Astronomy and Space

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Q. What are Gravity waves?
Q. Is energy always conserved?
Q. What are the effects of Finite Light Speed?
Q. What is the Twin Paradox?
Q. What is Olbers' Paradox?
Q. What is Dark Matter?
Q. What is Big Bang?

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Q. What are Gravity waves?

Ans. Gravitational Radiation is to gravity what light is to electromagnetism. It is produced when massive bodies accelerate. You can accelerate any body so as to produce such radiation, but due to the feeble strength of gravity, it is entirely undetectable except when produced by intense astrophysical sources such as supernovae, collisions of black holes, etc. These are quite far from us, typically, but they are so intense that they dwarf all possible laboratory sources of such radiation.

Gravitational waves have a polarization pattern that causes objects to expand in one direction, while contracting in the perpendicular direction. That is, they have spin two. This is because gravity waves are fluctuations in the tensorial metric of space-time.

All oscillating radiation fields can be quantized, and in the case of gravity, the intermediate boson is called the "graviton" in analogy with the photon. But quantum gravity is hard, for several reasons:

(1) The quantum field theory of gravity is hard, because gauge interactions of spin-two fields are not renormalizable. See Cheng and Li, Gauge Theory of Elementary Particle Physics (search for "power counting").
(2) There are conceptual problems - what does it mean to quantize geometry, or space-time?

It is possible to quantize weak fluctuations in the gravitational field. This gives rise to the spin-2 graviton. But full quantum gravity has so far escaped formulation. It is not likely to look much like the other quantum field theories. In addition, there are models of gravity which include additional bosons with different spins. Some are the consequence of non-Einsteinian models, such as Brans-Dicke which has a spin-0 component. Others are included by hand, to give "fifth force" components to gravity. For example, if you want to add a weak repulsive short range component, you will need a massive spin-1 boson. (Even-spin bosons always attract. Odd-spin bosons can attract or repel.) If antigravity is real, then this has implications for the boson spectrum as well.

The spin-two polarization provides the method of detection. All experiments to date use a "Weber bar." This is a cylindrical, very massive, bar suspended by fine wire, free to oscillate in response to a passing graviton. A high-sensitivity, low noise, capacitive transducer can turn the oscillations of the bar into an electric signal for analysis. So far such searches have failed. But they are expected to be insufficiently sensitive for typical radiation intensity from known types of sources.

A more sensitive technique uses very long baseline laser interferometry. This is the principle of LIGO (Laser Interferometric Gravity wave Observatory). This is a two-armed detector, with perpendicular laser beams each travelling several km before meeting to produce an interference pattern which fluctuates if a gravity wave distorts the geometry of the detector. To eliminate noise from seismic effects as well as human noise sources, two detectors separated by hundreds to thousands of miles are necessary. A coincidence measurement then provides evidence of gravitational radiation. In order to determine the source of the signal, a third detector, far from either of the first two, would be necessary. Timing differences in the arrival of the signal to the three detectors would allow triangulation of the angular position in the sky of the signal.

The first stage of LIGO, a two detector setup in the U.S., has been approved by Congress in 1992. LIGO researchers have started designing a prototype detector, and are hoping to enroll another nation, probably in Europe, to fund and be host to the third detector.

The speed of gravitational radiation (C_gw) depends upon the specific model of Gravitation that you use. There are quite a few competing models (all consistent with all experiments to date) including of course Einstein's but also Brans-Dicke and several families of others. All metric models can support gravity waves. But not all predict radiation travelling at C_gw = C_em. (C_em is the speed of electromagnetic waves.)

There is a class of theories with "prior geometry", in which, as I understand it, there is an additional metric which does not depend only on the local matter density. In such theories, C_gw != C_em in general.

However, there is good evidence that C_gw is in fact at least almost C_em. We observe high energy cosmic rays in the
10 ** 20-10 ** 21 evregion.
Such particles are travelling at up to ( 1 - 10 ** -18 ) * C_em.
If C_gw < C_em, then particles with C_gw < v < C_em will radiate Cerenkov gravitational radiation into the vacuum, and decelerate from the back reaction.
So evidence of these very fast cosmic rays good evidence that
C_gw >= ( 1 - 10 ** -18 ) * C_em, very close indeed to C_em.
Bottom line: in a purely Einsteinian universe, C_gw = C_em. However, a class of models not yet ruled out experimentally does make other predictions.

A definitive test would be produced by LIGO in coincidence with optical measurements of some catastrophic event which generates enough gravitational radiation to be detected. Then the "time of flight" of both gravitons and photons from the source to the Earth could be measured, and strict direct limits could be set on C_gw.

