.”  There is no observational difference between these two situations, and the laws of physics should capture that.

The critical point here is that if you are in free fall, you feel no force! Einstein’s great insight was that the central difficulty with gravitational theory up to that point was that it was anchored in thinking about forces. This thought experiment convinced him that the right thing to think about was not force, but the motion of things.

This thought experiment came to Einstein in 1907 on a languid afternoon in the Bern patent office. Later in his life, Einstein would recall that moment and this idea with great fondness, referring to it as the happiest thought of his life.  This experiment is known as the “universality of free fall,” which physicists like to give the moniker “the Equivalence Principle.”

This is how I often imagine Einstein’s life during the years he worked in the Bern patent office! [From "The Far Side" by Gary Larson]

This is how I often imagine Einstein’s life during the years he worked in the Bern patent office! [From “The Far Side” by Gary Larson]

I have a very strong memory of my father first telling me about the universality of free-fall in about fifth or sixth grade. When you’re not used to it, the notion that falling in an elevator is the same as floating in outer space engenders a spontaneous and vehement response: “That can’t be true!” We had many long debates about this (it was following hot on the heels of my meltdown over the existence of negative numbers — maybe my dad was trying to forestall another meltdown…), and I don’t think it ever quite sank in. I, of course, feigned understanding and dutifully repeated the tale of the falling elevator to my classmates, reveling in their confusion and indignant denial of the logic of it.  I was a tween — what did you expect?

The "Leaning Tower of Niles," a half-scale replica of the tower in PIsa. Located in Niles, IL (a suburb of Chicago).

The “Leaning Tower of Niles,” a half-scale replica of the tower in PIsa. Located in Niles, IL (a suburb of Chicago).

But now, many years later and with a LOT of physics under my belt, I know that that the outcome of these thought experiments derive from a very old result that we are all familiar with — that all objects fall identically, irrespective of their mass. Galileo taught that, at least in folklore, by dropping various masses off of the Leaning Tower of Pisa. The obvious question to ask is “how is Galileo’s experiment connected to Einstein’s thought experiments?”

For the moment, imagine the various parts of your body as having different masses.  Your head masses about 5 kg (a little bit more than the 8 pounds you learned from watching Jerry Maguire), where as a good pair of running shoes may mass about 1.5 kg.  If you are standing happily in the elevator when the cable snaps, the traditional explanation is that everything begins to fall.

If objects of different mass fell at different rates, then your head would be pulled down faster than your shoes — you would feel a force between your head and your shoes. That force could be used to deduce the existence of a pulling force.  But Galileo taught us that is not the way gravity works — your head and your shoes will get pulled down at strictly the same rate in a uniform gravitational field. Every little piece of you, from your head to your toes, your kneecaps to your freckles, falls at the same rate — there are no different forces between the different parts of your body and so you feel (you observe) yourself to be weightless. This is sometimes called the Galilean Principle of Equivalence, or the Weak Equivalence Principle.

Okay — so what? Apples and freefall, elevators and rockets. What does any of this have to do with developing a deeper understanding of gravity?

What this thought experiment reveals, what the Equivalence Principle tells us, is that thinking about forces is not the best way to think about the world because we can’t always be sure of what is going on! Instead, we should think about what we can observe — how particles move — and ways to describe that. That simple intuitive leap would, in the end, change the face of gravity. Particles move through space and time, which had brilliantly been unified by Einstein’s teacher and colleague, Hermann Minkowski, into a single unified medium called “spacetime.”

What is spacetime? It is the fabric of the Cosmos —  it can be stretched and deformed. The fundamental idea of general relativity is that gravity can be described not by a force, but by the curvature of spacetime, the medium on which particles move.  That will be the subject of our next little chat.

