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Against the Curvature of Space-Time

Manu N. Lakshmanan(Dated: April 8, 2010)

In my everyday life there’s nothing quite as strenuousas riding my bike up a steep hill and as thrilling as riding

down one. Gravity is what makes these experiences sosensational. But what is gravity? What causes it? Thestrength with which gravity acts was first described quan-titatively in 1726 by Isaac Newton, though he had no an-swer for what it actually   is . Newton’s law of gravitationdescribed it as an attractive force that two masses exerton each other, along a straight line connecting them. Soas I ride my bike up a hill, should I envision some sortof elastic tether extending straight from my body to theEarth’s massive core, pulling me down? In 1915, this pic-ture was completely repainted by Albert Einstein with atheory that prevails to this day.

Although both Newton’s and Einstein’s theories give

accurate quantitative predictions of gravity on Earth, inthe context of much heavier bodies such as the planets,Einstein’s theory of General Relativity continues to de-scribe events correctly where Newton’s cannot. In 1919,Einstein’s theory was first accepted by the scientific com-munity as a result of the work of the astrophysicist SirArthur Stanley Eddington; Eddington made measure-ments of the deflection of starlight during a solar eclipseand found that they agreed with the predictions of Gen-eral Relativity. Although it is possible to visualize anelastic tether somehow latching on to a light ray, causingit to deflect, a closer inspection of Einstein’s equationsgives a radically different picture.

In 1905, ten years before he introduced General Rela-tivity, Einstein formulated the theory of Special Relativ-ity. Special Relativity is based on the physical law thatfrom the point of view of any object, regardless of itsvelocity, one would perceive light as moving at its usualspeed of about 186,000 miles per second. Therefore, fora person moving at any speed, especially near the speedof light, in order for her to still perceive light moving atits usual speed of 186,000 miles per second, she will ex-perience an accelerated passage of time (time dilation)and perceive a contraction of space (length contraction).

Special Relativity gives rise to two important conse-quences that led to Einstein’s theory on gravity. Thefirst of these is that no object or entity can travel faster

than the speed of light. From the way gravity is describedby Newton’s theory, the entity that carries the gravita-tional information from one mass to another seems to besome sort of invisible elastic tether. But Newton’s the-ory doesn’t even describe how and at what speed thatelastic tether is formed. By omitting this informationabout the tether, Newton leaves us to assume that itsimply appears instantaneously out of thin air, travelingat infinite speed. Therefore, Newton’s theory is eithercontradicted by Special Relativity by assuming a faster

than light speed for the tether that transmits gravity, orit is incomplete because it doesn’t provide us with any

idea of the speed of gravity. Einstein was the first to real-ize how blatant an omission this was in Newton’s theory.If you have no understanding about the speed of moderncars, whether they move at 100 mph or 186,000mph, doyou really understand anything about cars?

The second consequence of Special Relativity that ledEinstein to pursue a new theory for gravity is that whenan object moves at some velocity, the way it experiencestime dilation and length contraction is that it trades dis-tance for time. If you were to visualize reality as not

 just being composed of the three spatial dimensions of forward-back, left-right and up-down, but of also havinga fourth dimension, past-future, then you’d see that space

and time are actually coupled into one entity: space-time.Let’s form a metaphor of the space-time terrain as a mapof some land, the type we deal with in our everyday lives.Let’s define “normal” movement along the map as beingin the northeast direction. If a traveler, halfway throughhis journey, notices that he’s moved farther to the eastand less toward the north than he would have expectedhad he been moving in the stipulated direction, then hispath must have been at some angle relative to the nor-mal, northeast direction. Similarly, in space-time some-one who is standing at rest experiences time and spacenormally; therefore his movement in space-time is alonga normal path. When he begins moving at some velocity,however, he will begin to experience accelerated time andcontracted space, in the same way that the traveler expe-rienced extra easterly and less northerly movement. Theperson who is moving at some velocity is actually movingthrough space-time along a path at some angle relativeto the normal. As the velocity of the object increases, sodoes the angle of its path in space-time relative to thenormal.