For more information, see Gravitational Radiation (NATO ASI - Les Houches 1982), specifically the introductory essay by Kip Thorne.

Q.Is energy always conserved?

Why? Every conserved quantity is the result of some symmetry of nature. This is known as Noether's theorem. For example, momentum conservation is the result of translation invariance, because position is the variable conjugate to momentum. Energy would be conserved due to time-translation invariance. However, in an expanding or contracting universe, there is no time-translation invariance. Hence energy is not conserved. If you want to learn more about this, read Goldstein's Classical Mechanics, and look up Noether's theorem.

Does Red-Shift lead to Energy Non-Conservation: Sometimes

There are three basic cosmological sources of red-shifted light:
(1) Very massive objects emitting light
(2) Very fast objects emitting light
(3) Expansion of the universe leading to CBR (Cosmic Background Radiation) red-shift
About each:

(1) Light has to climb out the gravitational well of a very massive object. It gets red-shifted as a result. As several people have commented, this does not lead to energy non-conservation, because the photon had negative gravitational potential energy when it was deep in the well. No problems here. If you want to learn more about this read Misner, Thorne, and Wheeler's Gravitation, if you dare.

(2) Fast objects moving away from you emit Doppler shifted light. No problems here either. Energy is only one part a four-vector, so it changes from frame to frame. However, when looked at in a Lorentz invariant way, you can convince yourself that everything is OK here too. If you want to learn more about this, read Taylor and Wheeler's Spacetime Physics.

(3) CBR has red-shifted over billions of years. Each photon gets redder and redder. And the energy is lost. This is the only case in which red-shift leads to energy non-conservation. Several people have speculated that radiation pressure "on the universe" causes it to expand more quickly, and attempt to identify the missing energy with the speed at which the universe is expanding due to radiation pressure. This argument is completely specious. If you add more radiation to the universe you add more energy, and the universe is now more closed than ever, and the expansion rate slows.

If you really MUST construct a theory in which something like energy is conserved (which is dubious in a universe without time-translation invariance), it is possible to arbitrarily define things so that energy has an extra term which compensates for the loss. However, although the resultant quantity may be a constant, it is of questionable value, and certainly is not an integral associated with time-invariance, so it is not what everyone calls energy.

Q.What are the effects of Finite Light Speed?

Ans. There are two well known phenomena which are due to the finite speed of electromagnetic radiation, but are essentially classical in nature, requiring no other facts of special relativity for their understanding.

(1) Apparent Superluminal Velocity of Galaxies

A distant object can appear to travel faster than the speed of light relative to us, provided that it has some component of motion towards us as well as perpendicular to our line of sight. Say that on Jan. 1 you make a position measurement of galaxy X. One month later, you measure it again. Assuming you know it's distance from us by some independent measurement, you derive its linear speed, and conclude that it is moving faster than the speed of light.

What have you forgotten? Let's say that on Jan. 1, the object is D km from us, and that between Jan. 1 and Feb. 1, the object has moved d km closer to us. You have assumed that the light you measured on Jan. 1 and Feb. 1 were emitted exactly one month apart. Not so. The first light beam had further to travel, and was actually emitted (1 + d/c) months before the second measurement, if we measure c in km/month. The object has traveled the given angular distance in more time than you thought. Similarly, if the object is moving away from us, the apparent angular velocity will be too slow, if you do not correct for this effect, which becomes significant when the object is moving along a line close to our line of sight.

Note that most extragalactic objects are moving away from us due to the Hubble expansion. So for most objects, you don't get superluminal apparent velocities. But the effect is still there, and you need to take it into account if you want to measure velocities by this technique.


Considerations about the Apparent 'Superluminal Expansions' in Astrophysics, E. Recami, A. Castellino, G.D. Maccarrone, M. Rodono, Nuovo Cimento 93B, 119 (1986).
Apparent Superluminal Sources, Comparative Cosmology and the Cosmic Distance Scale, Mon. Not. R. Astr. Soc. 242, 423-427 (1990).

(2) Terrell Rotation

Consider a cube moving across your field of view with speed near the speed of light. The trailing face of the cube is edge on to your line of sight as it passes you. However, the light from the back edge of that face (the edge of the square farthest from you) takes longer to get to your eye than the light from the front edge. At any given instant you are seeing light from the front edge at time t and the back edge at time t-(L/c), where L is the length of an edge. This means you see the back edge where it was some time earlier. This has the effect of rotating the image of the cube on your retina.