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This post is part of an ongoing series written for the General Relativity Centennial, celebrating 100 years of gravity (1915-2015).  You can find the first post in the series, with links to the successive posts in this series here: http://wp.me/p19G0g-ru

Gravity 1: Seeing the Invisible

by Shane L. Larson

General Relativity is only the latest refinement of our ever growing understanding of gravity. Most of us become aware of gravity at a very young age. It is a playmate when we throw balls, an accomplice when we knock our unwanted food off the table, and our Nemesis as we learn to stand up and walk. All things being equal, gravity is a source of much mayhem when we are children, but hidden in the chaos we learn a few things, and we learn them deeply. When we drop things, they fall. When we jump in the air, we always come back down. It is such a pervasive part of our lives, that we seldom give it a second thought. Once you start school, you learn that gravity is a thing, and that thing keeps you on the floor, makes rain fall from the sky, and makes planets go around the Sun.  Gravity is something you learn about in science class. But why is it a part of science class, when you learned about it as a toddler?

(TOP) Most of us develop an intimate relationship with gravity at a very young age. (BOTTOM) Whether we know it or not, gravity impacts all aspects of our every day life in subtle ways.

(TOP) Most of us develop an intimate relationship with gravity at a very young age. (BOTTOM) Whether we know it or not, gravity impacts all aspects of our every day life in subtle ways.

The job of a scientist (and a toddler) is to look at the world around us, take note of those “obvious” things that we don’t even give a second thought to, and ask “why is the world that way?” The answers to that question enable us to harness Nature by predicting the future. If I understand gravity, I can figure out how strong a building needs to be without tipping over (like the Leaning Tower of Pisa), I can figure out how much pressure a water tower will provide for a city, or I can figure out how big to make an airplane wing so it can fly.  There is other physics to be sure in all of these, but gravity is at the heart of it all, just as we learned as children.

The authors of our fundamental thinking about gravity. (L) Isaac Newton, who developed the Universal Law of Gravitation and (R) Albert Einstein, who developed General Relativity.

The authors of our fundamental thinking about gravity. (L) Isaac Newton, who developed the Universal Law of Gravitation and (R) Albert Einstein, who developed General Relativity.

So how do we think about gravity? After all, it is not like an orange or a Lego brick — it’s not something tangible that you can pick up. In fact, if anything, it is totally invisible! The discovery of the invisible and how to talk about it is still one of the greatest feats of the human imagination. The first person to do this for gravity, was Isaac Newton. Fundamentally, Newton put us on the path to describing gravity using mathematics, the language of science. He first wrote down the Universal Law of Gravitation in his 1687  book, the Principia, along with all the math you needed to work with the Universal Law (read: calculus). Einstein refined and extended our understanding of gravity by writing down general relativity using a new mathematical approach: curvature and tensor calculus.

gravityEquations

But learning how gravity works in the Cosmos from mathematics can take years of practice and patient study. Fortunately, we can develop some intuition about how gravity works by learning to draw some simple pictures.

Physicists describe the long range effect of gravity using the concept of a “force field,” or simply a “field.” As is often the case with spoken language, scientists adopt common words to mean very specific things that don’t always jive with what the rest of us think the word means. What do you think of when I say “field?” If you’re a country kid like me, you may imagine a vast expanse of rolling hillsides in eastern Oregon, stalks of wheat heavy with ripening grain rippling as the wind blows across the “field.” Others of you may imagine a late July afternoon, the hot sun shining down on the bleachers in Wrigley Field as the Cubs once again try to chalk up a win in the run to the pennant. Neither of these examples is what a scientist means by “field,” but they both have an important element to the scientific definition — big, open spaces.

Some other kinds of fields.

Some other kinds of fields.

An important part of understanding gravity is recognizing that no matter where you are in space, if there is a source of gravity somewhere (say a planet, or a star), then you feel the tug of that gravity. Gravity fills all of space. That simple fact leads to the concept of a “field” in physics. We are going to draw pictures of fields, but there is a robust and well understood mathematical treatment of fields that will give you the same intuition (and more) as our simple pictorial model.

