- Special Relativity
A brief introduction to special relativity and general relativity is needed before vanishing into the black hole.Special relativity was presented by Albert Einstein (1879--1955) in 1905. Certain aspects of special relativity had already discussed for some years, but everything was very cloudy.
Einstein in 1905???.
This was when he was a patent office clerk and discovering special relativity
Credit: ?; download site?
This picture was obviously taken in the first decade of the 20th century and by U.S. copyright law is now out of copyright. According to the informative, but not authoritative, source WebMuseum, Paris copyright in all other jurisdictions would have expired if the holder died more than 70 years ago.
Einstein developed special relativity starting from two basic postulates:
-
The Relativity Postulate:
The laws of physics should
be the same or, more exactly, have the same formulation
in all
inertial frames: they should be frame-invariant
Actually, many physicists accepted this postulate already in 1905.
But the problem was that some physical laws did NOT obey it.
Galilean transformations (which we will not detail here) are the classical way of changing from one frame of reference to another: they were accepted almost without question from the time of Newton.
Newtonian physics is frame-invariant under the Galilean transformations, but Maxwellian electromagnetism was NOT.
-
[
Maxwellian electromagnetism is just the ordinary
theory of electromagnetism which has been found extremely
adequate for macroscopic electric and magnetic phenomena.
It's a true approximate theory in the instructor's jargon.]
Most people then guessed it was Maxwellian electromagnetism that was wrong.
Einstein reasoned that it was the Galilean transformations and Newtonian physics that were wrong and Maxwellian electromagnetism was right---and he was right.
-
The Speed of Light Postulate: The vacuum speed of light
is the same for all
inertial frame
observers regardless of how they are moving.
This postulate upsets our usual ideas of relative motion.
Relative motion and the vacuum speed of light.
Something has to give if we accept this postulate.
Among other things we discover that MASS, LENGTH, TIME, and SIMULTANEITY become FRAME-DEPENDENT quantities.
But we will just discuss a few salient features.
- The weirdnesses of
special relativity
are pretty much unnoticeable at speeds much less than the speed of light,
but increase in size as speeds increase.
This is why we don't ordinarily notice SPECIAL RELATIVISTIC EFFECTS.
But they can be measured by precise measurements---and they have.
Special relativity is a very well verified theory.
Within its realm of validity (i.e., where you don't have to deal with strong gravity), it has always been verified.
There are no doubts that it is a true approximate theory.
- The
vacuum speed of light
is the ultimate physical speed, except
for some
quantum mechanical effects which we will avoid like the plague---though
they are very important.
No physical effect or information can be can be transferred faster than vacuum speed of light.
You can have faster geometrical speeds.
For example, say there are two rockets moving in opposite directions from you both with speed 0.75c.
You judge their relative speed to be 1.5c---and you are right.
But no physical effect or information is traveling at that speed and observers on the rockets would measure their relative speed at a bit less than c. Various SPECIAL RELATIVISTIC EFFECTS would always make it work out that way.
- The
Galilean transformations and
Newtonian
physics
are only approximations valid at low speeds, weak gravity, and
in the macroscopic realm.
The more correct transformations are Lorentz transformations.
Maxwellian electromagnetism is frame-invariant under the Lorentz transformations, and so already satisfies the relativity postulate.
Newtonian physics had to be modified to be correct in the framework of special relativity.
- Length is frame-dependent.
The length of a moving object is shorter along the direction of motion than that of one a rest.
The effect grows as the relative speed grows.
Now motion is, of course, relative in special relativity, and so two observers in relatively moving frames would measure meter sticks in the other observer's frame as less than one meter.
This is paradoxical.
But, in fact, the paradox is resolved.
A length measurement is one where the ends of an object are located SIMULTANEOUSLY.
But TIME and SIMULTANEITY are frame-dependent, and so a SIMULTANEOUS measurement of ends in one frame is not SIMULTANEOUS in the other.
This is not a full explanation, but it leads to one.
- Then there is
time dilation
which is mnemonicked---to verb a word---by ``moving clocks run slow."
All clocks observed in moving frames run slow. Time is literally running slower in a moving frame.
The effect grows as the relative speed grows.
