I’ve told you so far that the gravitational field is encoded in a \(4 \times 4\) matrix known as the metric.  Here it is, displayed as a nice table: $$ g_{ab} = \left( \begin{array}{cccc} g_{00} & g_{01} & g_{02} & g_{03}\\ g_{01} & g_{11} & g_{12} & g_{13} \\ g_{02} & g_{12} & g_{22} & g_{23} \\ g_{03} & g_{13} & g_{23} & g_{33} \end{array} \right)$$There’s 10 components because the matrix is symmetric when reflected diagonally.  The 4 diagonal components \((g_{00}, g_{11}, g_{22}, g_{33})\) tell you how to measure length-squared along the four coordinate axes.  For example, the length along the \(1\)-axis is given by $$\Delta s = \sqrt{g_{11}} \Delta x^1,$$ where \(\Delta x^1\) is the coordinate difference in the \(1\)-direction.  The remaining 6 off-diagonal terms keep track of the spatial angle between the coordinate axes.  If you know enough Trigonometry, you can figure out that the angle \(\theta\) between e.g. the \(1\)-axis and the \(2\)-axis is given by this formula: $$\cos(\theta) = \frac{g_{12}} {\sqrt{g_{11} g_{22}}}$$

However, I’ve also said that the metric depends on the choice of coordinates, which is arbitrary.  We can use this freedom to choose a set of coordinates where the metric looks particularly simple at any given point.   We can start by choosing our four coordinate axes to be at right-angles to each other.  This gets rid of all those funky off-diagonal components of the metric, which involve two different directions: $$ g_{ab} = \left( \begin{array}{cccc} g_{00} & 0 & 0 & 0\\ 0 & g_{11} & 0 & 0 \\ 0 & 0 & g_{22} & 0 \\ 0 &0 & 0 & g_{33} \end{array} \right)$$If any of the four remaining numbers happen to be 0, we say that the metric is degenerate.  This would correspond to a weird geometry in which you can move in one of the directions for free without it affecting your total distance travelled.  Since we all know that’s not the way the real world works, we’ll ignore this possibility.

We can also rescale the tick marks along any coordinate axis.  This allows us to multiply each diagonal component of the metric by a positive real number.  So if say \(g_{22}\) is positive, we can choose coordinates where it’s \(+1\), and if it’s negative, we can choose coordinates where it’s \(-1\).  This gives us:$$ g_{ab} = \left( \begin{array}{cccc} \pm 1 & 0 & 0 & 0\\ 0 & \pm 1 & 0 & 0 \\ 0 & 0 & \pm 1 & 0 \\ 0 &0 & 0 & \pm 1 \end{array} \right)$$Since it also doesn’t matter what order we list the four coordinate directions, all that matters is the total number of \(+\)’s and \(-\)’s.  This choice is called the signature of the spacetime.

Now if you remember my very first post on spacetime geometry, \(+\) directions in the metric correspond to spatial dimensions, while the funny \(-\) sign is what makes for a time dimension.  But the real world has one time dimension, everywhere.  No matter how far you travel, you’ll never find a place (so far as we know) where there isn’t any time direction, or where there are extra time dimensions.  So that means that the correct signature for spacetime has \((-, +, +, +)\) along the diagonal, which is called Lorentzian (a.k.a. Minkowskian) signature.  (If we had wanted to describe a timeless four-dimensional space, we would instead select the Riemannian (a.k.a. Euclidean) signature \((+, +, +, +)\).)  We conclude that for any point of spacetime, you can always choose a set of coordinates such that the metric takes a special form that we’ll call \(\eta_{ab}\): $$ g_{ab} = \left( \begin{array}{cccc}-1 & 0 & 0 & 0\\ 0 & +1 & 0 & 0 \\ 0 & 0 & +1 & 0 \\ 0 &0 & 0 & +1 \end{array} \right) = \eta_{ab}.$$In other words, if you zoom in on any point, you recover Special Relativity.  So after all this fidgeting around, we end up with a somewhat profound conclusion: in General Relativity, every point of spacetime looks the same as every other point.

This is related to what Einstein called the Equivalence Principle, which says that at short enough distances, the effects of acceleration are indistinguishable from being in a gravitational field.  We all know from personal experience that riding in an elevator can make us weigh more or less, and from TV that astronomers in the Space Shuttle are weightless when they’re in free fall.  In other words, you can always choose a coordinate system in which there is no gravitational force at any given point.