This does not mean that the cube itself rotates. The image is rotated. And this depends only on the finite speed of light, not any other postulate or special relativity. You can calculate the rotation angle by noting that the side face of the cube is Lorentz contracted to L' = L/gamma. This will correspond to a rotation angle of arccos(1/gamma).

It turns out, if you do the math for a sphere, that the amount of apparent rotation exactly cancels the Lorentz contraction. The object itself is flattened, but then you see behind it as it flies by just enough to restore it to its original size. So the image of a sphere is unaffected by the Lorentz flattening that it experiences.

Another implication of this is that if the object is moving at nearly the speed of light, although it is contracted into an infinitesimally thin pancake, you see it rotated by almost a full 90 degrees, so you see the complete backside of the object, and it doesn't disappear from view. In the case of the sphere, you see the transverse cross-section (which suffers no contraction), so that it still appears to be exactly a sphere.

That it took so long historically to realize this is undoubtedly due to the fact that although we were regularly accelerating particle beams in 1959 to relativistic speeds, we still do not have the technology to accelerate any macroscopic objects to speeds necessary to reveal the effect.

References: J. Terrell, Phys Rev. 116, 1041 (1959). For a textbook discussion, see Marion's Classical Dynamics, Section 10.5.

Q.What is the Twin Paradox?

The Twin Paradox

A Short Story about Space Travel:

Two twins, conveniently named A and B, both know the rules of Special Relativity. One of them, B, decides to travel out into space with a velocity near the speed of light for a time T, after which she returns to Earth. Meanwhile, her boring sister A sits at home posting to Usenet all day. When A finally comes home, what do the two sisters find? Special Relativity (SR) tells A that time was slowed down for the relativistic sister, B, so that upon her return to Earth, she knows that B will be younger than she is, which she suspects was the the ulterior motive of the trip from the start.

But B sees things differently. She took the trip just to get away from the conspiracy theorists on Usenet, knowing full well that from her point of view, sitting in the spaceship, it would be her sister, A, who was travelling ultrarelativistically for the whole time, so that she would arrive home to find that A was much younger than she was. Unfortunate, but worth it just to get away for a while.

What are we to conclude? Which twin is really younger? How can SR give two answers to the same question? How do we avoid this apparent paradox? Maybe twinning is not allowed in SR? Read on.

Paradox Resolved:

Much of the confusion surrounding the so-called Twin Paradox originates from the attempts to put the two twins into different frames --- without the useful concept of the proper time of a moving body.

SR offers a conceptually very clear treatment of this problem. First chose _one_ specific inertial frame of reference; let's call it S. Second define the paths that A and B take, their so-called world lines. As an example, take ( ct , 0 , 0 , 0 ) as representing the world line of A, and ( ct , f(t) , 0 , 0 ) as representing the world line of B (assuming that the the rest frame of the Earth was inertial). The meaning of the above notation is that at time t, A is at the spatial location ( x1 , x2 , x3 ) = ( 0 , 0 , 0 ) and B is at ( x1 , x2 , x3 ) = ( f(t) , 0 , 0 ) always with respect to S.

Let us now assume that A and B are at the same place at the time t1 and again at a later time t2, and that they both carry high-quality clocks which indicate zero at time t1. High quality in this context means that the precision of the clock is independent of acceleration. [In principle, a bunch of muons provides such a device (unit of time: half-life of their decay).]

The correct expression for the time T such a clock will indicate at time t2 is the following [the second form is slightly less general than the first, but it's the good one for actual calculations]:

........ (1)
where d \ t a u is the so-called proper-time interval, defined by


is the velocity vector of the moving object. The physical interpretation of the proper-time interval, namely that it is the amount the clock time will advance if the clock moves by dx during dt, arises from considering the inertial frame in which the clock is at rest at time t, its so-called momentary rest frame (see the literature cited below). [Notice that this argument is only of a heuristic value, since one has to assume that the absolute value of the acceleration has no effect. The ultimate justification of this interpretation must come from experiment.]

The integral in (1) can be difficult to evaluate, but certain important facts are immediately obvious. If the object is at rest with respect to S, one trivially obtains T = t2-t1. In all other cases, T must be strictly smaller than t2-t1, since the integrand is always less than or equal to unity. Conclusion: the traveling twin is younger. Furthermore, if she moves with constant velocity v most of the time (periods of acceleration short compared to the duration of the whole trip), T will approximately be given by

The last expression is exact for a round trip (e.g. a circle) with constant velocity v. [At the times t1 and t2, twin B flies past twin A and they compare their clocks.]

