Bending Spacetime in the Basement

One of the things I detested about being a little kid was that every time I thought of something really cool to do, I was invariably thwarted by my little brother shouting, "Mom! Kelvin's mixing rocket fuel in the bathtub again!" or "Mom! Kelvin's making a submarine out of the old refrigerator!". Well, middle age has its drawbacks, but at least you can undertake a project like this without fear of getting nipped in the bud at the cry, "Mom! Kelvin's down in the basement bending spacetime!". It's important to recall the distinction between "grownup" and "grown up". Let's us grownups head for the basement to bend some serious spacetime.

Matters of Gravity

This page presents a "basement science" experiment which reveals the universality of gravitation by demonstrating the gravitational attraction between palpable objects on the human scale. The experiment deliberately uses only the crudest and most commonplace materials, permitting anybody who's so inclined to perform it. Einstein's 1915 theory of General Relativity explains gravitation as spacetime curvature created by matter and energy. So, by demonstrating how every object in the universe attracts everything else, we're bending spacetime in the basement.

But, if gravitation is ubiquitous, why was it not discovered millennia before Newton's 1687 Philosophiæ naturalis principia mathematica? The reason lies in the extraordinary weakness of the gravitational force.

Feeble Attraction

The stark difference in the strength of the electromagnetic and gravitational forces is evident in the picture to the left. The bright square in the jaws of the pliers is a 4 mm cubical magnet. It is lifting a spherical steel pétanque (a lawn bowling game popular in southern France and Switzerland) ball which weighs 550 grams. Consider this picture in the following way: we're pitting our valiant little magnet, with a volume of 0.064 cubic centimetres, weighing less than one gram, pulling up with the electromagnetic force, against the entire Earth, pulling down with gravity. And the winner is...the magnet. A one gram magnet (I'm being generous: I don't have a scale on which its weight reads other than zero) out-pulls the Earth, which weighs 5.9736×10²⁷ grams and has a volume of more than 10²⁷ cubic centimetres.

(The apparent discrepancy between the ratio of masses of the Earth and the magnet and the 4.17×10⁴² strength ratio of electromagnetism and gravitation is due to the fact that only an infinitesimal fraction of the mass of the magnet contributes to the [electro]magnetic attraction on the ball, while every gram of the Earth's mass exerts gravitational force. To obtain the correct ratio of force strengths, one must compare the gravitational attraction between two electrons at a given distance with the electromagnetic repulsion resulting from their charge. This calculation arrives at the correct strength ratio for the two forces.)

If gravity were not so weak compared to the electromagnetic force, you wouldn't be reading this page; it's only because the electromagnetic force that bonds the atoms in your body together so easily defeats the Earth's gravity that you, along with all other solid objects, don't slump into a puddle and eventually merge into a perfectly spherical (actually, slightly ellipsoidal thanks to rotation) planet.

If your browser supports JavaScript, you can use the following calculator to determine the gravitational force between any two objects. The gravitational force between two masses m and m' whose centres of gravity are separated by a distance r is given by:

where G, the gravitational constant, is:

Given:
Mass 1: m = grams
Mass 2: m' = grams
Distance: r = centimetres

Gravitational force: F = dyne

Since most folks don't encounter forces in dynes on an everyday basis, not to mention small fractions of a dyne, it's useful to express this force in terms of the weight of something one can visualise. We can calculate the mass of an object from its weight at the Earth's surface merely by solving the gravitational force equation above for m and plugging in the Earth's mass, m_e = 5.9736×10²⁷ grams, and the Earth's equatorial radius, r_e = 6.37814×10⁸ cm:

The value below, then, is the mass of an object whose weight at the Earth's surface equals the gravitational force displayed in the box above. This value is automatically updated when you press the "Calculate" button.

Equivalent mass: grams

This mass is equivalent to the following length of nylon monofilament fishing line with a diameter of 0.02 cm (0.2 millimetres) like that shown at the left (the scale at top shows centimetres and millimetres; the micrometer holding the segment of monofilament is, regrettably, calibrated in English inches--one inch is 2.54 centimetres.

