General Relativity

Einstein’s Theory of Gravity

Paul Woodward

Now we discuss some of the aspects of Einstein’s theory of gravity, which revises Newton’s theory a little.

We need to know something about this theory in order to make sense of the topics in cosmology that we willbe discussing for the remainder of this course.

We need Einstein’s theory of gravity to understand the expansion of the universe – the “big bang” theory.

We also need it to understand our present quest to identify “dark matter” and to develop a theory of the formation of galaxies.

Einstein’s theory of gravity is called the general theory of relativity.

Einstein generalized his special theory of relativity to account for the presence of masses in the world.

The generalization here is that instead of only considering observers in states of constant motion in straight lines, now Einstein considers observers whose velocities are changing as the result of acceleration.

[For lack of time, we will not discuss Einstein’s special theory. You have probably heard about it already.]

In our everyday experience, we know two kinds of acceleration.

The first is the acceleration of the earth’s gravity, which we feel when we stand on our feet, which then feel the weight of our bodies, or when we sink into a chair.

We don’t feel accelerated in either case, because we do not perceive that our velocity is changing. Nevertheless, we are definitely being acted upon by a force, no question about it.

The second kind of acceleration we are familiar with seems more deserving of the name.

This is the acceleration we experience when we push down on the accelerator pedal in our cars.

So long as we are not wimpy about this, we not only see our velocity changing by noting the increasingly more rapidly passing scenery, but we also feel the acceleration by being pressed back against the seat.

Einstein suggests that if we remove the evidence of the passing scenery, we cannot tell the difference between these two very different reasons for being pressed against a seat.

Think about an elevator (no scenery).

This Einstein called the principle of equivalence.

He claimed that it was the best idea of his life.

We will see that it has many very important consequences.

It is always possible by performing an experiment to discover that you are in an accelerated frame of reference.

It is also possible to know when you are freely falling, because you are weightless.

It is impossible to know whether you are hovering in a gravitational field or if you are simply accelerating far from any gravitating masses.

Unlike Maxwell’s assertion that the speed of light is the same to all observers, Einstein’s principle of equivalence is really not weird at all.

But Einstein, of course, did not stop there.

He went on to generalize the concept of a straight line.

Not a straight line in space, but a straight line in space-time.

In Galileo’s world, and also in the world of special relativity, a straight line is the path followed by an object moving at a constant velocity and not acted upon by any force.

This definition, of course, is circular, since when turned around it is also the definition of a constant velocity. At least for Galileo.

But then Galileo had no particular reason to think too deeply about what is straight and what is not.

Einstein decided that the generalization of the concept of constant velocity motion, without being subjected to forces, is motion of observers in free-fall.

The purpose of this generalization is to account for the presence and influence of masses in our world.

So now you and Jackie may still be in your space ships, but you are no longer out in deep space far from any other objects.

The idea is that if you don’t turn on you rocket motors, both you and Jackie will be freely falling and you will travel in “straight” lines – not straight lines in space, but straight lines in space-time.

The forces exerted on you by masses through their gravity will show up through a redefinition of the concept of a straight line, that is, through a redefinition of space itself.

This concept lies at the heart of Einstein’s theory. Masses distort space and time, so that they can be considered as the “stuff” that ultimately defines space and time. Sort of.

We will see in lectures to come where this concept leads us.

Einstein decided that the generalization of the concept of constant velocity motion, without being subjected to forces, is motion of observers in free-fall.

In Einstein’s view, masses distort space-time, and you, while freely falling, just travel along straight lines in this distorted context.

This is, so far as we know, a perfectly OK way of thinking about gravity, but other ways of thinking about it are probably OK too.

But no one else has thought as deeply about gravity as Einstein using any other conceptual framework. (Others have tried, and the thoughts, for example, of Richard Feynman, a Nobel prize winner, on this subject were published a few years ago, but they did not get far enough to be called a theory.)

The distortion Einstein is talking about is in a 4-dimensional space, and to “see” this distortion properly, we would have to “embed” this 4-D space-time in a 5-dimensional space that was not distorted.

No one, to my knowledge, has ever achieved such “sight,” or perhaps we should say “insight.” Probably not Einstein either.

But we can get an idea by projecting pictures onto 2-D pieces of paper, exploiting the fact that our brains know how to build 3-D perceptions from 2-D perspective drawings.

So here’s representation #1.

