Warped Passages (19 page)

General relativity goes much further than Newtonian gravity because it allows us to calculate the relativistic gravitational field of any distribution of energy and matter. Moreover, the revelation that the geometry of spacetime encodes the effects of gravity permitted Einstein to close a major gap in his original formulation of gravity. Although physicists at the time knew how objects would respond to a gravitational field, they did not know what gravity was. Now they understood that the gravitational field is the distortion of the spacetime fabric caused by matter and energy. This distortion extends throughout the cosmos itself, or, as we will see shortly, throughout a higher-dimensional spacetime that might include branes. All of the gravitational effects of these more complicated situations can be embedded in the ripples and curves of a spacetime surface.

A picture gives perhaps the best description of how matter and energy distort the spacetime fabric to create a gravitational field. Figure 39 shows a sphere of matter sitting in space. The space surrounding the sphere is distorted: the ball makes a depression in the spatial surface whose depth reflects the ball’s mass or energy. A ball
passing nearby will roll towards the central depression, where the mass is located. According to general relativity, the spacetime fabric warps in an analogous fashion. Another ball passing through would be accelerated towards the center of the sphere. In this case, the result would agree with what Newton’s law would predict, but the interpretation and calculation of the motion would be very different. According to general relativity, a ball follows the undulations of the spacetime surface, and thereby implements the motion induced by the gravitational field.

Figure 39.
A massive object distorts the surrounding space, thereby creating a gravitational field.

Figure 39 is a bit misleading, so you should keep in mind several caveats. First of all, I’ve shown the space surrounding the ball as two-dimensional. But really, the full three-dimensional space and the full four-dimensional spacetime are warped. Time is warped because it too is a dimension from the vantage point of special and general relativity. Warped time is how special relativity tells us that clocks run at different rates in different places, for example. A further caveat is that a second ball rolling in the curved geometry around the first ball would also affect the geometry of spacetime; we have assumed that its mass is much smaller than the larger ball’s and neglected this small effect. The third thing that’s important to keep in mind is that the object distorting spacetime can have any number of dimensions. Later on, a brane will play the role of the sphere in this picture.

Nonetheless, in all cases matter tells spacetime how to curve, and spacetime tells matter how to move. Curved spacetime sets up the
geodesic paths along which, in the absence of other forces, things will travel. Gravity is encoded into the geometry of spacetime. It took Einstein the better part of a decade to deduce this precise connection between spacetime and gravity, and to incorporate the effects of the gravitational field itself—after all, the gravitational field carries energy, and is therefore bending spacetime.
^*
It was a heroic effort.

In his famous equations, Einstein specified how to find the universe’s gravitational field, given the contents of the universe. Although his best-known equation is
E
=
mc
²
, physicists use the term “Einstein’s equations” to refer to the equations that determine the gravitational field. The equations accomplish this formidable task by showing how to determine the metric of spacetime from a known distribution of matter.
⁹
The metric you calculate determines the spacetime geometry by telling you how to translate numbers associated with arbitrary scale units into physical distances and shapes that determine the geometry.

With the final formulation of general relativity, physicists could determine the gravitational field and calculate its influence. As with previous formulations of gravity, physicists use these equations to figure out how matter moves in a given gravitational field. For example, they can plug in the mass and position of a big spherical body, such as the Sun or the Earth, and calculate the well-known Newtonian gravitational attraction. In this particular example, the results wouldn’t be new—but their meaning would be. Matter and energy bend spacetime, and that bending gives rise to gravity. But general relativity has the further advantage that it incorporates any type of energy—including that of the gravitational field itself—into the distribution of matter and energy. This makes the theory useful even in situations where gravity itself contributes a significant amount of energy.

Because they apply to any distribution of energy, Einstein’s equations changed the outlook for cosmologists—historians of the cosmos. Now, if scientists knew the matter and energy content of the universe, they could calculate its evolution. In an empty universe, space would
be completely flat, with no ripples or undulations—no curvature at all. But when energy and matter fill the universe, they distort spacetime, producing interesting possibilities for the universe’s structure and behavior over time.

We most definitely do not live in a static universe: as we will soon see, we just might live in a warped, five-dimensional one. Fortunately, general relativity tells us how to calculate their consequences. Just as there are examples of two-dimensional geometries with positive, zero, and negative curvature, there are four-dimensional geometrical configurations of spacetime with positive, zero, and negative curvature, which could arise from appropriate distributions of matter and energy. Later on, when we discuss cosmology and branes in extra dimensions, the distortions of spacetime arising from matter and energy—both in our visible universe and on the branes and in the bulk—will be of critical importance. We’ll see that the three types of spacetime curvature (positive, negative, and zero) might be realized in higher dimensions as well.

General relativity has lots of consequences that you can’t calculate with Newtonian gravity. Among its many merits, general relativity eliminated the annoying action-at-a-distance of Newtonian gravity, which asserted that an object’s gravitational effects would be felt everywhere as soon as it appeared or moved. With general relativity, we know that before gravity can act, spacetime has to deform. This process does not happen instantaneously. It takes time. Gravity waves travel at the speed of light. Gravitational effects can kick in at a given position only after the time it takes for a signal to travel there and distort spacetime. That can never happen more quickly than the time it would take light, which travels as fast as anything we know, to get there. For example, you will never receive a radio signal or a cell phone call sooner than the time it would take for a light beam to travel to you.