Monofilament length: mm

This is small, but it is an object visible to the naked eye (easily with a modest magnifier), and heavy enough to fall under the influence of Earth's gravity, as opposed to smaller and lighter particles of dust and smoke which stay suspended in the air for long periods of time.

Do the Twist!

Measuring tiny gravitational forces would be easy if we were in deep space, far from any massive bodies. The only forces on objects in our space laboratory, then, would be those entirely under our control. As long as we made sure none of the objects we were experimenting with were magnetic or electrically charged (easily arranged, assuming they are conductive, simply by bringing them into contact so all excess charges equilibrate), the only force remaining between objects would be gravity, so however weak it be, we need only be sufficiently patient to observe its effects. (The other two forces, the strong and weak nuclear interactions, are limited in range to distances on the order of the size of an atomic nucleus and can be neglected on the human scale.)

What we'd like to do, then, is cancel the Earth's gravity so that the much smaller gravitational forces between objects that fit in the basement become evident. Fortunately, we don't need a 25th century WarpMan to accomplish this, only a modest helping of 18th century technology.

Differential Cleverness

The Torsion Balance

State of the art torsion balances have measured the gravitational constant to better than one part per million and confirmed the equivalence principle to more than 11 decimal places. This requires extraordinarily refined and delicate laboratory apparatus and experimental design, in which a multitude of subtle effects must be compensated for or canceled out. We, however, aren't going to measure anything--we're only interested in observing universal gravitation. This allows simplifying the torsion balance to something we can set up in the basement.

The principle of the torsion balance is extremely simple. Suspend a horizontal balance arm from a vertical elastic fibre. At each end of the balance arm are masses, much denser than material of the arm, which respond to the gravitational force. Once the suspending fibre, balance arm, and weights are set up and brought into balance, the downward force of gravitation acts equally on every component. The balance arm is then free to rotate without any hindrance from the Earth's gravity. It is constrained only by air friction and the torsional strength of the support fibre--its resistance to being twisted. We can then place test masses near the ends of the balance arm and observe whether the gravitational attraction between them and the masses on the arm causes the balance arm to move. When measuring the gravitational constant one must precisely calibrate the torsional strength of the fibre, but to simply observe gravitation we need only make sure the fibre is sufficiently limp to allow the gravitational force to overcome its resistance to twisting.

In practice, the balance arm is so free to move that once any force sets it into motion, it oscillates for a long period, spinning round and round if free or bouncing back and forth off the stops if constrained. To avoid this we need to damp the system so kinetic energy acquired by the bar is more rapidly dissipated. Well, nothing's more damp than water, so we add a water brake to the arm which turns in a fixed reservoir. The resulting drag as the balance arm moves is much greater than air resistance and frictional losses in the fibre, and reduces the oscillations to a tolerable degree.

The Gravitational Balance

"The time has come," the Hacker said,
   "To talk of many things:
Of plastic foam--and tuna cans--
   Of chunks of lead--and string--
And how the force of gravity--
   Will make the balance swing."

So here's the sophisticated, high-tech, big science apparatus we'll use to observe the subtle curvature of spacetime. An aluminium ladder serves as the support from which the balance arm is suspended. Nylon monofilament fishing line, as shown above, is knotted to the middle of the third cross-beam at the back of the ladder, one above the brace bearing the little white box, about which more later. Using a ladder or similar movable support frame allows setting up the balance in the middle of the room. This is important because we are bending spacetime in the basement, in this case an underground storage room at Fourmilab. Ground level is about even with the ceiling of this room, about 45 cm above the top of the ventilation window at the upper right of the picture. An underground room is ideal because it minimises temperature variations and vibration which might perturb the balance arm. Both walls shown in this picture are sunk into solid limestone rock--if you set up the balance near one of these walls, the gravitational field from all that rock will mask that of the test masses, and the balance will assume a "gravity gradient" position with one of the ends of the bar pointing toward the wall, and will budge only slightly under the influence of the test masses. With the bar in the middle of the room, the tidal influence of the mass of the wall and the rock behind it is reduced to a negligible value. The pipe on the wall at the right is part of the serpentine pressurised hot water heating system; it was disabled to prevent air currents from disrupting the balance arm. In fact, since the room is underground, the heating system is rarely engaged, and only in the depths of winter, never in June.