Again motion is relative in special relativity, and so two observers in relatively moving frames would measure clocks in the other observer's frame as running slow.
This is paradoxical.
But, in fact, the paradox is resolved.
One has to take into account all SPECIAL RELATIVISTIC EFFECTS.
This is not a full explanation, but it leads to one.
- The
twins paradox.
Take a pair of twins. One stays on Earth and the other is sent on a trip at a relativistic speed and returns.
The tripping twin's clocks have run slow and he/she is less aged than the stay-at-home twin.
-
Question: In this case, there is a definite asymmetry, but
how can this be if all motion is relative?
-
Special relativity
is wrong.
- One actually has to be a bit more careful in what one says. Not
all motion is relative.
Accelerated and unaccelerated motions are not the same.
The tripping twin had to have been accelerated. Thus his/her motion was physically distinct from the stay-at-home twin.
Answer 2 is right.Special relativity, like Newtonian physics, does distinguish unaccelerated and accelerated motions relative to inertial frames.
Two atomic clocks are synchronized and one is flown aroun the world on a jet. The jet clock shows less time has passed when the two clocks are compared.
This experiment has been done many times now and special relativity has always been verified to within experimental accuracy (HRW-928).
The twins paradox effect and other similar effects including some from general relativity must be accounted for by the Global Positioning System (GPS). Special relativity and general relativity do play roles in modern everyday life even if you don't know it.
-
Special relativity
is wrong.
- The
Einstein equation E=mc**2 falls rather naturally out of the
development of
special relativity.
Remember this equation means two things:
- All energy has mass: i.e., resistance to acceleration and
is affected by gravity.
-
Rest mass
(i.e., the mass of an object at rest in a particular
frame) is a form of energy (or has an associated energy).
Like all energy, rest-mass energy can be transformed into or created from other forms of energy.
Nothing with rest mass can be accelerated to the vacuum speed of light because, it turns out, as an object moves faster, it has more mass-energy and thus more resistance to acceleration.
The mass-energy of an object with rest mass goes to infinity as the speed goes to the vacuum speed of light, and so no such object reaches the vacuum speed of light. Modern particle accelerators demonstrate this. No matter what they do particles never reach the vacuum speed of light and no one has ever expected that they would.
- All energy has mass: i.e., resistance to acceleration and
is affected by gravity.
- Because
special relativity
shows that time and space are connected, we use the term
spacetime when we
want to discuss these 4 dimensions of reality
at the same time.
- General Relativity
General relativity (GR) was a generalization of special relativity that Einstein developed to give a more satisfactory treatment of gravity and accelerations under gravity.One point about Newtonian gravity that Einstein believed had to be wrong was that changes in Newtonian gravitational field propagate instantly through space and that violates the nature of special relativity in which the vacuum speed of light is the highest physical speed.
Another point was that the gravitational field is a form of energy, but special relativity that means it has mass, and therefore should exert gravity on itself (ST-107). This very complicating effect does not appear in Newtonian gravity.
There were other puzzles, of course, that led Einstein on.
General relativity was very hard to develop and understand.
Einstein had to go on a long excursion into very difficult math: tensors and differential geometry.
But he eventually touched down in 1915 ( St. Andrews' Einstein biography) with his complete theory summarized in the Einstein field equations collectively represented by
8*pi*G G_{ik}= ___________ T_{ik} which is a tensor equation. c**4 c is the vacuum speed of light, G is the gravitational constant from Newton's gravity law, G_{ik} is a tensor describing the geometry of space, T_{ik} is a tensor describing energy and momentum. Reference: CL-9). We will not worry what tensors are---but one thing they are is compact way of writing something complicated. The field equations are analogous to Newton's 2nd law (F=ma) with the force being gravity.What we need to know is that in general relativity mass-energy determines the geometry of spacetime.And the geometry of spacetime tells mass-energy how to move.
We will discuss the geometry aspect in the next section Geometry.
What the aforesaid discription implies is that problems in general relativity are CIRCULAR!
This makes it very hard to find solutions for physical systems in general relativity: e.g., how do to gravitating bodies interact?
Systems with exact analytic solutions (i.e., solutions that can be written down in formulae) are very rare: I recall that there are only 6????. I don't think Einstein found any of those solutions himself.