Now the big deal with SR, in the present context, is that T (or d \ tau, respectively) is a so-called Lorentz scalar. In other words, its value does not depend on the choice of S. If we Lorentz transform the coordinates of the world lines of the twins to another inertial frame S', we will get the same result for T in S' as in S. This is a mathematical fact. It shows that the situation of the traveling twins cannot possibly lead to a paradox within the framework of SR. It could at most be in conflict with experimental results, which is also not the case.

Of course the situation of the two twins is not symmetric, although one might be tempted by expression (2) to think the opposite. Twin A is at rest in one and the same inertial frame for all times, whereas twin B is not. [Formula (1) does not hold in an accelerated frame.] This breaks the apparent symmetry of the two situations, and provides the clearest nonmathematical hint that one twin will in fact be younger than the other at the end of the trip. To figure out which twin is the younger one, use the formulae above in a frame in which they are valid, and you will find that B is in fact younger, despite his expectations.

It is sometimes claimed that one has to resort to General Relativity in order to "resolve" the Twin "Paradox". This is not true. In flat, or nearly flat space-time (no strong gravity), SR is completely sufficient, and it has also no problem with world lines corresponding to accelerated motion.


Taylor and Wheeler, Spacetime Physics (An excellent discussion) Goldstein, Classical Mechanics, 2nd edition, Chap.7

Q.What is Olbers' Paradox

Olbers' Paradox

Why isn't the night sky as uniformly bright as the surface of the Sun? If the Universe has infinitely many stars, then it should be. After all, if you move the Sun twice as far away from us, we will intercept one-fourth as many photons, but the Sun will subtend one-fourth of the angular area. So the areal intensity remains constant. With infinitely many stars, every angular element of the sky should have a star, and the entire heavens should be a bright as the sun. We should have the impression that we live in the center of a hollow black body whose temperature is about 6000 degrees Centigrade. This is Olbers' paradox. It can be traced as far back as Kepler in 1610. It was rediscussed by Halley and Cheseaux in the eighteen century, but was not popularized as a paradox until Olbers took up the issue in the nineteenth century.

There are many possible explanations which have been considered.
Here are a few:

(1) There's too much dust to see the distant stars.
(2) The Universe has only a finite number of stars.
(3) The distribution of stars is not uniform. So, for example, there could be an infinitely of stars, but they hide behind one another so that only a finite angular area is subtended by them.
(4) The Universe is expanding, so distant stars are red-shifted into obscurity.
(5) The Universe is young. Distant light hasn't even reached us yet.

The first explanation is just plain wrong. In a black body, the dust will heat up too. It does act like a radiation shield, exponentially damping the distant starlight. But you can't put enough dust into the universe to get rid of enough starlight without also obscuring our own Sun. So this idea is bad.

The second might have been correct, but estimates of the total matter in the universe are too large to allow this escape. The number of stars is close enough to infinite for the purpose of lighting up the sky. The third explanation might be partially correct. We just don't know. If the stars are distributed fractally, then there could be large patches of empty space, and the sky could appear dark except in small areas.

But the final two possibilities are are surely each correct and partly responsible. There are numerical arguments that suggest that the effect of the finite age of the Universe is the larger effect. We live inside a spherical shell of "Observable Universe" which has radius equal to the lifetime of the Universe. Objects more than about 15 billions years old are too far away for their light ever to reach us.

Historically, after Hubble discovered that the Universe was expanding, but before the Big Bang was firmly established by the discovery of the cosmic background radiation, Olbers' paradox was presented as proof of special relativity. You needed the red-shift (an SR effect) to get rid of the starlight. This effect certainly contributes. But the finite age of the Universe is the most important effect.


Ap. J. 367, 399 (1991). The author, Paul Wesson, is said to be on a personal crusade to end the confusion surrounding Olbers' paradox.
Darkness at Night: A Riddle of the Universe, Edward Harrison, Harvard University Press, 1987

Q. What is Dark Matter?

The story of dark matter is best divided into two parts. First we have the reasons that we know that it exists. Second is the collection of possible explanations as to what it is.

Why the Universe Needs Dark Matter

We believe that that the Universe is critically balanced between being open and closed. We derive this fact from the observation of the large scale structure of the Universe. It requires a certain amount of matter to accomplish this result. Call it M.

You can estimate the total BARYONIC matter of the universe by studying big bang nucleosynthesis. The more matter in the universe, the more slowly the universe should have expanded shortly after the big bang. The longer the "cooking time" allowed, the higher the production of helium from primordial hydrogen. We know the He/H ratio of the universe, so we can estimate how much baryonic matter exists in the universe. It turns out that you need about 0.05 M total baryonic matter to account for the known ratio of light isotopes. So only 1 / 20 of the total mass of they Universe is baryonic matter.