The Balance Arm and Cradle

The Support Fibre

The Water Brake

The water brake consists of a flap which projects downward from the balance arm (in this case, a piece of aluminium cut with scissors from the tray of a "heat and eat" meal, fixed with white glue into a slot cut into the bottom of the balance beam). The flap projects into a reservoir (a tuna fish can) filled with water. A more viscous fluid such as salad oil would provide greater damping and less bouncing than water, but I opted for water since it's less icky to clean up when the inevitable spill occurs and can be disposed of when the experiment's done without a visit to the village recycling barrel.

If I were rebuilding the balance beam, I would use a longer and narrower flap and/or a larger and deeper water reservoir. If the flap is only slightly smaller than the inside diameter of the reservoir, you have to be very careful that the flap and reservoir are centred on the beam. Otherwise, the flap will touch the edge of the reservoir and freeze the beam in place, as that frictional force is many orders of magnitude greater than the gravitational force we wish the beam to respond to. The water reservoir can be as large as you like, as long as it doesn't interfere with placing the test masses; the larger it is, the less you have to worry about its being precisely centred.

Test Masses and Supports

Use the densest objects you can obtain for the ends of the balance beam and as test masses: lead sinkers, steel balls, plutonium hemispheres, etc. Density is important because the gravitational force varies as the inverse square of the distance between the centres of mass of two objects. With a dense substance, the centre of mass is closer to the surface, so you can get the centres of mass closer together and enhance the gravitational force. For example, consider two pairs of one-kilogram spheres, the first made of lead (density 11.3 g/cm³), the second of pine wood (density about 0.43 g/cm³), placed so the surfaces of each pair of spheres are 1 cm apart. A one kilogram lead sphere has a radius of 2.76 cm, so the centres of mass are separated by 1+2.76×2, or 6.52 cm. A one kilogram sphere of pine has a radius of 8.22 cm, by comparison, so the centres of mass of the two pine spheres are 1+8.22×2 = 17.44 cm apart. Taking the square of the ratio of these distances shows that the gravitational force between the lead spheres is more than 7 times that of the pine spheres. Since attraction is linear by mass but inverse square in distance, you're better off with a modest mass of high-density material than a large mass of a substance with lesser density.

It's best to use a nonmagnetic material like lead for the weights on the ends of the balance arm. The forces we're working with are so small that if you use, for example, steel ball bearings on the arms, you may end up accidentally reinventing the compass instead of detecting the force of gravity.

The Spy Cam

Gravitation in Action

There is, at this writing, no movie format supported by all Web browsers and computer systems. The movies are furnished in three different forms, in the hope one will prove compatible with your equipment and software. The links below the movie posters download the movie in the various formats. Each gives the size of the movie file, which varies dramatically depending on the format. If your browser supports MPEG, that's the best choice, since the files are much smaller than the other alternatives. After the movie plays, use your browser's "Back" button to return to this document.

Movie 1

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Movie 2

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Pay no attention to the plastic robot ant--she's just curious. It's a long story.

A Tide in the Affairs of Man

The universality of gravitation means that every object in the universe is interlinked in a web of mutual attraction; the universe is transparent to gravitation. The most distant galaxies exert a pull on you, as you do upon them--immeasurably tiny to be sure, but present just the same. From a practical standpoint, universality means there's no way to shield your torsion balance from the gravitational attraction of masses in its vicinity; you can only set it up sufficiently far from other massive objects so the attraction of the test masses predominates. One interesting massive object to consider is yourself (I use "massive" only in the sense of "possessing mass", not pejoratively; if you took it that way, perhaps you should check out my on-line diet book).