(Lewis Carroll actually described this principle several decades before Einstein in Sylvie and Bruno, which includes a description of a tea party taking place in a freely-falling house.  Then he describes what happens if the house is being pulled down with a rope faster than gravity would accelerate it, and explains how you could have a normal tea party as long as you have it upside-down.  I like this book better than his more famous classics, but don’t read it unless you can withstand LD20 of Victorian sentimentality about fairy children.  Also, Carroll didn’t go on to discover a revolutionary theory of gravity based on this principle.)

It might seem now like everything has become too simple.  If the metric looks the same at every single point, then why did we even bother with it?  Where’s the information in the gravitational field?  Well, it’s true that for any one point, there’s a coordinate system where the metric looks just like \(\eta_{ab}\).  But there’s no coordinate system for which the metric looks like \(\eta_{ab}\) everywhere at once.  (Unless there’s no gravitational field anywhere, in which case Special Relativity is true).  If you make the metric look simple in one place, it has to look complicated somewhere else.

So in order to describe the gravitational field properly, we have to find a way to compare the metric at different points.  We can do this using something called parallel transport.  I’ll give more details later, but basically it tells us how an object moves in a gravitational field when we carry it along a path through spacetime.  When we carry the object around a tiny loop so that it returns to its original position, we might find that it comes back rotated compared to its original orientation.  If so, we say that the spacetime contains curvature.  If the spacetime contains curvature, this is a fact about the gravitational field which is invariant, i.e. objectively true.  You can’t eliminate it just by changing your coordinates.

In my last post about spacetime, I explained how the geometry of spacetime is determined at each spacetime point by a set of 10 numbers.  These 10 numbers are packaged together into a \(4 \times 4\) matrix called the metric, which is written as \(g_{ab}\).  The subscripts \(a\) and \(b\) stand in for any of the 4 coordinate directions (in a 4-dimensional spacetime).  Since the metric is symmetric, i.e. \(g_{ab} = g_{ba}\), there are 10 possible numbers in this matrix.  The value of these 10 numbers depends on your position and time,which makes them a field, specifically the gravitational field.

However, there is an important caveat in all this.  The coordinates which you use to describe a given spacetime are totally arbitrary.  For example, a flat 2-dimensional Euclidean plane can be described using Cartesian coordinates \(-\infty < x < +\infty\) and \(-\infty < y < +\infty\).  In this coordinate system, the distance-squared is given by the Pythagorean formula $$(ds)^2 = (dx)^2 + (dy)^2,$$ which can be written in terms of the metric as $$g_{xx} = 1; \qquad g_{yy} = 1; \qquad g_{xy} = 0.$$

On the other hand, for applications involving rotations, it’s often useful to use polar coordinates: \(0 \le r < +\infty\) (the distance from the origin) and \(0 \le \theta < 2\pi\) (the angle around the origin, measured in radians).  They’re related to the original coordinate system by $$x = r \sin \theta;\\ y = r \cos \theta.$$  In polar coordinates, the distance-squared is given by $$(ds)^2 = (dr)^2 + r^2 (d\theta)^2,$$where the extra \(r^2\) factor comes in because circles that are a greater distance from the origin have a larger circumference, so there’s more space as you move outwards.  This can be written in terms of the metric like this: $$g_{rr} = 1; \qquad g_{\theta\theta} = r^2; \qquad g_{r\theta} = 0.$$

(Note: I’ve given these coordinate systems their traditional coordinate names to make them look more familiar, but this is actually just a redundancy to make it easier for humans to think about it.  I could have written the two coordinates as \((x^0, x^1)\)—the superscript being a coordinate index, not an exponent—and then you could tell whether it was Cartesian or polar coordinates just by inspecting the formula for the metric.)

Now the point is, these two coordinate systems describe the same geometry in a different coordinate system.  If we were playing pool (or billiards) on a planar surface, and you wanted to describe how billiard balls bounce off of each other, you could equally well describe it using either coordinate system.  The physics would be the same.