So how do we draw a picture of the “gravitational field?” The rules are:

  • Draw arrows to represent the “gravitational force.” Those arrows fill all of space, and point toward the source of gravity (the direction that gravity is trying to pull you); they are usually called “lines of force.”
  • Big, massive objects have more gravity than small objects, so they have more arrows pointing toward them — they exert more gravitational force on their surroundings.
  • The gravity any object experiences is understood by how closely packed the field lines are in the vicinity. Lots of field lines near you equates with stronger gravity in your vicinity.

Demonstration of the drawing of gravitational "fields." [TOP] The field lines (lines of force) point toward the mass creating the gravity. The number of field lines depends on the mass of the object; more mass, more field lines. [BOTTOM] What you feel in terms of gravitational forces depends on how many field lines are around you. The gravitational force felt far away from the source of gravity is weaker, evidenced in our picture view by fewer field lines.

Demonstration of the drawing of gravitational “fields.” [TOP] The field lines (lines of force) point toward the mass creating the gravity. The number of field lines depends on the mass of the object; more mass, more field lines. [BOTTOM] What you feel in terms of gravitational forces depends on how many field lines are around you. The gravitational force felt far away from the source of gravity is weaker, evidenced in our picture view by fewer field lines.

So how do we use this pictorial approach to gravity in practice? Let’s imagine a trip to one of the most picturesque destinations in our solar system: Jupiter. Spacecraft from Earth have visited Jupiter nine times so far.  They have returned stunning pictures and made astonishing discoveries about Jupiter and it’s ragtag group of moons.

The planet itself is enormous, comprised mostly of gas surrounding a small rocky core. Deep beneath the clouds the pressure and temperature soar, making Jupiter glow in the infrared, cloaked in the light of its own inner heat.

On the top of the clouds, an enormous cyclonic storm has roiled and churned for at least the last 400 years, sometimes growing to three times the size of the Earth. We call it “The Great Red Spot.”

Among Jupiter’s entourage of moons is a wild and unpredictable world with volcanoes that spew molten sulfur 500 kilometers into space. This is the most volcanically active world we know, called Io.

In 1992, Comet Shoemaker-Levy 9 strayed too close to Jupiter and was torn apart into 22 fragments. In 1994, as we watched from the relative safety of Earth, each of those 22 chunks of rock and ice pummeled into Jupiter one after another.  Any one of them could have leveled our cuvilization; they burned and scarred the clouds of Jupiter, but over time even those marks faded into memory and Jupiter kept on about its business as if nothing had happened.

Wonders of the Jupiter system, all ultimately connected to Jupiter's strong gravity. (L to R) Jupiter glows in the infrared; the 400+ year old storm known as the Great Red Spot; the volcanic moon Io; the scars left by the impact of Comet Shoemaker-Levy 9, crushed by Jupiter's gravity.

Wonders of the Jupiter system, all ultimately connected to Jupiter’s strong gravity. (L to R) Jupiter glows in the infrared; the 400+ year old storm known as the Great Red Spot; the volcanic moon Io; the scars left by the impact of Comet Shoemaker-Levy 9, crushed by Jupiter’s gravity.

Each of these wonders, each of these  strange and wonderful things we have discovered at Jupiter, are a consequence of Jupiter’s enormous gravity.

Let’s draw the picture of Jupiter’s gravitational field. The number of field lines is related to the mass of the planet. Suppose we drew 10 lines to represent the gravity of the Earth. Jupiter is 318 times more massive than Earth, so we should draw 3180 lines to represent the gravitational field of Jupiter!  That’s too many to easily see, so let’s just think about Jupiter’s own gravity, and decide it can be represented by 8 lines.

The gravitational field fills all of space, so no matter where you are, you feel the tug of Jupiter pulling on you, from wherever you are, directly toward Jupiter. Far from the planet, the lines are more widely spaced, so gravity is weaker than it is down close to the planet where the lines are closer together.

Rodin's "The Thinker" is probably engaged in a gedanken experiment.