We draw a flat space as a flat rubber trampoline, and we illustrate the distortion of this flat space by masses by showing them sinking down in it and making the surrounding regions of the trampoline curve downward.

In this analogy, bigger masses create greater distortions of the surrounding space, and this is just what Einstein’s view of gravity says they should do.

The analogy works quite well, even for orbits of other small masses, as shown on the next slide.

Just as Kepler says for planets orbiting the sun, a marble rolling on the surface of the distorted trampoline will roll faster when near the masses that distort its surface.

You can verify this at the Minnesota Science Museum, where a set-up just like on the next slide is on display.

The diagrams on the following slide illustrate this concept of the distortion of space by a concentration of mass in a slightly different way.

At the left, we see images similar to the one taken from your textbook that appears on the previous slide.

To the right of each of these images is a view of the same situation from a vantage point directly above the object in the third dimension that we are using to make its distortion of the surrounding space visible to us.

These views on the right give us the classic pictures that we would plot by applying Newton’s theory of gravity, much as he did in his lifetime.

This slide is to illustrate how Einstein was able to obtain precisely the same result as Newton, in this simple case, but from a rather more peculiar way of thinking about the problem.

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Orbits about a massive object can be viewed as particles simply following the straightest possible trajectories in a curved space.

The curvature of this space can be measured by comparing the length of a radial line with the circumference of a circular orbit.

In Einstein’s view, the large mass has distorted the space around it, so that freely falling particles, particles on which no forces additional to gravity are acting, follow paths that are curved by the distortion of the space they are moving through.

For Einstein, if we can calculate the curvature of space-time everywhere from a knowledge of the positions of the gravitating masses in the space-time, then we can find the paths of all freely falling particles. Such paths, for example, are the orbits of the planets or of space craft in our solar system.

For Newton, we imagine that space is neatly ruled out with straight lines, with fiducial markings, like on graph paper. From the positions of the masses, we can use his gravitational force law and add up all the gravitational tugs on a freely falling particle and, with some calculus, then compute its path, or orbit.

Both approaches get the same result in all circumstances except for some rather exotic cases in which Einstein, so far, seems to be right and Newton wrong.

Einstein says that it is possible to tell if the space around you is distorted. All you need to do is to rule out in the best way you can three straight lines that meet at the 3 corners of a triangle. Then you get out your protractor and measure the three inner angles and add them up. If space is flat, the sum will be 180°. If not, it will be more or less than that. The distortion of space by the mass of the earth is extremely small, so here in the classroom, you will get 180°, but right up close to a pulsar, for example, you would not.

This concept is not really so outrageously peculiar. Think about the two-dimensional space that is the surface of the earth. If we draw triangles on this surface using straight lines that, of course, remain on the surface, then those straight lines will be “great circles.” These give the shortest distances between two points on the surface and are taken by airplanes (which is why the shortest path from here to Tokyo will nearly take you up to Alaska). For a big triangle, like Minneapolis to Tokyo to Cancun and back, we will not get 180°.

In the example using the earth’s surface as our space, we said the straight lines would be great circles, which are the paths that airplanes take. These are just what Einstein would choose, because, if we get rid of the air resistance and get rid of all the mountains and valleys, space ships would orbit (in free fall) precisely along these paths just above the earth’s surface. So these are indeed Einstein’s straight lines.

You can take an especially simple example by building the triangle using the north pole and two cities on the equator, say someplace in Ecuador and someplace in equatorial Africa. The two angles at the equatorial cities will be precisely 90° each, and the angle at the pole will certainly be greater than zero, so Einstein’s way (actually Riemann’s way, to be more precise, since he figured this out before Einstein) of determining the curvature of the space definitely works.

This trampoline analogy also works as a way to understand the effects of compressing a given mass into smaller and smaller volumes of space.

As, for example, we make the sun progressively more compact, going from main sequence star to white dwarf, or even to neutron star or black hole, the outer region is unaffected, but the inner region becomes ever more greatly distorted.

Let’s look again at the first diagram with the orbits, because it helps to bring out another important point.

Notice that two of the orbits cross at two points.

If these orbits are “straight” lines, how can they intersect with each other twice?

The resolution of this question comes from recognizing that in this analogy we are projecting things from our 4-D space-time onto a 2-D image.

In space-time, the orbits do not intersect more than once, since it takes different amounts of time to go along the different orbits between the 2 intersection points.