Unfortunately, the best estimates of the total mass of everything that we can see with our telescopes is roughly 0.01 M. Where is the other 99% of the stuff of the Universe? Dark Matter!

So there are two conclusions. We only see 0.01 M out of 0.05 M baryonic matter in the Universe. The rest must be in baryonic dark matter halos surrounding galaxies. And there must be some non-baryonic dark matter to account for the remaining 95 % of the matter required to give omega, the mass of universe, in units of critical mass, equal to unity.

For those who distrust the conventional Big Bang models, and don't want to rely upon fancy cosmology to derive the presence of dark matter, there are other more direct means. It has been observed in clusters of galaxies that the motion of galaxies within a cluster suggests that they are bound by a total gravitational force due to about 5 - 10 times as much matter as can be accounted for from luminous matter in said galaxies. And within an individual galaxy, you can measure the rate of rotation of the stars about the galactic center of rotation. The resultant "rotation curve" is simply related to the distribution of matter in the galaxy. The outer stars in galaxies seem to rotate too fast for the amount of matter that we see in the galaxy. Again, we need about 5 times more matter than we can see via electromagnetic radiation. These results can be explained by assuming that there is a "dark matter halo" surrounding every galaxy.

What is Dark Matter

This is the open question. There are many possibilities, and nobody really knows much about this yet. Here are a few of the many published suggestions, which are being currently hunted for by experimentalists all over the world:

1. Normal matter which has so far eluded our gaze, such as
  • dark galaxies
  • brown dwarfs
  • planetary material (rock, dust, etc.)

    2. Massive Standard Model neutrinos. If any of the neutrinos are massive, then this could be the missing mass. Note that the possible 17 KeV tau neutrino would give far too much mass creating almost as many problems as it solves in this regard.

    3. Exotica

    Massive exotica would provide the missing mass. For our purposes, these fall into two classes: those which have been proposed for other reasons but happen to solve the dark matter problem, and those which have been proposed specifically to provide the missing dark matter.

    Examples of objects in the first class are axions, additional neutrinos, supersymmetric particles, and a host of others. Their properties are constrained by the theory which predicts them, but by virtue of their mass, they solve the dark matter problem if they exist in the correct abundance.

    Particles in the second class are generally classed in loose groups. Their properties are not specified, but they are merely required to be massive and have other properties such that they would so far have eluded discovery in the many experiments which have looked for new particles. These include WIMPS (Weakly Interacting Massive Particles), CHAMPS, and a host of others.


    Dark Matter in the Universe (Jerusalem Winter School for Theoretical Physics, 1986-7), J.N. Bahcall, T. Piran, & S. Weinberg editors. Dark Matter (Proceedings of the XXIIIrd Recontre de Moriond) J. Audouze and J. Tran Thanh Van. editors.

    Q. What is Big Bang?

    Stated most simply, the big bang theory holds that the universe had its beginning in a colossal explosion some 8 to 20 or so billions years ago.

    Some people seem to think that the big bang theory of the origin of the universe is "dead." Some think it a viable theory but think it holds that the universe arose in the big bang from truly "nothing." Some folks here seem to think that, if true, the big bang theory constitutes some sort of proof or support for the existence of a creator God.

    Based on present scientific knowledge and what the theory (theories, actually) "call for,", none of the above propositions are necessarily true.

    While the big bang theory has its disputants in the scientific community, it is by no means "dead," it remains as presently the leading cosmological theory, and is called "The Standard Cosmological Model" by Weinberg and P.J.E. Peebles (PRINCIPLES OF PHYSICAL COSMOLOGY, Princeton University Press, NJ, 1993, page 6).

    The big bang theory of the origin of our universe is presently the leading cosmological theory because of the compelling nature of the diverse and abundant scientific evidence corroborating it. Moreover, while the big theory does hold that the universe arose in a colossal explosion (or perhaps "inflation") from nothing like we know, it does NOT hold or require the arising of the universe from TRULY "nothing" (-ex nihilo-). Maybe it did arise truly -ex nihilo-, but the theory does not call for or necessitate that, it postulates the arising of the universe from mass/energy is a different state and form from the various states and forms of mass/energy with which we today are familiar or at least know exists.

    Even though it is presently the leading scientific theory of cosmology, the big bang theory is not unarguably established or universally accepted today. There are some other theories that are radically different from the big bang theory, and others yet which are signifi- cant variations on the big bang theme; all have at least some level of empirical corroboration, but none are presently as well-corroborated by crucial evidence as the big bang theory.

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