Rodin’s “The Thinker” is probably engaged in a gedanken experiment.

Now, let’s conduct a gedanken experiment — a thought experiment — together. This is a time honored method in theoretical physics to try and understand how the world works. The basic idea is this:

(0) Suppose you have some aspect of Nature you are trying to understand; in this case, the “field description” of gravity.

(1) Imagine a situation to which the law of physics should apply. This could be a situation that could legitimately be addressed in the laboratory with an experiment, given enough time and money, or it could be a physical situation that we can’t recreate but might encounter in Nature. This second case is the one we would like to consider, as it involves the gravitational field of an entire planet.

(2) Apply the law of physics to your situation, and examine all the possible outcomes that would result if you could actually do the experiment for real.

(3) Lastly, you examine the consequences of your gedanken experiment by asking legitimate questions and answering them. Do the predicted outcomes make sense? Do any of the outcomes violate the laws of physics? Are there observational consequences that we might be able to see that would confirm our gedanken experiment?

For our thought experiment, let’s imagine we had the ability to simply squeeze Jupiter and make it smaller. We don’t want to take any mass away, or add any mass to it, we simply want to squeeze it into a smaller, denser ball of stuff, and ask what happens to the gravitational field.

If we follow our rules for drawing fields: (1) The number of field lines won’t change, because the mass of Jupiter doesn’t change. (2) The field lines fill all of space. When we squeeze Jupiter smaller, the field lines in the picture already fill space far away from Jupiter, so we just need to extend down toward the new, smaller Jupiter.

jupiterFields

(L) Jupiter’s gravitational field is stronger near the surface, and weaker far away. (R) If I shrink Jupiter without changing its mass, the field stays the same far away, but it gets stronger at the surface!

Now we examine the consequences of our experiment. Far away from Jupiter, nothing has changed. The same number and spacing of field lines are present with the big or the small Jupiter. If you’re an astronaut, drifting aimlessly in orbit around Jupiter, nothing noticeable happens.  But in close, things are a bit different. In the case of the big Jupiter, if we hovered over the clouds we felt some pretty strong gravity. If we compare that to the case of hovering over the clouds of the new small Jupiter, we feel even stronger gravity! How do we know this? Because near the new small Jupiter, the field lines are closer together.

So what do we conclude from this? The “surface gravity” of an astronomical body depends on the compactness (or, more properly, the density) of the planet/star/thing in question. Far away from the astronomical body, the gravitational field depends only on the total mass of the object.

Can we observe these effects for real, somewhere in the Cosmos? Yes!

White dwarfs are the size of the Earth, but the mass of the Sun. The result is a huge gravity at the surface!

White dwarfs are the size of the Earth, but the mass of the Sun. The result is a huge gravity at the surface!

When a star like the Sun reaches the end of its life, it does not explode. Instead, it shrinks to a husk of its former self, a shriveled skeleton known as a white dwarf.  White dwarfs are about the size of Earth, but have the same mass as the Sun. We observe atoms moving in their atmospheres, just over the surface and find that the surface gravity is a staggering 10,000 times greater than the surface gravity of the Sun. By a similar token, many white dwarfs orbit companion stars, and some have been observed to have planets (perhaps long ago, and we just didn’t realize it), all of which are far from the white dwarf but careen happily along their orbits as if they were orbiting an ordinary, Sun-like star.

These observations agree handily with our gedanken experiment.  We used our pictorial model to deduce that if you squeeze an object smaller without changing its mass, the surface gravity changes, but the gravity far away does not!

The Cosmos, and our brains, have not let us down. We’ll put these ideas to the test again, as we delve into the development of General Relativity and encounter even stranger and denser objects — black holes.

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This post is part of an ongoing series written for the General Relativity Centennial, celebrating 100 years of gravity (1915-2015).  You can find the first post in the series, with links to the successive posts in this series here: http://wp.me/p19G0g-ru