In space time, a circular orbit maps into something that we might better visualize as a helix, although even then it is hard to think of this as a straight line.

It is possible to have many straight lines passing through any particular point in space-time, and these would be the paths of freely-falling objects that have different velocities (both speeds and directions).

But the paths of light rays through this point are special, because light goes at the fastest possible speed.

The trampoline analogy shows how even light will be deflected as it passes by a condensed mass, like the sun or like a galaxy.

The following diagrams show that not only can light be deflected, but it can reach an observer along multiple possible paths simultaneously, producing what is called an Einstein ring.

Both the deflection of light by the sun and the phenomenon of the Einstein ring have been observed, so this aspect of the theory checks out.

Above are shown a couple of actual Einstein rings.

An entire cluster of galaxies acts as a gravitational lens. The combined

gravitational field of the yellow elliptical and spiral galaxies has dramatically distorted the image of a more distant

blue galaxy into 5 separate parts. The galaxy is about twice as far away as the

cluster CL0024+1654. The galaxy’s spiral shape has been considerably

changed by the lensing effect, but several details are still visible. Its clumpiness,

for instance, indicates its youth and active star formation.

Colley et al. & NASA.

Now let’s think about a further consequence of the Principle of Equivalence: clocks run slower in gravitational fields.

A mysterious arc of light found behind a distant cluster of galaxies has turned out to be the biggest, brightest, and hottest star-forming region ever seen in space. The so-called Lynx arc is 1 million times brighter than the well-known Orion Nebula, a nearby prototypical star-birth region visible with small telescopes. The newly identified super-cluster contains a million blue-white stars that are twice as hot as similar stars in our Milky Way galaxy. It is a rarely seen example of the early days of the universe where furious firestorms of star birth blazed across the skies. The spectacular cluster's opulence is dimmed when seen from Earth only because it is 12 billion light-years away.

Text from Space Telescope Science Institute via European Space Agency.

The Lynx Arc, shown here, is thought to be the brightest star cluster ever observed, seen with the aid of this cosmic telescope.

This is the artist’s concept of what the Lynx Arc really looks like. Wow!

The idea is that the first stars were much, much more massive than stars that form today.

An international team of astronomers may have set a new record in discovering what is the most distant known galaxy in the universe. Located an estimated 13 billion light-years away, the object is being viewed at a time only 750 million years after the big bang, when the universe was barely 5 percent of its current age. The primeval galaxy was identified by combining the power of NASA's Hubble Space Telescope and CARA's W. M. Keck Telescopes on Mauna Kea in Hawaii. These great observatories got a boost from the added magnification of a natural "cosmic gravitational lens" in space that further amplifies the brightness of the distant object.

Text from Space Telescope Science Institute via European Space Agency.

The distant galaxy certainly doesn’t look like much in this picture.

Where is that artist who draws those fabulous concept’s when you need her?

Because you are at the front of the space ship, the acceleration is always taking you away from the point at which Jackie’s light flashes originate, thus the light from each flash takes a little longer to reach you than it would in unaccelerated motion. Therefore Jackie’s clock will seem to be running slower than yours.

This prediction that clocks in stronger gravitational fields run slower has been tested, and it checks out.

In the two situations shown here, Einstein’s principle of equivalence implies that clocks (andhence timeitself) runsslower forclocks instrongergravitationalfields.

We talked about black holes before during our discussion of binary star systems (well, at least we intended to do so).

We will just recapitulate some of that discussion here.

Later, in discussing the expanding universe we will encounter further ramifications of Einstein’s theory of gravity, his general theory of relativity.

We can think of a black hole as mass that is so concentrated that gravity is so strong near it that even light is deflected so greatly that it orbits the mass concentration rather like the way the planets orbit the Sun.

A black hole has what we call an event horizon, beyond which events are unobservable to us.

Inside the event horizon, gravity is so strong that the escape velocity exceeds the speed of light.

Even light cannot escape from within the event horizon.

Light loses energy upon working its way out from a strong gravitational field. This energy loss takes the form of a redshift, and we call it the gravitational redshift. It is related to the slower rate at which clocks run in strong gravitational fields.

Light emitted directly outward from the location of the event horizon is infinitely redshifted, that is, it reaches locations far from the black hole redshifted to infinite wavelength (zero energy), and is thus invisible.

