ASTR1010
- INTRODUCTION TO ASTRONOMY
The Solar System
February 25,
2009
The
Special Theory of Relativity led to some amazing conclusions about the behavior
of moving objects. It also ensured
that the laws of electromagnetism remained the same for any observer moving at
a constant velocity. Minkowski
realized that the relationships between the three spatial coordinates and time
could be expressed in a very elegant way if Nature was thought to be a
4-dimensional surface with time being the 4th coordinate. The combination of the three spatial
coordinates with time is called spacetime or the spacetime continuum. Mathematicians think of it as a
four-dimensional ŌsurfaceĶ called a manifold. With the Special Theory of
Relativity completed, Einstein could now focus on making the results of that
theory more general; in other words, what would the theory look like if the
condition of constant velocity were dropped? Could he derive a theory that would ensure that observers
would have the same fundamental laws regardless of their state of motion? In other words, all observers would have
the same laws even if they were accelerating with respect to the each
other. Because gravity produces
acceleration in objects, Einstein was going to apply his ideas from Special
Relativity to create a new theory of gravitation.
The General Theory of Relativity
(1915)
Newton
had realized that in his system, three-dimensional Space played a key role: it
was an absolute, real entity against which or with respect to which you could
determine the state of motion of an object. So, Space could affect matter. Einstein wondered if matter could affect Space. In addition, he realized that an
observer could not tell the difference between the effects of a gravitational
field and an accelerated motion at the same acceleration as that of the
gravitational field but with the acceleration produced by another mechanism
entirely. Think of the elevator
accelerating upwards in empty space that we discussed in class. This was the Principle of Equivalence
and it led Einstein to realize that gravity was not a force between the
gravitational mass quantity of two objects, but rather, a geometric property of
four-dimensional spacetime. He
postulated that matter could have an effect on spacetime equivalent to creating
a curvature on the surface of the four-dimensional manifold. Thus, he postulated that mass ŌwarpsĶ
space in particular ways (described by his equations). A moving particle entering this region
of space no longer moves in what we consider to be a straight line in flat
space. Instead, it responds to the
curvature of space by deflecting from the straight line, flat space path and
now moves in what looks like a curved path. This looks to us like one body has tugged on another, but
all that has happened is one body has made space curve and the other body as it
moved into this warped region responds to the curvature by changing its
trajectory. It turns out that the
body is still moving in a ŌstraightĶ path, but a straight path in curved spacetime
does not look like a straight line in our beloved flat space. Think of our discussion in class about
traveling to Europe by going northeast instead of east.
The solutions of the equations of General Relativity for the Sun and Mercury immediately solved the long-standing discrepancy about the precession of the perihelion of the planetÕs orbit. But there were startling predictions that the General Theory of Relativity made: the first one to be tested was the bending of light near a massive object. NewtonÕs Theory said that this was not possible because there is no gravitational force between two objects if one of them (or both of them) has no mass. Light has no mass (at least no gravitational mass) and so it could not respond to the gravitational mass of another object. But under EinsteinÕs Theory, both light and matter would respond to a curved space and so light should curve or bend or deflect near a massive object. This was tested during a solar eclipse in 1919 and EinsteinÕs General Theory came through with flying colors. General Relativity revolutionized our ideas about what gravity is, but there were even more startling predictions about Nature that the Theory would make.
The
first two successful predictions of EinsteinÕs Theory of General Relativity
were 1) getting the correct value for the precession of the perihelion of
MercuryÕs orbit (although this wasnÕt a prediction so much as a resolution of a
long-standing problem) and 2) the bending of light by a massive object. With these two successes and the fact
that NewtonÕs Laws of Motion were contained within the equation of General
Relativity as an approximation in those instances where the curvature of
spacetime was very small (like on the EarthÕs surface), the sheer elegance of
the General Theory of Relativity swept through the physical community and
became one of the two fundamental pillars of modern physics (the other is
Quantum Mechanics which describes physics at the atomic and subatomic scale and
was developed in the mid 1920Õs).
For the record and just so that you can say that you have seen it, here
is the fundamental equation in General Relativity:
The terms on the left describe the curvature of spacetime and the terms on the right describe the mass (really, the energy-momentum density) which produces the curvature.
There
were other tests of this theory, some of which couldnÕt be done for decades,
but for each test that has been devised, the theory has come through with
flying colors. The most important
tests after the perihelion of MercuryÕs orbit and the bending of light are:
3)
Gravitational Redshift – light emanating from a massive body has its
frequency increases as it moves through the severe curvature of
space-time. Similar to this is the
effect that slows clocks down in a region of curved space. In other words, in the presence of a
massive body, clocks run slower.
This is a different slow-down of time than the one predicted by the
Special Theory of Relativity.
4)
The decay of the orbits of two massive objects like the Binary Relativistic
Pulsars. This decay is produced by
gravitational radiation.
5)
The General Theory of Relativity allows for the existence of regions of
spacetime where the curvature is so pronounced that light itself cannot make
its way out of them. These regions
are called black holes and modern extra-solar system astronomy has confirmed
their existence. Such beasts
cannot exist under Newtonian Mechanics.
6)
This last one is not really a prediction because Einstein was basically too
afraid to make it. The Theory of
General Relativity, when applied to the Universe, says it should be expanding
(or contracting). When Einstein
saw this consequence to his ideas, he refused to believe it and fudged his
equations to remove the expansion.
In the 1910Õs nearly everyone believed the Universe was static. It wasnÕt until the late 1920Õs that
the expansion was discovered. When
Einstein found out about the expansion of the Universe, he realized he had made
his greatest blunder in not believing the original results that General
Relativity had given him.
Nearly
100 years after it was proposed, the General Theory of Relativity is our modern
theory of gravitation. It has
passed each test it has been subjected to and seems to be as robust as any
physical theory. However, just as
was the case for NewtonÕs Laws of Motion, someday, the General Theory of
Relativity may be superseded by a greater, more-encompassing theory. That would not minimize the greatness
of General Relativity. The mark of
a great theory is that some of its consequences let you discover things about
Nature you never would have suspected in the absence of that theory. Who would have ever thought of looking
for the bending of light during a Solar Eclipse if General Relativity hadnÕt
predicted that this should happen?
Whether EinsteinÕs Theory of General Relativity is still the correct
theory of gravitation one thousand years from now or not is irrelevant: It is
and always will be a great theory because of the insights on Nature it
provided.
Problems:
1) You should be able to
describe the experimental tests of the General Theory of Relativity.
2) What is the basic idea
behind General Relativity?