Einstein noticed that there must be a similar space-time effect due to gravity when he had his self-proclaimed“happiest thought” as he watched a man fall off of his lad-der. What he realized is that free fall is the only scenarioin which one feels free from all forces, even from gravity.In contrast, while you are standing, you feel the force of 

gravity as you bear the weight of your body. A thirdsituation is when you are accelerating, such as in an ele-vator that begins to move upward or in a car that beginsto accelerate rapidly forward. In these cases of acceler-ation, you experience an additional force. In the case of the elevator when you are accelerating up, in the oppo-site direction to the force of gravity, if you were standingon a weight scale at the time, the scale would give ahigher reading than if the elevator were at rest. Einsteinrecognized that acceleration and gravity were equivalent:

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in the case of free fall, you accelerate in the directionof gravity causing the forces to cancel each other out sothat you feel truly free; and in the case of the elevator, theforces add together, making you appear heavier on thescale. This realization of the equivalence of accelerationand gravity is called the equivalence principle. Recall-ing that acceleration is defined as a change in velocityand that Special Relativity is a theory about what hap-

pens when you move at some velocity, Einstein realizedthat the equivalence principle forms a bridge connectingSpecial Relativity and gravity.

So if moving at constant velocity causes one to experi-ence space-time at some tilted angle, such as in SpecialRelativity, then what happens to the space-time whenone is accelerating, or equivalently, is experiencing somegravitational force? Keeping in mind that acceleration issimply a change in velocity and that the angle of space-time tilt is determined by how fast one is moving, thenexperiencing acceleration would result in a space-timethat has a constantly changing angle of tilt. A line thathas a constantly changing angle of tilt is a curve.   There-

 fore, gravity causes space-time to curve . One can imaginea simplified picture of a curved space-time as a tram-poline that becomes warped when an extremely heavyobject such as a boulder is dropped on its center. Thisis a simplified picture because the surface of the tram-poline is just two-dimensional, while the space-time of reality is four-dimensional, consisting of: forward-back,left-right, up-down and past-future. When the trampo-line surface is warped by the boulder, any ping-pong ballspresent will begin moving along the trampoline surfaceand following this warping of its shape. More precisely,Einstein found that in the warped geometry of space-time, masses travel along the shortest path available, inthe same manner that the shortest path for an airplane

to take between two points on the curved surface of theEarth is a curved route. On a curved surface such asthe Earth or the geometry of space-time, the shortestpath between two points will always be a curved one, aso called “geodesic.” This tendency for objects to movealong geodesics in space-time becomes apparent in theexample of a canon ball and a penny being launched into

the air at the same angle and same initial speed; as theequations of elementary physics confirm, you’d find thatboth objects despite their differences in mass will havethe same trajectory. They both follow the curvature of the space-time at that location in space and that momentin time.

Returning to the question of how gravitational infor-mation is transmitted, that is, what entity actually car-

ries it from one mass to another, one can see that theanswer to this question is not an elastic tether. Instead,it is the same as for light and sound: gravitational infor-mation is transmitted in the form of waves. Gravitationalwaves are formed in the same way as if you were to dropa pebble into a pond: that change in shape of the pond’ssurface is first observed as ripples, or waves, emanatingfrom where the pebble was dropped into the pond. Inthe case of gravity, if a planet were to suddenly appearin space-time, it’s appearance would cause a ripple in thefabric of space-time, that emanates from the planet out-wards, eventually reaching other planets which will thenexperience the familiar pull that we know to be “gravity.”From Einstein’s equations, it turns out that the speed of these gravitational waves is exactly equal to the speed of light, or electromagnetic waves. In the case of a pond,because the pebble subsequently sinks into the pond, itdisappears from the surface of the pond after the initialstrike; consequently, the surface is eventually restored toits smooth form prior to being struck by the pebble. Inthe case of the planet appearing in space-time, assumingthe planet remains embedded in the space-time fabric,the ripples that it formed when it appeared will reach anequilibrium state, forming some permanent re-shaping of the geometry of the space-time fabric.

Gravity is not just a force - as I used to think of it.Gravity is the geometry of the space-time in which we

exist. The tether metaphor could not be more of anover-simplification. When I reach the peak of a hill onmy bike, I pause to relax my muscles and take a firmgrasp of the handlebar. Then I give myself a little nudgeforward, launching into descent, allowing myself to flowalong the curves of space-time.