Furthermore, physicists were able to use Einstein’s equations to explore other types of gravitational field. With general relativity, scientists could describe and study black holes. These fascinating, enigmatic objects form when matter is highly concentrated within a very small volume. In black holes the geometry of spacetime is extremely distorted, so much so that anything entering a black hole
gets trapped inside. Even light cannot escape. Although the German astronomer Karl Schwarzschild discovered that black holes were a consequence of Einstein’s equations almost immediately after general relativity’s development,
^*
it was not until the 1960s that physicists took seriously the idea that they could be real things in our universe. Today, black holes are well accepted in the astrophysical community. In fact, it looks as though there is a supermassive black hole at the center of every galaxy, including our own. Moreover, if there are hidden dimensions then there exist higher-dimensional black holes which, when big, look like the four-dimensional black holes that astronomers have observed.

Coda

To conclude the story of the GPS system, it turns out that to calculate position to within a meter, we must measure time to better than one part in 10
¹³
. The only possible way to get this accuracy is with atomic clocks.

But even if we had perfect clocks, time dilation would slow them down by about one part in 10
¹⁰
. This error, if not corrected, would be a thousand times too big for our desired GPS system. We also have to account for the gravitational blueshift, a general relativity effect associated with a photon traveling in a changing gravitational field, which gives an error at least this great. This and other general relativity deviations would give errors that, if ignored, would build up at a rate greater than 10 km per day.
^†
Ike (and current GPS systems) must correct for these relativistic effects.

Although by now relativity has been well tested and even gives rise to effects that need to be accounted for in practical devices, I do find it fairly remarkable that anyone listened to Einstein at first. He was completely unknown, working in the Bern patent office because he
couldn’t get a better job. From this unlikely location he proposed a theory that went against the beliefs of all other physicists of his time.

Gerald Holton, a Harvard historian of science, tells me that the German physicist Max Planck was Einstein’s first champion. Without Planck, who immediately recognized the brilliance of Einstein’s work, it might have taken much longer for it to be recognized and accepted. Following Planck, a few other notable physicists knew enough to listen and pay attention. And shortly afterwards, so did the world.

What to Remember

• The speed of light is constant. It is independent of the speed of an observer.
  
• Relativity
  modifies our notions of space and time and tells us that we can treat them together as a single
  spacetime
  fabric.
  
• Special relativity relates the values of energy, momentum (which tells how an object responds to a force), and mass. For example,
  E
  =
  mc
  ²
  , where
  E
  is energy,
  m
  is mass, and
  c
  is the speed of light.
  
• Mass and energy make spacetime curve, and you can think of that curved spacetime as the origin of the gravitational field.
  

6

Quantum Mechanics: Principled Uncertainty, the Principal Uncertainties, and the Uncertainty Principle

And you may ask yourself,

Am I right?…Am I wrong?

Talking Heads

Ike wondered whether Athena was making him watch too many movies or Dieter was talking too much about physics. But whatever the reason, the previous night Ike dreamed he met a quantum detective. Dressed in a fedora, a trench coat, and with a stone-faced expression, the dream detective spoke:

“I knew nothing about her except her name, and that she was standing there before me. But from the moment I set eyes on her I knew Electra
^*
would be trouble. When I asked her where she came from, she refused to say. The room had two entrances, and she must have come through one. But Electra whispered hoarsely, ‘Mister, forget it. I’ll never tell you which.’

“Although I saw that she was shaking, I tried to pin this lady down. But Electra paced frenetically when I started to approach. She begged me to come no closer. Seeing she was agitated, I kept away. I was no stranger to uncertainty, but this time it had me beat. It looked like uncertainty was going to stick around here for a while.”

Quantum mechanics, counterintuitive as it is, fundamentally altered the way scientists view the world. Much of modern science evolved
from quantum mechanics: statistical mechanics, particle physics, chemistry, cosmology, molecular biology, evolutionary biology, and geology (through radioactive dating) were all either invented or revised as a result of its development. Many conveniences of the modern world, such as computers, DVD players, and digital cameras, wouldn’t be possible without the transistor and modern electronics, whose development relied on quantum phenomena.

I’m not sure I fully appreciated how weird quantum mechanics is when I first studied it in college. I learned the basic principles and could apply them in various contexts. But it wasn’t until I taught quantum mechanics many years later and carefully worked through quantum mechanical logic that I came to see just how fascinating it is. Although we can now teach quantum mechanics as part of the physics curriculum, it is nonetheless truly shocking.

The story of quantum mechanics beautifully exemplifies how science is supposed to evolve. Early quantum mechanics was done with a model building spirit—it addressed confusing observations even before anyone had formulated an underlying theory. Both experimental and theoretical advances happened fast and furiously. Physicists developed quantum theory to interpret experimental results that classical physics could not explain. And quantum theory, in turn, suggested further experiments with which to test hypotheses.