Of course, the language you use to describe the system differs.  Suppose that I analyze a collision using Cartesian coordinates, while you use polar coordinates.  And suppose we had to communicate to each other what happened.  If you say to me, “The cue ball had a velocity in the \(x^1\) direction”, then I’ll get confused because \(x^1\) means something different to me than it does to you.  These kind of statements vary under a change of coordinate system, they are “relative” to your coordinate-perspective.  So if you want to communicate with me, you have to find a way to describe what’s going on which does not refer to coordinates in any way.  For example, you could say “The cue ball hit the 3 ball, which knocked the 8 ball into a pocket.”  Since the two balls and the pocket are unique physical objects, we can all agree on whether or not this happened, no matter what coordinate system we use.  These kind of statements are invariant under a change of coordinate system.  The goal of coordinate-invariant physics is to describe everything in this sort of way.

Here’s another way in which coordinate systems can let you down: when you use polar coordinates, there are places where the coordinates go kind of funny.  For example, when you’re going around the origin clockwise in the direction of increasing \(\theta\), and you arrive at \(\theta = 2\pi\), you immediately teleport back to \(\theta = 0\) since you’ve come full circle.  Even stranger, space seems to come to an end at \(r = 0\) (the origin) since there’s no such thing as negative \(r\).  And if you’re sitting right at \(r = 0\), the different values of \(\theta\) all refer to the same point as each other.  However, in reality we know that nothing weird is happening to the geometry at any of these points, since nothing strange happens in Cartesian coordinates.  (A similar issue comes up in black hole physics.  The original set of coordinates found by Schwarzschild blow up at the event horizon, but actually nothing unusual happens there in classical general relativity.)

The upshot of all this for general relativity is the following: I told you above that you can describe general relativity using the metric \(g_{ab}\), which involves 10 numbers at each point.  But this description actually has some redundancy in it, since there’s infinitely many possible coordinates systems you could use (one for each way of labelling the points uniquely with four numbers), and the metric looks different in each one—it isn’t an invariant object.

When a theory has redundancy like this, we say there is a gauge symmetry.  A regular symmetry says that two different states (i.e. configurations) of a system behave in the exact same way as each other.  A gauge symmetry is stronger than a regular symmetry: it says that the two configurations are actually the same physical state of affairs.  In general relativity, the choice of coordinates is a gauge-symmetry.  It is a mere human convention which doesn’t correspond to any actual physical thing in Nature.

Of course, even if you aren’t doing general relativity, you can still use whatever coordinate system you like!  Most games of billiards can be understood in the approximation where space is flat (unless you like to spice up your games with black holes and gravity waves, like the cool kids do!)  In flat space time, all coordinates are equal, but some are more equal than others.  Although nothing stops you from calculating in horrible coordinates, the laws of physics look especially simple in ordinary Minkowski coordinates, where the symmetries of spacetime look especially simple.  Since Newton’s First Law of motion holds in these coordinates, we call it an inertial frame.  (Here I’m ignoring the downward pull of gravity, since in billards we’re only interested in horizontal motions.)

However, if you’re doing general relativity, then there’s a property of spacetime which forces you to describe physics in a coordinate-invariant way; at least if you want the equations of the theory to look elegant and lovely instead of like horrendous cludge.  This property is called curvature—but we’re out of time for today.

In Time as the Fourth Dimension?, I explained how to calculate the distance (or duration) squared between any two points of spacetime, using a spin-off of the Pythagorean theorem: $$s^2= (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 – (\Delta t)^2.$$Then I explained the Ten Symmetries of Spacetime, i.e. ways to shift or rotate the coordinate system \((t,\,x,\,y,\,z)\) that don’t change the formula for \(s^2\).

Well, it turns out that I lied.  The formula isn’t actually true, except in the special case that there is nothing in the universe.  A significant reservation, I know.  Instead, what’s true is that the geometry of space is a field, meaning that it varies from place to place, depending on where you are!  However, if you zoom in really close at any particular point, it looks similar to the formula I told you.

The field that says what geometry is like at any given place and time is called (brace yourself) the gravitational field.  In order to describe it, we use something called the metric, which indicates what the geometry of spacetime looks like at any given point.  The way this works is, suppose we have two points \(p\) and \(q\) which are very close to each other.  Suppose we want to know the distance between these points.