For all other systems, one must make approximations or grind solutions out by a computer.
-
Question: If
general relativity
is so hard to understand and use, why do we use it?
- We don't use it. Mathematicians just like to exercise their brains with it.
- We like pain.
- It gives physical predictions that are right where Newtonian physics gives wrong predictions.
Answer 3 is right.That was an easy one. But you could make a case for all of the above.
It's amazing actually given that Einstein winged it up with very little experimental guidance beyond well known results fully explained by Newtonian physics.
Let us just briefly review the verified predictions of general relativity:
- First, one should say that the low-velocity, weak-gravity
limit of
general relativity
is
Newtonian physics
(ABS-345).
The vanishing gravity limit is
special relativity.
That these things should be so was built into general relativity by Einstein from the beginning since Newtonian physics and special relativity. works excellently well in the right limits.
Newtonian physics and special relativity are true approximate theories.
- In the 19th century, there was a problem in understanding the
orbit of
Mercury.
General relativity accounts for the extra perihelion shift of Mercury
(FK-537;
ABS-209,213).
The resolution of the problem with the perihelion shift of Mercury, was the first observational test passed by general relativity.
For many years years---up to the 1960s???---it was probably the strongest evidence for general relativity.
-
[Incidently, in the 19th century people tried to solve the
extra perihelion shift by postulating a planet inward of
Mercury
that gave an extra gravitational perturbation.
This hypothetical planet was never found and rendered otiose by the general relativistic explanation.
The planet has continued to live on in myth: it was named Vulcan (ABS-200).]
- Gravity bends light beams.
It takes strong gravity to make a noticeable bending: e.g., the gravity near a star.
The light-bending effect is not predicted by pure Newtonian physics although various extensions can make some predictions---not verified ones though.
Gravity bends light
(FK-537;
ABS-219).
I believe Einstein himself???? made the first prediction of this phenomenon.
For light-bending by the Sun, the prediction originally could only be verified in the visual band during total solar eclipses.
The bending was first observed in 1919 and the announcement was the event that made Einstein world famous.
Actually, the measurements in 1919 and were rather inaccurate, and so although the bending was confirmed in 1919, exact numerical prediction was not confirmed until the 1960s??? (ABS-219).
Nowadays the BENDING OF LIGHT as predicted by general relativity has been well verified.
Since light can be bent by gravity, gravitational sources can cause gravitational lensing (FK-600--601; CK-406--407,422--423; HI-432,450,451).
In gravitational lensing light from distant objects is focused into BRIGHTENED IMAGES and/or ARCS.
Nowadays gravitational lensing is a very important tool in determining the masses of galaxies and clusters of galaxies, looking for star-size gravity sources (e.g., MACHOS which are discussed in IAWL Lecture 27: The Milky Way). One sees the gravitational lensing and infers the mass of the lens.
The brightening effect of gravitational lensing allows one to observe objects that are too remote to be seen otherwise. This effect is becoming an important tool in studying the evolution of the observable universe.
- Clocks run slow in
gravitational wells.
More precisely, the deeper in a
gravitational well
the slower clocks run: i.e., the slower time passes.
A gravitational well is any localized source of gravity: e.g., a planet, star, black hole, etc.
This has been experimentally verified with terrestrial clocks at different altitudes (HRW-928).
- Gravitational wavelength shifts.
Light beaming from a gravitational well is redshifted and loses energy.
Light beaming into a gravitational well is bueshifted and gains energy.
This effect is usually called just gravitational redshift even though blueshifts happen too: I guess the blueshifts are just thought of as negative redshifts.
The effect is similar to the Doppler effect, but it is not that.
The gravitational redshift was first verified terrestrially in 1960 using a 72-foot shaft (i.e., a shaft of about 20 meters) and the Moessbauer effect for gamma-rays (ABS-140; FK-538).
-
General relativity
predicts that there should be
gravitational waves or radiation traveling at the
vacuum speed of light (in a vacuum) produced by accelerated
mass-energy
(ABS-317,326;
FK-538,539;
CM-348)
somewhat analogous to
electromagnetic waves or radiation.
