Classical theories (including SR and QM) treat space as continuous nothingness; hence the term space-time continuum. In this view, objects exist in continuous space and interact with each other in continuous time.

Although this notion of space time continuum is intuitively appealing, it is, at best, incomplete. Consider, for instance, a spinning body in empty space. It is expected to experience centrifugal force. Now imagine that the body is stationary and the whole space is rotating around it. Will it experience any centrifugal force?

It is hard to see why there would be any centrifugal force if space is empty nothingness.

GR introduced a paradigm shift by encoding gravity into space-time thereby making it dynamic in nature, rather than empty nothingness. Thus, mass gets enmeshed in space (and time), space becomes synonymous with the universe, and the spinning body question becomes easy to answer. Yes, it will experience centrifugal force if it is the universe that is rotating around it because it is equivalent to the body spinning. And, no, it won’t, if it is in just empty space. But “empty space” doesn’t exist. In the absence of mass, there is no space-time geometry.

So, naturally, before the Big Bang (if there was one), there couldn’t be any space, nor indeed could there be any “before.” Note, however, that the Ashtekar paper doesn’t clearly state why there had to be a big bang. The closest it gets is that the necessity of BB arises from the encoding of gravity in space-time in GR. Despite this encoding of gravity and thereby rendering space-time dynamic, GR still treats space-time as a smooth continuum — a flaw, according to Ashtekar, that QG will rectify.

Now, if we accept that the universe started out with a big bang (and from a small region), we have to account for quantum effects. Space-time has to be quantized and the only right way to do it would be through quantum gravity. Through QG, we expect to avoid the Big Bang singularity of GR, the same way QM solved the unbounded ground state energy problem in the hydrogen atom.

What I described above is what I understand to be the physical arguments behind modern cosmology. The rest is a mathematical edifice built on top of this physical (or indeed philosophical) foundation. If you have no strong views on the philosophical foundation (or if your views are consistent with it), you can accept BB with no difficulty. Unfortunately, I do have differing views.

My views revolve around the following questions.

• What is space?
• Why is the speed of light important in it?
• Where does the Heisenberg Uncertainty Principle come from?

These posts may sound like useless philosophical musings, but I do have some concrete (and in my opinion, important) results, listed below.

• Are GRBs and Radio Sources Luminal Booms? (An article published in IJMP-D, which became one of the “Top Accessed Articles” of the journal. )
• Light Travel Time Effects and Cosmological Features (Trying to get this one published.)

There is much more work to be done on this front. But for the next couple of years, with my new book contract and pressures from my quant career, I will not have enough time to study GR and cosmology with the seriousness they deserve. I hope to get back to them once the current phase of spreading myself too thin passes.

I picture the earthquake (in cross-section) as a cantilever spring being held down and then released. The spring then transfers the energy to the tsunami in the form of potential energy, as an increase in the water level. As the tsunami radiates out, it is only the potential energy that is transferred; the water doesn’t move laterally, only vertically. As it hits the coast, the potential energy is transferred into the kinetic energy of the waves hitting the coast (water moving laterally then).

Given the magnitude of the energy transferred from the epicenter, I am speculating what would’ve happened if there was no mechanism for the transfer. Any thoughts?

In answering my question, Martin and Swansont (who I assume are academic phycisists) point out my misconceptions and essentially ask me to learn more. All shall be answered when I’m assimilated, it would appear!

This debate is published as a prelude to my post on the Big Bang theory, coming up in a day or two.

But what will happen if the twins both accelerate symmetrically? That is, they start from rest from one space point with synchronized clocks, and get back to the same space point at rest by accelerating away from each other for some time and decelerating on the way back. By the symmetry of the problem, it seems that when the two clocks are together at the end of the journey, at the same point, and at rest with respect to each other, they have to agree.

Then again, during the whole journey, each clock is in motion (accelerated or not) with respect to the other one. In SR, every clock that is in motion with respect to an observer’s clock is supposed run slower. Or, the observer’s clock is always the fastest. So, for each twin, the other clock must be running slower. However, when they come back together at the end of the journey, they have to agree. This can happen only if each twin sees the other’s clock running faster at some point during the journey. What does SR say will happen in this imaginary journey?

(Note that the acceleration of each twin can be made constant. Have the twins cross each other at a high speed at a constant linear deceleration. They will cross again each other at the same speed after sometime. During the crossings, their clocks can be compared.)

In the animation above, the laser pointer is at the bottom-center of the figure. I’m turning it so fast that the laser dot on the ceiling should travel at 10c (which is indicated by the white circle moving across the ceiling.) But it takes a while for the laser to reach the ceiling. The light is indicated by the small red dots which move towards the ceiling (10 times slower than the white circle). The dot appears when the light hits the ceiling.