Since the points are really close to each other, we call the distance between them \(ds\), where the \(d\) is just a reminder that we’re using Calculus to study infinitesimal quantities.  If you don’t know Calculus, just pretend these are really small numbers.  We want to figure out what \(ds\) is, if we know the infinitesimal coordinate differences \((dx,\, dy,\,dz,\,dt)\).  The way we do this is by generalizing the heck out of the Pythagorean theorem.  I’ll write it down, and then explain what it means: $$(ds)^2 = g_{xx}\,(dx)^2 + g_{yy}\,(dy)^2 + g_{zz}\,(dz)^2 + g_{tt} \,(dt)^2 + \\ 2[ g_{xy}\,dx\,dy + g_{xz}\,dx\,dz + g_{xt}\,dx\,dt + g_{yz}\,dy\,dz + g_{yt}\,dy\,dt + g_{zt}\,dz\,dt].$$The right-hand side of the equation consists of every possible way of multiplying two of the coordinate distances \((dx,\, dy,\,dz,\,dt)\).  There are 4 different ways to pick the first \((dx,\, dy,\,dz,\,dt)\), and 4 different ways to pick the second, which gives \(4 \times 4 = 16\) possible combinations in all.  However, multiplication is commutative so e.g. \(dx\,dy = dy\,dx\).  So I added terms like that together; that’s where the factor of 2 came from.  Taking that into account, there’s 10 terms in all.

The funny \(g\) things with subscripts are just functions of spacetime, i.e. they are just numbers that depend on where you are, i.e. they are fields.  In the special case where we pick these numbers to be \(g_{xx} = g_{yy} = g_{zz} = +1,\,g_{tt} = -1\) and the rest zero, we get the geometry I told you about, which goes by the aliases “Minkowski space”, “flat spacetime”, and “Special Relativity”.  In all other cases we have what is colloquially called “curved spacetime” which is the province of “General Relativity”.

The formula above looks kind of ugly, but we can prettify it by choosing good notation.  We collectively refer to all ten of these gravitational fields as the metric, denoted \(g_{ab}\), where subscripts like \(a\) and \(b\) can refer to any of the four coordinate labels.  (People often call these labels \((0,\,1,\,2,\,3)\) instead of \((x,\,y,\,z,\,t)\) to avoid confusion, since the metric itself says which of the coordinate directions behave more like space, and which behave more like time, and this can vary from place to place!)  Then we write the four coordinate differences \((dx,\, dy,\,dz,\,dt)\) collectively as \(dx^a\), where the superscript says which of the four it is.  Finally, we make up a rule called the Einstein summation convention, that if we ever see the same letter as both a subscript and as a superscript, we add up all of the four possible ways for them to be the same (i.e. both 0, both 1, both 2, or both 3).  These are just changes in how we write things, not substantive changes, but they let us rewrite that long ugly equation like this:$$ds^2 = g_{ab}\,dx^a\,dx^b.$$There, isn’t that much prettier?

Suppose we want to find the distance (or duration) between two points which are NOT infinitesimally close to each other.  In that case, we have to choose a path between the two points, since the amount of distance (or duration) depends on which path you choose, and in a curved spacetime there’s not necessarily one “best” path.  This shouldn’t seem that strange, since even in everyday life we know perfectly well that the distance between San Francisco and L.A. depends on which highway you take, and the distance between Tokyo and New York depends on which way around the globe you fly.  (It’s totally intuitive for distances, but when the duration depends on the route you take through spacetime, people call it the Twin Paradox and act all shocked!)

So this is the first main idea of General Relativity: the geometry of spacetime is a field which varies from place to place.  This field affects matter by determining the paths that things take through space and time, but it also is affected by matter—we call this gravity.  The second main idea is that coordinates are an arbitrary choice; I’ll tell you about this later.  The third main idea is the Einstein equation which says how matter affects the metric.  I haven’t told you anything about this equation yet, but once I do, you would in principle be able to calculate everything about the gravitational field from that one equation.

There can also be distortions of the spacetime geometry which exist independently of matter.  These gravity waves are to gravity what light is to electromagnetism, ripples in the field which travel through empty space, and can be emitted and absorbed.  The propagation of these waves is also determined by the Einstein equation.  Since gravity comes from massive objects, gravity waves are emitted when extremely large masses oscillate, for example when two neutron stars orbit each other.  We know gravity waves are there, but we haven’t detected them directly.  However, we hope to detect them soon with the LIGO experiment.

