Dark matter

As we discussed in Chapter 3, about 26% of the mass-energy density of the Universe at present is in the form of non-relativistic gravitating “matter”, and the rest is in some form of “dark energy”. Considering for now this 26% non-relativistic matter component, about 1/7th of this, or 4% of the grand total, is in the form of known particles, mostly baryons (the leptons, neutrinos and photons represent much less mass than the baryons). The other 6/7th of the non-relativistic matter, or 22% of the grand total, is something which we call “dark matter”, colloquially referred to as DM.

What we know about the dark matter is frustratingly little, but enough to convince us of its existence and to give us an idea of some of its overall properties, as outlined in Chapters 3 and 4. We know that it is there, because its gravity makes itself felt in the dynamics of our galaxy and that of other galaxies, as well as in the dynamics of the expansion of the Universe, and its presence is directly mapped via the “gravitational lensing” effect which distorts the paths of the light rays coming to us from distant objects through foreground dark matter-dominated clusters of galaxies. The DM also plays an important role in determining at what epoch in the expansion of the Universe proto-galaxies start to form and assemble. We also know that it is extremely weakly interacting, if not downright inert. It does not emit, block or reflect light or electromagnetic waves, nor does it seem to interact with other particles, at least not enough to have been detectable so far.

Importance of the GeV–TeV range

The GeV–TeV gamma-ray range holds a strategically important role in astrophysics, by providing the first high quality surveys of most classes of very high and ultra-high energy sources, including sufficiently large numbers of objects in each class to be able to start doing statistical classifications of their properties. The number of photons collected for individual sources in this energy range extends in some cases into the tens of thousands, leading in many cases to quite high signal-to-noise ratios.

The GeV–TeV photon emission provides not only important information about the photon emission mechanisms and the source physical properties, but also clues for the importance of the corresponding very high energy (TeV and up) neutrinos and even higher energy cosmic rays which may be emitted from such sources [44]. In addition to the discrete astrophysical sources, instruments in this energy range also provide information about the diffuse gamma-ray emission, such as that associated with cosmic rays interacting with the gas in the plane of our galaxy, the diffuse emission from our galactic center, and the extragalactic emission component, all of which could yield information or constraints about possible dark matter annihilation processes, in addition to the astrophysical processes and the sources involved.

Ripples in space-time

The gravitational force field, as discussed in Chapter 2, is described in General Relativity as a distortion of space-time caused by the masses in it, which results in any small test mass in this space-time moving along the curvature of the space-time. If the position of the large source mass (or masses) which dominate a certain region of space-time is varying, the space-time structure readjusts itself to reflect the changed positions of the source masses, after a delay caused by the fact that the information about this change of position of the source masses cannot be communicated faster than the speed of light. That is, the space-time at some location r away from the source mass which has moved can respond to this change only after a time t = r/c. This traveling information about changes in the space-time structure is the basis of the phenomenon of gravitational waves, which can be thought of as ripples in the texture of spacetime that travel at the speed of light.

One can visualize this also in a simpler quasi-Newtonian picture, provided one accepts the relativistic principle that information travels at most at the speed of light. Imagine two equal masses M in a circular orbit of radius d around each other, in a plane parallel to the line of sight to the observer, with the center of mass of the orbit (the mid-point of the line separating the two) being a fixed point in space at a distance D from the observer (see Fig. 9.1).

The dark and the light

The Universe, as we gaze at it at night, is a vast, predominantly dark and for the most part unknown expanse, interspersed with myriads of pinpricks of light. When we consider that these light spots are at enormously large distances, we realize that they must be incredibly bright in order to be visible at all from so far away. Occasionally, some of these specks of light get much brighter, and some of them which were not even seen with the naked eye before become in a few days the brightest spot in the entire night sky, their brightness having increased a billion-fold or more against the immutable-looking dark background. Thus, we have come to realize that the Universe is characterized by what Renaissance artists called chiaroscuro, referring to the contrast between light and dark, which is both stark and subtle at the same time. In the case of the Universe, the contrasts can be enormous and surprisingly violent, as well as of a subtlety which beggars the imagination. In this book we will focus on these contrasts between the vast, unknown properties of the dark Universe and its most violent outpourings of energy, light and particles.

According to current observations and our best theoretical understanding, the Universe is made up of different forms of mass, or rather of mass-energies, since as we know from special relativity, to every mass there corresponds an energy E = mc2 and vice versa, where E is energy, m is mass and c is the speed of light.

Einstein's celebrated Theory of Relativity is one of those scientific theories whose name is so famous that most people have heard of it, but very few people actually know what the theory says, or even what the theory is about. You, too, have probably heard the name, perhaps referred to in a science fiction novel or movie, even if you do not know much about it. And you may have received the impression that it is a very esoteric and difficult theory that could only be understood and appreciated by a select few.

The aim of this book is to show you that that impression is wrong. The Theory of Relativity comes in two flavors, the Special and the General, and if we limit our attention to the Special Theory of Relativity (SR), which is a theory of motion, it is not a particularly difficult theory at all and can be understood by anyone, perhaps “even a child.” By “be understood” here, I do not mean that anyone can develop a vague idea of what the theory is saying, but that anyone can understand it in its full glory beginning from its basic tenets to all of its logical consequences.