Fig. 17.12 (a) A 2-D representation of “flat” space-time. The circumference of each circle is 2π times its radius.

(b) A 2-D representation of the “curved” space-time around a blackhole. The black hole’s mass distorts space-time, making theradial distance between two circles larger than it would be in a“flat” space-time.

3 Solar mass black hole with photon sphere, event horizon, Schwarzschild radius, incoming light.

Black holes may sound like very violent things, but they need not be.

It has been popular for many years, although it is less popular today, to think of our universe as closed. This would mean that if we sent off a beam of light in any direction, it could not escape from the universe, no matter how long it traveled. We might fire off such a beam of light, it might go all the way around the universe, and ultimately come back to us from the opposite direction to that in which we launched it. Such a closed universe would be very much like the inside of a black hole. The round-about travel of the light beam is like the light that orbits a black hole. In fact, we could live quite happily inside a black hole, if it were large enough so that its tides did not tear us apart and the orbits of its photons were so large that they would appear nearly straight and infinite to our human scale of perception.

Your textbook has a whole section about people in rocket ships orbiting a black hole and probing it in various ways.

This is well worth reading, but there is no point to recap it here.

Fig. 17.13 Time runs more slowly on the clock nearer to the black hole, and gravitational redshift makes its glowing blue numerals

appear red from your spaceship orbiting the black hole further out.

At the left is an image, taken in visual light, of the region of the Galactic center, and at the right is a radio image, which is enlarged on the next slide.

Most of the circular features in this view of the central region of our galaxy are supernova remnants.

The density of stars is very high in this region, so it is not surprising that some of them have exploded recently enough that their remnants are still very noticeable.

Radio close-up of the central region. We see a “hypernova” remnant toward the left, and a spiral of gas centered on the very center of the Galaxy.

Still closer view of the central region. We see a “hypernova” remnant toward the left, and a spiral of gas centered on the center.

Stars in the very central region orbit the center, and Kepler’s laws tell us how massive the region they orbit must be: about 2.6 million solar masses.

Stars in the very central region orbit the center, and Kepler’s laws tell us how massive the region they orbit must be: about 2.6 million solar masses.

Stars in the very central region orbit the center, and Kepler’s laws tell us how massive the region they orbit must be:about 2.6 million solar masses.

With high frequency radio observations, we can measure the size of the central object, or at least a size that it cannot exceed.

With high frequency radio observations, we can measure the size of the central object, or at least a size that it cannot exceed. It turns out to be no larger than the diameter of the earth’s orbit about the sun.

Einstein’s view of gravity in terms of a distortion of space-time that is experienced by any particles travelling through has some interesting consequences.

First, it follows that if the masses in the universe move, the distortion of space-time must change accordingly, but this change cannot happen everywhere in the universe instantly.

This argument implies that a wave of space-time distortion change must propagate outward (at the speed of light, of course, what else?) from a moving mass.

These propagating modifications of the distortion of space-time are called gravitational waves.

An apparently somewhat less than honest professor at the University of Maryland claimed to have observed such waves in the 1960s (what was he smoking?).

However, since the 60s, several very careful, expensive, and, until just a few years ago unsuccessful, experiments have looked for gravitational waves.

The latest is called LIGO, a set of gravitational wave observatories in unlikely states (Washington and Louisiana) costing the taxpayers some 60 million dollars.

LIGO, after being increased in sensitivity by about an order of magnitude over its original design, observed its first gravitational waves in 2015.

The problem with gravitational waves is that they are weak. Like, really weak.

You can sense their passage by observing a couple of incredibly massive aluminum cylinders moving toward and away from each other by a microscopic, laser-sensed amount. The trouble is that there are so many other reasons why such cylinders might do this.

Indirect evidence of gravitational waves is more plentiful.

In the 1970s, a search for pulsars (rotating neutron stars) came up with a binary pulsar. This is 2 pulsars in a close orbit about each other.

The gravitational force between these 2 collapsed objects is so strong that their orbiting must send out gravitational waves of significant strength (we think we saw this kind of signal directly a few years ago as 2 neutron stars merged).

The emission of the gravitational waves removes orbital energy from the system, so that the pulsars spiral in toward each other.

This orbital decay has been observed and checks out with the predicted amount of gravitational wave energy loss. The guy who worked this out won a prize for this.

Indirect evidence of gravitational waves, however, does exist

Black Holes dot org has great movies. check it out.