UPDATE: I realized that I never said how you would calculate the distance between two points, once you choose a path.  The answer is that you chop the path into lots of tiny little line segments, and find the length of each line segment using the metric.  Then you add them all up.  If you know Calculus, this can be done using an integral.

There’s been a huge kerfuffle in the quantum gravity community since this summer, when some people here at UCSB published a paper arguing that (old enough) black holes may actually be surrounded by a wall of fire which burns people up when they cross the event horizon.  This is huge, because if it were true it would upset everything we thought we knew about black holes.

General relativity is our best theory of gravity to date, discovered by Einstein.  This is a  classical theory.  (In the secret code that we physicists use, classical is our code-word for “doesn’t take into account quantum mechanics”.  Don’t tell anyone I told you.)

In my other posts on physics, I’ve been trying to explain the fundamentals of physics in the minimum number of blog posts.  This post is out of sequence, since I haven’t described general relativity yet!  But I wanted to say something about exciting current events.

In classical general relativity, a black hole is a region of space where the gravity is so strong that not even light can escape.  They tend to form at the center of galaxies, and from the collapse of sufficiently large stars when they run out of fuel to hold them up.  A black hole has an event horizon, which is the surface beyond which if you fall in, you can’t ever escape without travelling faster than light. The information of anything falling into the black hole is lost forever, at least in classical physics.

In the case of a non-rotating black hole, without anything falling into it, the event horizon is a perfect sphere.  (If the black hole is rotating, it bulges out at the equator.)  If you fall past the event horizon, you will inevitably fall towards the center, just as in ordinary places you inevitably move towards the future.

At the center is the singularity.  As you approach the singularity, you get stretched out infinitely in one direction of space, and squashed to zero size in the other two directions of space, and then at the singularity time comes to an end!  Actually, just before time comes to an end, we know that the theory is wrong, since things get compressed to such tiny distances that we really ought to take quantum mechanics into account.  Since we don’t have a satisfactory theory of quantum gravity yet, we don’t really know for sure what happens.

Now it’s important to realize that the event horizon is not a physical object.  Nothing strange happens there.  It’s just an imaginary line between the place where you can get out by accelerating really hard, and the place where you can never get out.  Someone falling into the black hole just sees a vacuum.  If the black hole was formed from the collapse of a star, the matter from the star quickly falls into the singularity and disappears.  The black hole is empty inside, except for the gravitational field itself.

We don’t know how to describe full-blown quantum gravity, but we have something called semiclassical gravity which is supposed to work well when the gravitational effects of the quantum fields are small.  In semiclassical gravity, one finds that black holes slowly lose energy from thermal “Hawking” radiation.  This radiation looks exactly like the random “blackbody radiation” coming from an ordinary object when you heat it up. Here’s the important fact: You can prove that the radiation is thermal (i.e. random) just using the fact that someone falling across the horizon sees a vacuum (i.e. empty space) there.

The Hawking radiation comes from just outside the event horizon.  It does not come from inside the black hole, so in Hawking’s original calculation it doesn’t carry any information out from the inside.   Nevertheless, for various reasons I can’t go into right now, most black hole physicists have convinced themselves that the information eventually does come out.

As the black hole radiates into space, it slowly evaporates, and eventually probably disappears entirely (although knowing what happens at the very end requires full-blown quantum gravity).  If the outgoing Hawking radiation carries all the radiation out, then for a black hole at a late enough stage in its evaporation, the radiation must not be completely random, because it actually encodes all the information about what fell in.

The gist of what Almheiri, Marolf, Polchinski, and Sully argued, is that if we take both of these statements in bold seriously, then it follows that the black holes are NOT in the vacuum state from the perspective of someone who falls in.  Instead you would get incinerated by a “firewall” as you cross the horizon.  (It’s not clear yet whether this is only for really old black holes, or if it applies to younger ones too.)  That’s if we still believe there is an “inside” at all.  The argument shows that semiclassical gravity is completely wrong in situations where we would have expected it to work great.

If this is right, then it’s devastating to the ideas of many of us who have been thinking about black holes for a long time.  As a reluctant convert to the idea that information is not lost, I’m wondering if I should reconsider.  At the end of this month, I’m going to Stanford for a weekend, since Lenny Susskind has invited a bunch of us to try to get this worked out.  Exciting times!

What is the world made out of?  In the most usual formulations of our current best theories of physics, the answer is fields.  What are those?

Well, if you know what a function is, you’re already most of the way there.  A function, you will recall, is a gadget where, for any number you input, you can get a number out as an output.  We can write \(f(x)\) where \(x\) is the number you input, and \(f(x)\) is the number you output.  The function \(f\) itself is the rule for going from one to the other, e.g.  For example \(f(x) = \sqrt{\sin x^2 + 1}\).

Now, nothing stops you from having a function that depends on multiple numbers as input; for example the function \(f(x,\,y) = xy^2 + x^3y\) depends on two input variables, \(x\), and \(y\).  If there are \(D\) input numbers, then the \(D\)-dimensional space of possible combinations of input numbers is called the domain of the function.

Also nothing stops you from having the output be a set of several numbers.  In this case we would need some sort of subscript \(i\) to refer to the different possible output numbers.  For example, if we had a function with one input number \(x\) and three output numbers \(y\), then we could write \(f_i(x)\), where \(i\) takes the values 1, 2, or 3.  Then \(f_i(x)\) would really be just a package of three different functions: \(f_1(x)\), \(f_2(x)\), and \(f_3(x)\).  So if you specify the input \(x\), you get three output numbers \((f_1, f_2, f_3)\).  If there are \(T\) different output numbers, the \(T\) dimensional space of possible outputs is called the target space.

Now a field is just a function whose domain is the points of spacetime.  For example, the air temperature in a room may vary from place to place, and it may also change with time.  So if you imagine checking all possible points of space in the room at all possible times, you could describe this with a temperature field \(T(t, x, y, z)\).  However, the temperature field isn’t a fundamental entity that exists on its own.  It subsists in a medium (air) and describes its motion.  When the air molecules are moving around quickly in a random way, we say it’s hot, and when they start to move around slower, we say it’s getting chilly.  An example of a field which actually is fundamental (as far as we know) would be the electromagnetic field.  This has 6 output numbers, since the electric field can point in any of the 3 spatial directions, and the magnetic field also has 3 numbers.

For a while in the 19th century, scientists were confused about this.  They thought that electromagnetic waves had to be some sort of excitation of some sort of stuff, which they called the aether.  That’s because they were assuming (based on physical intuitions filtered through Newtonian mechanics) that matter is something solid and massy, which interacts by striking or making contact with other things.  The 20th century scientific advances partly came from realizing that its okay to describe things with abstract math.  Any kind of mathematical object you write down satisfying logically consistent equations is OK, as long as it matches experiment.  So electromagnetic waves don’t have to be made out of anything.  They just are, and other things are (partly) made out of them.

In our current best theory of particle physics, the Standard Model, there are a few dozen different kinds of fields, and all matter is explained as configurations of these fields.  I can’t tell you exactly how many fields there are, because it depends on how you count them.  Not counting the gravitational field, there are 52 different output numbers corresponding to bosons, and 192 different output numbers corresponding to fermions (Don’t worry about what these terms mean yet).  So you could say that there are 244 different fields in Nature, each with one output number.

That sounds awfully complicated.  But there’s also a lot of symmetries in the Standard Model which relate these output numbers to each other.  This includes not only the Poincaré group of spacetime symmetries, but also various internal symmetries related to the dynamics of the strong, weak, and electromagnetic forces.  They are called internal because they don’t move the points of spacetime around.  Instead they just mutate the different kinds of output numbers into each other.

So normally, particle physicists just package the output numbers into sets, such that the numbers in each set are related by the various kinds of symmetry.  (For example, the 6 different numbers of the electromagnetic field are related by rotations and Lorentz boosts.)  Each of these sets is called a field.  In future posts I’ll give more details about the different kinds of fields.  As always, questions are welcome.

UPDATE: I forgot to include the 4 vector components of the spin-1 gauge bosons, so the numbers of degrees of freedom of the bosons were wrong before.  Note to Experts: These are the “off-shell” degrees of freedom before taking into consideration constraints or gauge symmetry.  Note to Non-Experts: the numbers in this post are just for flavor, in order to give you the sense that there are a LOT of different fields in Nature.  You won’t need to understand how I got these numbers in order to enjoy future posts!