Using the gravitational force calculator earlier in this document, we can compute the gravitational attraction between the 338 gram mass at the end of the balance beam and the 740 gram test mass at the 14 cm distance when the beam is at the midpoint between the masses to be 0.000085 dynes. Now suppose you're crouching down in order to move the test masses, with your centre of gravity one metre from the closer test mass, and that you weigh 65 kg. Plugging these numbers into the calculator shows that your own gravitational attraction on the nearer end of the beam is 0.000147 dynes, 1.7 times as great as that of the test mass. Your actual influence on the motion of the balance arm is less, however, since what matters is the difference in force exerted on the masses at the two ends of the balance arm. Since your centre of gravity is more distant than the test masses, the difference is less.

Let's work it out. Assume the centres of gravity of the two masses on the balance arm are 25 cm apart, and that you're crouching so the arm makes a 45° angle with your centre of gravity, one metre from the centre of the arm. The nearer mass is then 17.68 cm closer than the more distant one and the difference in gravitational attraction (or tidal force) on the two masses is the difference in attraction on a mass 91.16 cm distant and one 108.84 cm away. The calculator gives the attraction on the near end of the arm as 0.0001764 dynes and the far end as 0.0001238 dyne, with a difference of 0.0000527 dynes. Now recall that the force exerted by the test mass was 0.000085 dynes, only 1.6 times as large, so even taking into account the reduced tidal influence due to your greater distance, the force you exert on the balance cannot be neglected. This makes it essential to remotely monitor the experiment so your own mass doesn't disrupt it.

In practice, air currents due to your motion and resulting from convection driven by your body's temperature being above room temperature may exert greater forces on the balance arm than the gravitational field generated by your mass. In any case, it's best to let the experiment evolve on its own, observed from elsewhere.

Enlightenment Deferred:
An Historical Speculation

To do this, he would have had to suspect that attraction was universal, suggest an experiment to confirm this, and perform that experiment, with results validating the hypothesis. Here is information in Archimedes' possession which might have suggested the universality of gravitation.

• The Earth is a sphere. The shape of its shadow on the Moon during a lunar eclipse demonstrated this, and was confirmed by the next item. The assertion that "the ancients thought the world was flat" is nonsense--Columbus didn't "discover the world was round": he discovered that his own estimate of the diameter of the world was wrong by a factor of two compared to that available to Archimedes; if he hadn't inadvertently discovered the New World, he and his unfortunate crew would have died of starvation far from the coast of China.

• The approximate radius of the Earth. Around 250 B.C., by measuring the difference in the angle of sunlight at noon on the June Solstice, which illuminated the bottom of a well at Syene (now Aswan) Egypt near the Tropic of Cancer, with the length of the shadow cast on the same date and time by a vertical pillar in Alexandria, a known distance to the North, Eratosthenes determined the Earth's circumference. Archimedes corresponded extensively with Eratosthenes and other scholars in Alexandria, and knew of this result. Archimedes himself calculated the value of Pi as between 3 10/71 and 3 1/7, with a mean value of 3.14185, allowing accurate computation of the Earth's radius from the circumference.

• The approximate mass of the Earth. Assuming the Earth to have the same density as common rocks such as limestone (2.7 g/cm³) gives an estimate within a factor of two of the correct value. The actual density of the Earth is 5.52 g/cm³.

• How to calculate with very large numbers. In The Sand Reckoner in 215 B.C., Archimedes invented a positional number system which allowed writing and calculating with arbitrarily large quantities, which he demonstrated by calculating not only how many grains of sand would fill the volume of the Earth, but how many grains of sand would fill the entire universe (which the Greeks estimated to be about one light year in diameter). The latter number, about 10⁶³, is comfortably larger than any of the quantities associated with gravitation.

• The existence of electrostatic and magnetic forces which appeared to act at a distance. The inverse square behaviour of these forces was not known in antiquity, however.

Suppose then that, given these facts, Archimedes embarked upon the following chain of reasoning.

• Objects fall, not in a fixed direction, but toward the centre of our world. If they fell in a fixed direction, if I dropped a rock down a well in the south of Egypt and a well in Syracuse, separated by a substantial percentage of the world's circumference, one would hit the wall of the well before striking the bottom. This doesn't happen, so objects fall everywhere toward the centre of the world.