All the problems in this chapter are qualitative and you will be able to solve them if you can read spacetime diagrams. Try them out to test your understanding of Special Relativity.

Reading the spacetime diagram

Street lamps

Five street lamps, numbered 1 through 5, are located on a straight line along the x-axis equal distance apart as shown in the figure. They turn on at points A, B, C, D, and E, respectively, on the spacetime diagram.

1. In what order do the lamps turn on in the ground-frame?

2. In what order does the light from the lamps reach the observer at the origin x = 0?

3. A car is moving to the right at constant speed relative to the ground. At t′ = t = 0, it is at x′ = x = 0. The space- and time-axes in the moving frame of the car are tilted with respect to those of the rest frame as shown in the spacetime diagram. In what order do the lamps turn on in the car-frame?

4. In what order does the light from the lamps reach the observer riding the car?

5. Where is the car when the light from street lamp 4 reaches it?

The mass-momentum vector

Situations that can be addressed within Newtonian dynamics are cases in which the velocities of objects are much slower than the speed of light c, and the Galilei transformation suffices as the transformation from one inertial frame to another.

Now, we would like to answer the questions posed in the previous section using diagrams. But for that we must be able to represent pictorially what we mean by the term “same impact.” In the current case, the “impact” we are talking about is that which accelerates the baseball from “a state in which it is at rest” to “a state in which it is traveling at +1 meters per second.” But this in turn means that we must first be able to represent pictorially the “state of the baseball moving at velocity υ” for generic velocities υ.

So how can we do this? Of course, the motion of any object is described by its worldline on the spacetime diagram, and the object's velocity is encoded in the slope of the worldline. But there are two reasons why the worldline is not an appropriate representation of the “state of motion” of an object:

1. Depending on whether the object is a baseball, a ping-pong ball, or a bowling ball, the amount of “stuff” that is moving is different, but the worldline does not give you that information.

2. […]

This brings us to the end of my exposition of Einstein's Special Theory of Relativity (SR). I have attempted to explain everything that is usually explained using equations using drawings only so that you can literally see what I am talking about. I hope you have found this approach more tractable, eye-opening, and fun.

Partly because of its name, Einstein's “Theory of Relativity” is often misunderstood to have discarded Newton's notions of space and time that were both “objective” and “absolute,” and to have pronounced that both space and time were “relative,” and even “subjective” concepts. In truth, Einstein was a firm believer in objective reality, and SR assumes the existence of an “objective” and “absolute” spacetime. All SR is claiming is that when the motion of objects in spacetime is observed from different inertial frames, things like velocity and length will be frame-dependent. And this dependence comes about because the way the time- and space-axes are introduced into the “absolute” spacetime differs from inertial frame to inertial frame. The frame-dependence of the time-axis already existed in Newton's theory, and as a consequence velocity, not surprisingly, was frame-dependent. In SR, however, in addition to a frame-dependent time-axis, the concept of simultaneity depends on the frame and results in a frame-dependent space-axis also.

This book explains Einstein's Special Theory of Relativity (SR) using diagrams only. Readers who are used to thinking of physics as a vast labyrinth of equations may feel somewhat uneasy about this unconventional approach and fear that it risks losing important information about SR that can only be conveyed via equations. However, this fear is not only unfounded but actually reversed: it is the equations that fail to convey the essence of SR that diagrams can easily display right in front of your eyes. After all, SR, and also the General Theory of Relativity (GR), are about the geometry of the spacetime that we inhabit, and what can best describe geometry if not diagrams? Equations are simply inadequate, to wit, one diagram is worth a thousand equations.

So if you are a reader for whom equations are anathema, rest assured that you will get much more out of this book than any physics student will get out of a textbook full of equations. If you are a physics student, this book will provide you with a deep enough understanding of SR that will enable you to reproduce any equation you may need from scratch, if such a need ever arises, and also prepare you for GR as well.

Before and after

As we have seen, the only way to reconcile the experimentally observed fact that the speed of light does not depend on the inertial frame with our belief in an objective reality, was to abandon the notion that “at the same time” meant the same thing for all observers. Two events that are “at the same time” in one frame may not be “at the same time” in another frame, and vice versa.

Now, some of you may have already realized that this could lead to a problem with the notion of causality, namely, the notion of cause and effect. If an event A is the cause of another event B, then A must happen before B, and B must happen after A. But according to Einstein, the chronological order in which two events happen can depend on the frame of reference!

To make our discussion concrete, consider again the tree planted firmly in the ground, the car moving to the right with speed ½c in the tree-frame, and the ball moving to the right with speed ⅘c in the tree-frame (which corresponds to ½c in the car-frame).

Recall the questions about motion that I listed as the “more advanced” ones in section 2.1:

Let us ask these questions of the motion of a baseball whose worldline is shown here on the spacetime diagram. The worldline has a kink at point A, the time-coordinate of which is t = 3 seconds, so we can tell that the velocity of the baseball changed at t = 3 seconds. The baseball has been hit by a bat at this point. The worldline before A is vertical, so the baseball was at rest before being hit. From the slope of the worldline after A, we can tell that the velocity of the baseball after being hit was +1 meters per second. So the answers to the above questions in this case are: