# Space Itself Is Expanding: Gravity and General Relativity Explained

The theory of General Relativity — the prevailing theory of Gravity and Electromagnetism — is one hundred years old this month. I’ve wanted to understand where gravity comes from, how Einstein’s theory of General Relativity describes it, and why we never hear a simple explanation. I wound up discovering how reluctant we all have been — even Einstein himself — to accept his theory. It’s disruptive and disturbing. It turns out earth’s Gravity is a “fictitious” force — not a pull downward, but a push upward by the locally ascending, expanding surface of the earth. It’s easy to confirm this with a smart phone. It’s been called positive spacetime curvature for a hundred years now, and it’s actually much simpler, more beautiful, and more accurate than Newton’s theory of gravity. Rulers, rigid objects, and space itself expand over time in the presence of matter. But we can’t fathom it because we can’t accept that our planet, our bodies, and everything around us could actually be stretching.

#### UNDERSTANDING GRAVITY

In 1915, Albert Einstein’s General Relativity famously recast and improved Isaac Newton’s ideas by describing gravity in terms of Curvature of SpaceTime. But if you ask today, most people have trouble saying what that means. A physicist may outline a nonlinear 10 dimensional tensor field equation that’s hard to visualize. Prof Brian Greene puts a bowling ball on a rubber sheet to explain spacetime curvature, but it’s more a metaphor than a full explanation. And when he uses terms like “geodesic” rather than “straight line” to describe motion, he obscures a key point: when we throw a ball above the earth it travels in a straight line — it only looks like it’s curving downward to us because we, and the ground underneath us, are accelerating upward.

In fact, most of us barely understand spacetime curvature — but we’ll see, that’s not because it’s complicated. For gravity, the core idea is very simple. The problem is, when it’s spelled out simply, it sounds so crazy and unfamiliar, it’s hard to accept or believe.

The two big barriers to understanding gravity are not mathematical, but psychological.

First, we must accept the ready experimental proof that earth’s Gravity is exactly the opposite of what we usually think — that it’s NOT a pull downward, toward the surface of the earth — it’s a push upward, by the surface of the earth, against objects in contact with it, and toward objects nearby. This isn’t what Newton thought, and it’s not what most people intuitively believe, but it’s easy to prove using any modern smart phone or tablet with an accelerometer — as we’ll see in a figure below.

Second, we must accept that according to relativity, the rulers and rigid objects around us aren’t fixed in size. With relativity, metric tensors and non-Euclidean geometry describe how rulers can expand, so earth’s volume can increase and push outward as described above, without the earth’s radius in meters getting any bigger. Since rulers and rigid matter expand identically, rulers won’t directly show the length expansion — only the forces exerted by the expanding objects reveal how fast the expansion is happening, and the variations in the expansion rate over space and time. We call those forces Gravity.

So, understanding relativity and gravity is mainly a matter of overcoming our resistance to a peculiar, embarrassingly implausible idea: that while you read this, the space that contains our planet — including every molecule in our bodies, everything we can see out to the horizon, and every ruler on its surface — will double in size.

#### MEASURE GRAVITY

One of many ways to confirm gravity is a fictitious force is to launch Apple’s sensor test app for developers, Accelerometer Graph, on any iPhone or iPad; hold the device vertically; toss it up vertically in the air; and catch it. While predictably the device appears to slow down, reach a maximum height, reverse direction, and fall faster and faster downward toward the floor, the accelerometer reading will be zero in every direction throughout its flight — showing that in fact the device is moving at a constant velocity in a straight line — until it lands or you catch it.

But hold the device in your hand, or set in down on a table or the ground, and you’ll measure a constant upward acceleration of 1 g. That’s why the device appears to change direction and fall — we and the surface of the earth are accelerating upward, catching up with it and outpacing it.

The green +Y axis is defined to go vertically down the page, so the negative value for Y indicates acceleration upward. Gravity is a push upward by the earth, not a downward pull on other objects.

Once we understand it’s the surface of the earth accelerating, not the “falling” objects, our gravity theory becomes vastly simpler. Newton’s idea that the force of gravity on an object is proportional to its mass (so first multiply by M) but its acceleration is inversely proportional to its mass (now divide by M) seems pointlessly complicated. It masks the fact that the “falling” object isn’t really what’s accelerating — that its apparent acceleration is independent of its mass. By Occam’s razor, Einstein’s is a vastly better, simpler theory of gravity — before even addressing its utility in describing electromagnetism or time dilation. Expressed as local expansion, relativity also has better answers than Newton for how light is effected by gravity, and whether it operates instantaneously at a distance (it doesn’t).