• Why does the world attract falling objects? Is there something special which endows it with this property? Yet the world seems to be made of the same substances as everything else. What is the difference between the world and a rock?

• ! The world is much larger than a rock! Perhaps every object attracts every other. We only feel the world's attraction because it is so large.

• But if this is so, might the celestial bodies be objects no different from the world and its inhabitants, and subject to the same forces?

• If attraction is universal, might an artificer be able to build a device to show it?

• Such a device must be isolated from falling down. Perhaps a horizontal balance, free to turn in either direction, with weights at each end to be attracted to objects in their vicinity....

The Archimedes Apparatus

Let's try to redesign the torsion balance using only materials available in antiquity.

The Balance Arm

Instead of plastic foam, we use a strip of pine wood, 2 cm wide, ½ cm high, and 30 cm long. Notches are cut in the edges of the beam near each end to secure the support cradle. For masses at the ends of the balance beam we may continue to use lead, which was produced in Egypt in the 2nd millennium B.C. and in Europe no later than the 6th century B.C. As the discoverer of specific gravity, Archimedes would understand the merit in using the densest substance available. (Gold, almost twice as dense as lead, would be an even better choice. Perhaps King Hiero of Syracuse, grateful to Archimedes for exposing the goldsmith who adulterated the gold in his crown with silver, might have contributed gold weights for the balance beam, thereby taking the first small step down the road to government-funded Big Science. Wishing to remain in the domain of basement science, we shall forgo royal subsidies and soldier on with lead.)

The Cradle

To support the balance arm, we substitute twine made of vegetable fibre for telephone wire. Actually, since copper was known for thousands of years before the Greeks, a lightweight copper cradle could have been made, but it would have been more work to fabricate and has no advantage compared to the twine. Thread, string, and rope were made from a variety of natural fibres by all ancient cultures.

The Support Fibre

The support fibre is the most difficult component to replace with a 3rd century B.C. analogue. Nylon monofilament so closely approaches the ideal of a massless support free of torsional resistance that doing without it requires experimenting with a variety of alternatives and compromising with the shortcomings of whatever is selected. I finally settled on a very thin vegetable fibre support "peeled" from a piece of twine by unwinding it. The fibres you find in rope or twine are a variety of lengths--you have to separate them and then select individual fibres long enough to support the balance arm. To obtain sufficient strength, I used four separate fibres selected from the twine. If the rope or twine has been twisted or braided, you'll have to let the fibre hang for an extended period of time (three or four days at least) to release its internal stresses.

In choosing and using any natural fibre support, you have to approach the project with a willingness to learn by trial and error and a great deal of patience. Each kind of fibre has its own "personality", and the quirks can take some time to understand. For example, many plant and animal fibres are sensitive to moisture--if a summer thunderstorm increases the relative humidity from 50% to 99% in the space of an hour, your balance arm may start to swing wildly as the fibre absorbs moisture from the air. Further, plant fibres tend to tear, both under tensile stress and when twisted. This can cause your balance beam to "spontaneously" shift to a different equilibrium point or, after having been displaced, return to a different location than the starting point.

Would Archimedes have appreciated the importance of choosing a supple and well-behaved support fibre? I think so. From the radius of the Earth, which he knew, and assuming its density to be the same as rocks such as limestone (about half the actual density of the Earth), the ratio of the Earth's mass to that of whatever test masses were employed could be estimated within a factor of two. Making the simplest assumption (which has the additional merit of being correct) that attraction is proportional to mass, it is clear that the force acting on the masses at the ends of the balance arm is minuscule, so a fibre which offers the least possible resistance to twisting should be employed.

Test Masses

Lead or gold (Monarch! Archimedes is doing natural philosophy in the bathtub again!) test masses would be preferable, but to show how robust this experiment is I opted for a couple of rocks--two kilogram paving stones like those which border every highway in Switzerland, where roads are so built to last that Julius Cæsar would shake his head in admiration.