These waves are very weak and have NEVER been directly detected, there is a major project LIGO (CM-348) dedicated to their detection from astrophysical sources which could include the collapses of star cores in supernovae to black holes, and coalescing binary compact objects.
Only sources like these might produce waves strong enough to be detected with the current-state-of-the-art detectors.
Maybe any day LIGO will report that they have detected gravitational waves---I'm not holding my breath.
There is indirect evidence for gravitational waves.
A system called the BINARY PULSAR discovered in 1974 consisting of two neutron stars, one of them a pulsar, has slowly decaying orbit with the neutron stars spiraling inward toward each other (ST-479ff).
Because the system does contain a pulsar very precise measurements can be made of the orbital decay.
The loss of gravitational energy from the BINARY PULSAR is exactly in accord with the energy that should be radiated away in the form of gravitational waves.
-
General relativity
along with certain assumptions predicted the
expansion of the universe before this was discovered observationally.
See the discussion in IAWL Lecture 31: Cosmology.
- Black holes
are a prediction of
general relativity
and, as we will see in the section
Do Black Holes Exist?,
there is strong evidence for their existence.
But in the opinion of the instructor and others, the evidence is NOT yet conclusive.
But, in fact, it is NOT believed to be a fundamental theory.
This is because it is NOT consistent with quantum mechanics and quantum mechanics is very well verified and very strongly believed to be on the right path to true fundamental physics.
People do not expect general relativity to hold in the microscopic realm of quantum mechanics.
It is believed that there must be a quantum gravity theory that applies in microscopic and super-dense conditions and that has a limiting form that closely approximates general relativity in those realms where general relativity is well verified.
This quantum gravity theory probably has implications for black hole theory as we discuss below in the sections Schwarzschild Black Holes and Kerr Black Holes.
Besides the quantum mechanical problem with general relativity, it is possible that both general relativity and newtonian gravity are wrong in the realm of very low accelerations: i.e., below 10**(-10) m/s**2.
Such low accelerations are unmeasurable in the laboratory, but occur in the outer parts of galaxies and in galaxy clusters, where discrepancies in motions have conventionally been ascribed to dark matter.
The dark matter is matter only observed through its gravitational effect up to now. Dark matter is discussed in IAWL Lecture 27: The Milky Way.
However, it is possible that there is no dark matter and that both general relativity and Newtonian gravity need correction for very small accelerations.
The idea for this corrected gravity law has conventionally been called MOND (for MOdified Newtonian Dynamics), and has been around since 1983.
Until recently MOND has been regarded as interesting idea that had to be wrong because it did not seem possible to make a MOND theory consistent with the relativity postulate.
But recently, a relativistic MOND theory has been developed ( Bekenstein, J. D. 2004, An Alternative to the Dark Matter Paradigm: Relativistic MOND Gravitation, astro-ph/0412652).
This is a remarkable development since it might cause a revolution in our ideas about large-scale structure and cosmology. It is still very early days, and whether such a revolution will happen is unknown.
We won't say much more about MOND theory and relativistic MOND theory since they are rather beyond the scope of yours truly.
But we will mention MOND again in IAWL Lecture 27: The Milky Way and in IAWL Lecture 31: Cosmology.
- Geometry
As we remarked above in the section General Relativity, in general relativity mass-energy determines the geometry of spacetime and the geometry of spacetime tells mass-energy how to move.We will not do a full exposition of what this means, but we can get a bit of insight.
Everyday life we are used to thinking of space as exhibiting Euclidean or flat geometry which is just the geometry we learn in high school: the one in which for example:
- parallel lines never meet,
- the sum of the angles of a triangle add up to 180 degrees, and
- space can be mapped by Cartesian coordinates.
Cartesian coordinates in 2 and 3 dimensions.
But, in fact, we are familiar with non-Euclidean or curved spaces too.
The surface of a sphere is a curved space.
The surface of a sphere: a curved 2-dimensional space.
We have a bit of difficulty picturing 3-dimensional curved spaces.
-
[But actually do we even really picture 3-dimensional flat spaces I
wonder.
What the eye shows us is a 2-dimensional image of the world. Experience tells how things are arranged in 3 dimensions and how they would look from different perspectives.
Maybe if we lived in a curved 3-dimensional space we would just get used to it similarly.]