As you can see, the light from the laser first hits the ceiling at a point close to the top (indicated by the black dot), and subsequently, light on either side starts hitting the ceiling, making two dots (in yellow and green). This is the how one laser pointer creates two dots appears at two places at the same time. Note how the dots slow down considerably as they move away from the center. Light travel time effects dominate at shallow angles.

But CPL.Luke is right. If the ceiling was a spherical shell and the laser was at its center, there would be only one dot moving at 10c. (At least, that’s what I get when I try to work it out.) In effect, by having a spherical ceiling, you are cutting out the shallow angles; the laser is always perpendicular to the ceiling. In this case you can treat the laser as a solid rod, but with a constant delay equal to r/c (which differentiates to zero, thus not affecting the speed of the dot).

If a superliminal object did exist, would it appear as two identical objects to us?

Okay, I will start with an animation first to make it interesting. I will post the notations and algebra in the next post.

In the figure below, we have a (purely hypothetical) superluminal object – the white circle flying across the animation at ten times the speed of light. As it flies by, it emits light. We consider the light rays (the red lines with small red circles at the end) coming towards the observer at the bottom-center of the animation. As we can see, the first ray of light that reaches the observer is emitted at a point close to the point of closest approach to the observer, indicated by a black dot that appears when the ray reaches him, say at time=. The rays emitted before this first ray reach the observer after . This reversal of the order in which the rays reach the observer gives rise to the perception of two objects moving away from the black dot. (If the object doesn’t change during its flight, the two “phantom” objects are identical to each other.)

Now, my question is, if we see two objects in a symmetric formation in the night sky, can we be sure that they are really two, and not our perception of one object in motion? Of course we can if we say that nothing can really travel faster than light. Assuming hypothetically that we didn’t know about SR and its constraint on the speed, is there any way we could work out the “real” speed from our observation of the rate of angular separation? My feeling is that there are at least two configurations (one superluminal object going in one direction or two objects – superluminal or otherwise – going in opposite directions) which will result in the same observation.

Algebra behind the animation

This post gives the algebra behind the animation. First, let’s define the notations used using the following figure.

Here, the object is traveling along the thick horizontal line at a speed . The black dot in the animation (where the object first appears to the observer) is B’. B is the point of closest approach. Let’s set the time when the object is at the point B. The line of flight (at its closest point B) is at a distance of y from the observer at O. A is a typical point at a distance x from B. is the angle between the line of flight and the observer’s line of sight. is that the angle that the object subtends at the observer’s position O with respect to the normal. Let’s set to simplify the algebra, so that , the observer’s time is . (A- is another representative point where and are negative.)

With these notations, we can write down the following equation for the real position of the object at time :

Or,

A photon emitted by the object at A (at time ) will reach O after traversing the hypotenuse. A photon emitted at B will reach the observer at , since we have chosen . We have defined the observer’s time such that , then we have:

which gives the relation between and .

Expanding the equation for to second order, we get:

(Call this equation Q.)

The minimum value of occurs at (which defines the position of the black dot in the animation, the point B’) and it is . To the observer, the object first appears at the position . Then it appears to stretch and split, rapidly at first, and slowing down later.

The quadratic equation Q above can be recast as:

which will be more useful later in the derivation. (Call this equation U.)

The angular separation between the objects flying away from each otheris the difference between the roots of the quadratic equation Q:

making use of the “useful” equation U above. Thus, we have the angular separation either in terms of the observer’s time () or the angular position of the object () as illustrated in the next figure, which illustrates how the angular separation is expressed either in terms of the observer’s time () or the angular position of the object ().

The rate at which the angular separation occurs is:

Again, making use of the useful equation U. Defining the apparent age of the formation and knowing , we can write:

Note that in order to go from the angular rate to the speed (even the apparent speed), we need to estimate , which is model-based.

Now, can you tell me why the object doesn’t appear at two places if ?

In fact, there is much more to this puzzle.

• The reason the dot on the right doesn’t move faster than c is closely related to way the relativistic speed limit is derived.
• One should also consider that saying that the dot is on the ceiling at a particular point at a certain instant of time is not good enough. When will the observer (presumably at the laser gun) see it? There is one more leg of light travel (from the dot back to the gun) that needs to included. This consideration may be behind the definition of simultaneity (using the round trip travel of light) in SR.
• To the left of the point of separation (the black point), the flow of time is reversed. In other words, if you changed the laser color as you scanned from the left edge of the figure to the center, the change in the color of the dot would appear in the reverse order (in time).
• The Doppler shift also is reversed in this region. This is why I was intrigued by the left-handed material, where the group velocity and the Doppler shift are reversed.