No Water Brake

After experimenting with a variety of vegetable fibres, I decided to proceed without a water brake for this experiment. The balance arm is more prone to oscillation, but the friction in the support fibre, much greater than in synthetic nylon monofilament, damps the oscillations adequately. Allowing the lead weights on the balance arm to collide with the stone test masses also dissipates substantial energy, further reducing the need for a brake.

Unreconciled Residua

I didn't bother to replace the aluminium ladder with a support Archimedes might have used. Any carpenter could fashion a more suitable replacement for the ladder. In fact, a wooden saw-horse would have been better for all these experiments, but I don't have one and didn't feel like making one from various pallets and spare lumber in the High Bay.

The concrete floor would also seem strange to Archimedes, but it is irrelevant to the experiment. A smooth stone floor, as existed for millennia before, would produce identical results.

The Archimedes Experiments

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The Archimedes Enlightenment?

References

Click on titles to order books on-line from

Archimedes. The Sand Reckoner. English translation in Newman, James R. The World of Mathematics . Redmond, Washington: Microsoft Press, 1988. ISBN 1-55615-148-9.

You can always rely on Microsoft, who have allowed this essential reference to go out of print.

Carroll, Lewis [Charles Dodgson]. Alice's Adventures in Wonderland and Through the Looking Glass . New York: New American Library, 1995. ISBN 0-451-52320-2.

Cavendish, Henry. "Experiments to determine the density of the Earth". Philosophical Transactions of the Royal Society of London, Part II (1798), pp. 469-526.

Eötvös, Lorand von. "Über die Anziehung der Erde auf verschiedene Substanzen." Math. Naturw. Ber. aus Ungarn 8, 65-68 (1889).

Eötvös (his Hungarian surname is pronounced like "ut-vush" in English) improved the original Cavendish torsion balance to its modern form and used it to test the equivalence principle (in his final publication on the topic in 1922) to better than one part in 5×10¹⁰.

Feynman, Richard P., Robert B. Leighton, and Matthew Sands. The Feynman Lectures on Physics Vol. 1 (Chapter 7). Reading, Massachusetts: Addison-Wesley, 1963. ISBN 0-201-02116-1.

This lecture, including an audio recording of the original lecture at Caltech, also appears in:

Feynman, Richard P. Six Easy Pieces (Chapter 5). Reading, Massachusetts: Addison-Wesley, 1995. ISBN 0-201-40896-1.

Gamow, George. The Great Physicists from Galileo to Einstein . Mineola, New York: Dover, 1988. ISBN 0-486-25767-3. (Originally published by Harper in 1961 as Biography of Physics.)

Hogben, Lancelot. Mathematics for the Million (Chapter 12). New York: W.W. Norton, 1937, 1967. ISBN 0-393-30035-8.

Icikovics, Jean-Pierre and Nicolas Journet, eds. "Archimède." Les Cahiers de Science&Vie: Les pères fondateurs de la science 18 (December 1993). ISSN 1157-4887.

Kutz, Myer, ed. Mechanical Engineers' Handbook , 2nd ed. New York: Wiley, 1998. ISBN 0-471-13007-9.

Lide, David R., ed. CRC Handbook of Chemistry and Physics , 73rd ed. Boca Raton, Florida: CRC Press, 1993. ISBN 0-8493-0473-3.

Newton, Isaac. Philosophiæ naturalis principia mathematica. London: Streater, 1687. English translation by A. Motte, revised by A. Cajori, Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World , 1729. A modern edition in two volumes is published by the University of California Press as ISBN 0-520-00928-2 (Vol. 1) and ISBN 0-520-00929-0 (Vol. 2).

Rucker, Rudy. Mind Tools . Boston: Houghton Mifflin, 1987. ISBN 0-395-46810-8.

Trifonov, D. N., and V. D. Trifonov. Chemical Elements: How they Were Discovered . Translated from the Russian by O. A. Glebov and I. V. Poluyan. Moscow: Mir Publishers, 1982.

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