#### HOW EXPANDING VOLUME PRODUCES GRAVITY

Imagine, in the absence of gravity, we could gradually inject and flood our 1 trillion cubic kilometer planet earth with an extra 5 million cubic kilometers of liquid in the course of a single second. The tides would rise, accelerating upward. A sailor far from shore would be pushed upward by the tide. The extra 5M cubic km would be pushed through the planet’s 510M square km surface area, resulting in an upward acceleration of 9.8 meters per second per second.**

If the sailor released an object as the tide rose this way, the object would appear to fall relative to the boat. He would feel a force against his feet and in his back, indistinguishable from his weight on our earth. And if he didn’t know the tides were rising, he would swear he was being pulled downward — that he was experiencing earth-like gravity.

Gravitation by relativistic positive spacetime curvature is like that, except instead of adding liquid, matter causes the volume of the space that contains the earth to increase at an accelerating rate proportional to its mass. The formula could hardly be simpler: the rate of expansion is 4πGM, where M is earth’s mass and G is Newton’s gravitational constant:

G = 6.67 x 10**-11 meters^3 / sec / sec / kg

Note that Newton’s G is already in units of accelerating expansion of volume in proportion to mass: cubic meters per second per second, per kilogram. The continuous expansion of space and rulers in the vicinity of matter easily explains Newton’s gravitational Inverse Square Law, as shown in a figure below. The derivation of a simple gravity field equation like Newton’s or Gauss’s* from Einstein’s tensor field equation can be complicated, but we needn’t track the details here, beyond knowing that positive space-time curvature means space, and rulers, expand over time.

We used a 5 million cubic kilometer/sec/sec expansion rate in our liquid example above because in fact that’s 4πGMe for planet earth, resulting in the same apparent surface gravity.**

#### EINSTEIN’S ELEVATOR EPIPHANY, AND GREATEST BLUNDER

During a 1907 elevator ride, Einstein had what he would call his all-time “happiest thought”: he realized that if we were in an opaque box and could not see the earth’s surface, we couldn’t tell whether we were in earth’s supposed gravitational field, or in outer space being pulled in an elevator or rocket accelerating upward — that gravity might be a ‘fictitious force’. Einstein pondered his weight change in the elevator, and his weightlessness if the cable was cut or he leapt from the building, and concluded that standing on earth’s surface is equivalent to being pulled upward in an elevator, at the beginning of a ride up. Beginning with these insights, and using Riemann’s and Minkowski’s non-Euclidean mathematics of spacetime curvature, he and his team had developed the General Relativity theory by 1915. Improbably, a rigid ruler can measure different lengths in a different place or at a different time — an extension of Einstein’s prior Special Relativity theory.

Fast forward: many years later Einstein had to admit that, for well over a decade, he had disastrously failed to accept the continuous expansion of space predicted by his own 1915 curvature theory! He called this his “greatest blunder”: if only he had accepted the fact that Space is truly expanding, he wouldn’t have erroneously added a Cosmological Constant to the theory in 1917, later retracting it; and he would have correctly predicted the red-shifted atomic spectra from distant galaxies that Edwin Hubble’s telescope saw in 1930, instead of being surprised by them.

#### WE OFTEN MISUNDERSTAND EXPANDING “SPACE”

When rulers and rigid objects continuously lengthen, space is expanding. But that phrase is at the core of general confusion about relativity that has persisted for a century. In relativity, the phrase means very nearly the opposite of what we usually, casually think it means.

When ordinary people talk about “space”, we naturally think of the empty space between solid objects; of outer space; of galaxies with stars, planets, and lots of empty space between them. In this context, “space is expanding” seems to suggest objects are moving apart, as more space separates them.

But to a physicist, space is the measurement framework we use to compare lengths, areas and volumes. It’s about rulers. Space includes space inside of planets and dense objects, as well as relatively empty space between them. Our casual, intuitive use of “space” isn’t compatible with ideas like: the space inside of a planet like earth expands much faster than the space outside it. But that’s exactly how general relativity and positive spacetime curvature work. Physicists often refer to the fabric of spacetime to be clearer.