Consider a 4-dimensional sphere in 4-dimensional flat space. We can't picture it, but it's equation is
w**2 + x**2 + y**2 + z**2 = r**2 , where w, x, y, and z are the 4 coordinates and r is the radius.The surface of this hypersphere is a curved 3-dimensional space: it is a finite, but unbounded space.More complicated curved spaces arise in solutions to the general relativity field equations. These, of course, are curved physical spaces.
A given mass-energy distribution gives rise to some curved space. In the absence of any mass-energy one has a flat 3-dimensional flat space and general relativity reduces to special relativity.
Since general relativity has been shown to be an accurate theory for many effects, we believe real space is curved in a complicated way due to the complicated mass-energy present in real. In most regions, the curvature is too small to notice---a microbe living on a beach ball thinks its space is flat---and over short distances it is correct.
The path of test particle particle of vanishingly small mass-energy acting under gravity is a geodesic (i.e., shortest path in a special sense that we will not fully explain here) in curved spacetime caused by mass-energy (ST-114). In general relativity the force of gravity is replaced by the effect of the curvature of spacetime.
-
[A geodesic
on a sphere is a
great circle:
i.e., a circle that cuts a sphere in half.
Going in one direction along a
great circle
is the shortest distance on the curved surface of the sphere between
two points.
This is why flights Europe are often over Greenland.]
If the moving object does not have vanishingly small mass-energy, it will affect the geometry of spacetime which will affect the motion which ... and so. This is what makes general relativity circular and hard to solve.
Really we should say curved spacetime in the above discussion, not just curved space since time is involved too---but again we will NOT go into what that really means---yours truly isn't all the that sharp on it anyway.
- History of the Idea of Black Holes
In pure Newtonian physics it is NOT absolutely clear how gravity affects light.-
[By saying ``in pure
Newtonian physics''
we mean imagine that
Newtonian physics
was a fundamentally true theory and see what things that implies.]
But on the other hand, Newton himself regarded light as made of particles---classical particles, not like modern photons---and this idea was current throughout the 18th century.
Thus, at least some people regarded light as possibly having some mass or being affected by Newtonian gravity.
If so, then it was possible to imagine an object sufficiently massive and compact that light could not escape from its intense gravity in Newtonian physics.
The equation for escape velocity in Newtonian physics is (without derivation)
v_escape = sqrt{2GM/r} , where G is the gravitational constant, M is the mass of a spherically symmetric gravity source that has radius =< r, and r is the radius from which the escaping object starts. The escaping object ideally has vanishingly small mass: i.e., it is a test particle. The escape is to infinity: the particle will not return. For example the escape velocity from the Earth's surface is 11.2 km/s (HRW-305). The direction of escape makes no difference provided only gravity acts: i.e., the test particle does NOT hit a planet or have to contend with an atmosphere. The formula is of high accuracy if the mass of the test particle is much less than M. Say the test particle has the speed of light initially. If v_escape = sqrt{2GM/r} > c, there will be no escape. Thus if r < 2GM/c**2 for the particle, there will be no escape. Anticipating, we call 2GM/c**2 the Schwarzschild radius R_sch and the spherical surface it defines the event horizon: the surface from which light cannot escape. Having an event horizon is the defining characteristic of a black hole in whatever theory of physics one uses. Because light can't escape from event horizon, black holes are very, very black: there is no light at all coming from within the event horizon. As early as 1795, Pierre-Simon Laplace (1749--1827) considered black holes in pure Newtonian physics, but NOT using the term black hole: but before 1968 no one used it.Newtonian black holes attracted little interest, because even their existence as theoretical objects was NOT certain and certainly no real object seemed to correspond to them.After Einstein published his general relativity field equations in 1915, Karl Schwarzschild (1873--1916) quickly found an exact analytical solution for spacetime OUTSIDE of a spherically symmetric mass distribution.
Schwarzschild at workCredit: early 20th century photographer; download site: St. Andrews Archive: Karl Schwarzshild.
Schwarzschild's solution is the famous Schwarzschild solution.
Schwarzschild found it while serving at the Russian front: he was soon invalided home to die.
The Schwarzschild solution is very important because the behaviors of objects in vicinity of spherically symmetric mass distributions is very important in astrophysics: e.g., planets around the Sun (ABS-194).