We’re terrible at visualizing rulers that vary in length — or that continuously expand. In fact, it’s much worse than that: when we hear “space is expanding” we picture objects moving further apart, explosions, and big bangs — practically the opposite of gravitational attraction. But when we think it through, we realize that when space and rulers are expanding, the distances between things — as measured by those rulers — are shrinking. The expansion of space means objects moving inertially in straight lines appear to be attracting and orbiting one another, not exploding apart.

So this relativistic idea of “space expanding” is not about growing distances between galaxies. It’s much more disruptive and even disturbing. It’s about the sizes of the atoms, rulers and objects all around us that we consider fixed in size — including our bodies and our planet — actually continuously growing. And one result is everyday gravity.

If we trust our accelerometers, it’s easy to measure how fast the earth is expanding geometrically. How long does it take to travel one earth radius when moving at 1 g? If we’re accelerating at g = 9.8 meters/sec/sec and earth’s radius Re = 6371 kilometers, it will take sqrt(Re/g) — about 805 seconds. So the earth’s radius and the value of g imply that locally, earth doubles in size every 13 minutes and 25 seconds, or about 107 times each day.

Without directly challenging ideas like the big bang here, we should consider how this proven theory of local space expansion fits with other ideas of whether and how “the universe” might be expanding. Before we adopt a theory about violent expansion that may have happened billions of years ago, we should acknowledge that the space our planet inhabits will expand by 2**107 today — a factor of many trillions of trillions — though, viewing it as attraction and orbit rather than expansion, we barely notice. When rulers expand at the same rate we do, we don’t see them doing it.

#### RULERS AND EXPANSION ARE LOCAL

It’s natural to wonder whether, amid continuous geometric expansion of volume, the expanding earth shouldn’t ultimately and inevitably say, crash into the moon. But that question is founded on a premise of fixed, uniform space between them. In fact, space far above the earth or near the moon is expanding much more slowly than near the earth. Though the ladder in the figure below is rigid, its rungs accelerate more slowly near the top than near the bottom, because the space there stretches more slowly. So the geometrically expanding earth never hits the moon. Instead the moon, traveling orthogonal to the expansion in a straight line, appears from the earth’s expanding surface to be in a nearly circular orbit.

#### EARTH GRAVITY AND NEWTON’S INVERSE SQUARE LAW

Completing our earth gravity example: earth’s volume is about 1 trillion cubic kilometers. For earth’s mass Me, 4πGMe comes to 5 million cubic kilometers per second per second, about 1/2000 percent of earth’s total volume each second, resulting in 9.8 meters per second per second or 1 g of outward, upward acceleration of the earth’s surface.**

The figure shows constant acceleration of volume at 4πGMe as a function of spherical radius or distance from the earth — on the surface (at Re) or on a ladder above it at any larger radius r. (Not drawn to scale.) The two purple areas shown in the cross section are equal in volume: 5M km**3. At greater distances from the center, the same volume of space expansion is spread thinner in proportion to the surface area of the sphere, thus decreasing the vertical acceleration by 1/r**2 in accord with Newton’s Inverse Square Law. Since rulers and rigid objects expand identically, earth’s measured radius remains fixed.

#### A SIMPLE EXPLANATION OF GRAVITY

One reason we rarely see gravity discussed this way in lessons on relativity is, educators naturally focus on how spacetime curvature accounts for the differences between results using Newton’s gravity theory as compared to Einstein’s, such as the astronomical cases that proved Einstein’s theory is more accurate. So a relativity class on positively curved spacetime may discuss tiny relativistic time dilation effects, but rarely how curvature and space dilation account for our weight and everyday gravity. A course may point out that a twin living in a basement will be 90 billionths of a second younger than his brother upstairs at the end of their lives, if only they had wristwatches accurate enough to measure it. But often the same lesson won’t mention the brothers’ easily measured weight, and how spacetime curvature causes it — though it’s much more dramatic. After all, 1 g or 9.8 meters/sec/sec equals 22 miles per hour per second — the acceleration of a race car.

Here we took the opposite approach: setting aside hairy tensor equations, and tiny but real relativistic timing effects, to instead describe the core of how Einstein’s simple, weird spacetime curvature theory accounts for our weight and for Newton’s gravity as we experience it.

So our simplified account of volume expansion at 4πGM won’t explain time dilation — how much an astronaut needs to adjust his watch on returning to earth; that requires Minkowski’s math, where to track a clock’s rate one must factor in the speed of light. And it doesn’t explain electricity and magnetism the way Einstein’s full tensor equations for general relativity do.

It just explains gravity.

SUMMARY AND CONCLUSIONS

This short article has explained the core of General Relativity with regard to gravitation: the positive curvature of spacetime means space itself expands in the vicinity of matter. Rigid bodies like the earth exert forces as if they were expanding outward at a constant volume acceleration, in proportion to their mass.

First, we confirmed by experiment — via mobile computing device with accelerometer and display — that gravity is a fictitious force, perceived by an observer who is accelerating — in our case, an observer on the upwardly or outwardly accelerating the surface of the earth. This corresponds with our experience if not our intuition. We will have noticed our weight vanishes when we’re not in contact with the earth, though we appear to accelerate downward.

Amid positive spacetime curvature, space containing matter expands over time. Setting aside electromagnetic energy and focussing on matter and gravity, a volume V containing mass M expands the space that contains it, and its contents, at a constant volume acceleration of:

If our accelerometers are correct, we’re being pushed upward at 1 g or 9.8 meters/sec per second — in which case, every sqrt(Re**3/GM) = 13.4 minutes we have in effect been pushed upward one full earth radius, amid non-uniform, geometrically expanding rulers.

This is simple but counterintuitive. Whatever our accelerometers and other measurement devices say, it’s hard to stop thinking of The Ground we stand on as our intuitive ‘stationary’ reference. Few of us are comfortable performing the experiment shown in the figure below: of a scientist (perhaps a graduate student or intern) stepping off a cliff to confirm that when a ball is thrown or a cannon ball is fired, it isn’t curving toward earth in a parabola — it’s simply moving at constant velocity in a straight line.

So even as we understand General Relativity, we may be tempted to dismiss our accelerometer readings. We prefer to believe that apples fall toward the earth — not that earth’s surface pushes us upward, and pushes trees upward until an apple’s stem snaps — while the ground continues to accelerate upward until it collides with stationary apples.

We’ve seen that even Einstein, the originator of the general relativity theory, was confused about the expansion of space, and was ultimately forced by evidence to retreat from a notion of fixed size space, in favor of an expanding model.

Accelerating expansion of space containing non-uniform matter is perceived by surface dwellers as attraction between denser objects, and as apparently curved trajectories for objects actually moving unaccelerated in free fall.

In this sense, we have explained the illusion of gravity.

References
Misner, Charles W., Thorne, Kip S. and Wheeler, John Archibald. Gravitation. W. H. Freeman, 1973.

Thorne, Kip S. Black Holes and Time Warps: Einstein’s Outrageous Legacy. W. W. Norton & Company, 1995.

Isaacson, Walter. Einstein: His Life and Universe. New York: Simon and Schuster, 2007.

* Gauss’s Law for Gravity

The simple formula for expansion of 3 dimensional volume of space by matter given here,

is similar in form and equivalent to Gauss’s Law for Gravity, derivable from Newton’s theory. Developed before Einstein’s theory that space itself can expand, Gauss described forces in terms of gravitational fields and flux rather than as accelerating volume, the natural form implied by the units of G. Gauss’s Law is shown here in differential and integral form:

The main difference is the negative sign, which can be construed as an assumption that gravity was a pull downward rather than a push upward.

**We avoided math in the body of the article. Here’s how we calculated the earth’s surface gravity as a function of its mass, surface area, and the expansion of space:

G = 6.67 x 10**-11 cubic meters/sec/sec/kg
Newton’s gravitational constant

Me = 5.92 x 10**24 kg
the earth’s mass

Ae = 510 million square kilometers
the surface area of the earth

d2V/dt2 = 4πGMe = 4 * 3.14* 6.67 * 5.92 x 10**(24–11)
= 5.0 x 10**15 cubic meters / sec / sec
= 5 million cubic kilometers/sec/sec
earth’s rate of accelerating volume expansion

So earth’s vertical surface gravity is:
g = 4πGMe/Ae = (5.0 x 10**15 meters^3/sec^2) / (5.1 x 10**14 meters^2)
= 9.8 meters/sec/sec
vertical acceleration of gravity at earth’s surface

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