The Schwarzschild solution does NOT apply inside the spherically symmetric mass distributions: e.g., inside the Sun.
Schwarzschild noted that there was a special length scale in the Schwarzschild solution which we now call the Schwarzschild radius:
R_sch = 2GM/c**2 , which first came out of Newtonian physics as noted above.Funny things would happen if a mass were compacted to within its Schwarzschild radius: i.e., within its event horizon as we now call it: i.e., the object became what we now call a black hole.Schwarzschild himself thought this aspect of the Schwarzschild solution was physically meaningless.
J. Robert Oppenheimer (1904--1967) and H. Synder in 1939 seem to be the first to seriously consider black holes (without using that term) and consider how a star-like object could collapse to one (ST-338).
There was not much interest in black holes until the 1960s when the discovery of quasars and pulsars (which are radio-pulse emitting neutron stars) forced people to consider seriously the existence of exotic compact objects (ST-338).
John A. Wheeler (1911--), one of the last surviving scientists to work with Einstein, finally coined the term black hole in 1968 (ST-338) by which time it was badly needed.
Black holes as they are now understood theoretically are very simple objects.
There is a famous aphorism of Wheeler's ``black holes have no hair'' which just means they do NOT have a lot of complicated features.
If one can ignore perturbations from other masses, then aside from its location in spacetime a black hole is fully specified by just three parameters: mass, angular momentum (a measure of rotation), and net charge.
There are three kinds of ideal black holes (i.e., ones where you can ignore perturbations):
-
Schwarzschild black holes:
These have zero angular momentum and zero net charge.
These are the black holes that follow from the Schwarzschild solution.
Since most objects in the universe are rotating with respect to inertial frames, exact Schwarzschild black holes are unlikely to exist, but low angular-momentum black holes probably approximate Schwarzschild black holes.
-
Kerr black holes:
These have non-zero angular momentum, but zero net charge.
New Zealander Roy Kerr (1934--) discovered the Kerr solution for the general relativity field equations for the spacetime outside a rotating spherically-symmetric body in 1963 (ABS-237).
The Kerr black hole emerged as one of the results if the rotating mass was compacted to within the Kerr-Schwarzshild radius which is Kerr generalization of the Schwarzshild radius (ABS-265).
Kerr black holes are the most likely black holes to be realized in nature since almost all objects in space are rotating to some degree.
-
Kerr-Newman black holes:
These have non-zero angular momentum and net charge.
(ST-357).
Macroscopic bodies in the universe are nearly neutral because any charge imbalance quickly attracts neutralizing charge.
Thus black holes that show significant Kerr-Newman black hole behavior are probably not likely exist.
Kerr-Newman black hole are theoretically interesting, but we will not consider them further.
- Schwarzschild Black Holes
Recall Schwarzschild black holes have zero angular momentum and zero net charge.Recall they have the defining property of a black hole is that it have an event horizon.
For Schwarzschild black holes, the event horizon is the the surface defined by the Schwarzschild radius.
Recall that because light can't escape from event horizon, black holes are very, very black: there is no light at all coming from within the event horizon.
-
[Actually, there can be escaping emissions from very near the
event horizon as
we will discuss in the section
Hawking Radiation.]
Schwarzschild black hole
(FK-545;
ABS-195,222).
The formula for the Schwarzschild radius is
R_sch = 2GM/c**2 = 2.95423 (M/M_Sun) km = approximately 3 (M/M_Sun) km .If any object is compressed to within event horizon, then the object according to general relativity must become a black hole: e.g., the Sun compressed to with a 3-km event horizon would become a black hole.Let us consider some of the features of Schwarzschild black holes:
- In
Schwarzschild solution
there is NOTHING
to stop the mass compressed to within its
event horizon
from collapsing to a point of infinite density: the
singularity.
-
In mathematics a point where a function goes to infinity is
a singularity: e.g.,
1/x has a singularity at x = 0 .In black-hole jargon, THE singularity is the region of infinite density inside the event horizon.
-
The Relativity Postulate:
The laws of physics should
be the same or, more exactly, have the same formulation
in all
inertial frames: they should be frame-invariant
