Blog Entry 11.19.07


Spacetime geometry
The endless pursuit of something I may never know.

“For if each Star is little more a mathematical point, located upon the Hemisphere of heaven by Right Ascension and Declination, then all the Stars, taken together, tho’ innumerable, must like any other set of points, in turn represent some single gigantick Equation, to the mind of God as straightforward, as say, the Equation of a sphere, - to us unreadable, incalculable, a lonely, uncompensated, perhaps even impossible Task, - yet some of us must ever be seeking, I suppose.”

- Thomas Pynchon, Mason & Dixon

That is perhaps one of the most beautiful quotes I have read in a physics book. For those familiar with the illustrations, it wouldn’t take long for you to conclude that it is with reference to the subject of general relativity (GR), also a beautiful field in physics I must add. To see the connections Einstein made between space, time, light and gravity is simply breathtaking. But before you can enjoy splendor in the concepts used, you need to of course understand what GR is. Now that’s the hard part.

“Argh!!!” Sorry, I just have to let that out. I have been spending close to a month studying GR from a book written by James B. Hartle and I’m still stuck at figuring out how this spacetime diagram works. Not to blame it on the author, it is a great book after all, my slow progress in learning is possibly due to the fact that I’m studying a third year university module (okay, maybe a second year module for Ivy League schools) without a solid foundation in freshman physics. Still, I am not about to give up.

For the GR students, here is where some real physics talk comes in*. The preliminaries for spacetime are quite easy to grasped. We have to depart from a Newtonian structure of spacetime because at speeds close to light, such laws of mechanics fail. Since we live in a world where objects in our lives are nowhere near that speed, we have grown accustomed to such laws. But where is the fun in that? We want to know what happens when we approach the speed of light. So we bring in the spacetime concept to set the background where the study of both SR and GR takes place. Instead of a spatial three-dimensional set with an absolute notion of time, we replace it with spacetime, a four-dimensional set in a given inertial frame where ‘time’ is no longer absolute and it is treated as any other axis, just that we can only move forward in time. Did I get this right so far?


Moving swiftly along in my study, I see the justification for this. Probably the first example would be the thought experiment of two people in a space ship emitting a light at different times but received by a middle person in the same time. A separation notion of time is needed for each person. Then, I get equipped with fundamental concepts of differential geometry, taking a line element and integrating to get the distance. This logically leads to the definition of a spacetime interval given by

which is invariant for whatever frame we are using. Things are still manageable up to the point where the spacetime diagram comes in.

This spacetime diagram, or Minkowski diagram named after the mathematician H. Minkowski, is the tool used to analyzing the interval between events. We usually draw it as the x and y axis indicating the spatial dimension (leaving the z axis for simplicity), and the time t as the axis perpendicular to both of them. For a certain event p, we can generate a light cone to indicate the other events, which can affect p. This light cone is the set of points that are connected to p by straight lines moving at the speed of light.

Here is where my intellectual ability falters. By the definition of the spacetime interval, we can have a position, negative or zero value for the interval. We give certain names for this interval namely

and can also indicated them on the spacetime diagram accordingly. This classification of three different time intervals simply baffles me.

Firstly, while the interval is zero for null separated events, I am having hard time distinguishing whether this means the object is stationary or whether the object is moving. I have read that null separated events are connected by a light ray. Still, what does connected by a light ray mean? Does it mean this: I stand still with a stopwatch in my hands, press the start button, trace a light emitted from face that travels to another observer, the observer receives this light ray, and then I stop the stopwatch. Are the two events, me pressing the stopwatch and the observer receiving the light, null separated? If so, I didn’t move a step. But we require  for this to happen. This confusion arises because I am stuck at thinking ‘physical distances’ between objects as opposed to ‘event distances’ between events.

Just when I thought that was hard enough to understand, the other two terms stirs more pandemonium. I understand that the speed of light is the maximum speed we can travel and so if two events were to affect each other, they must be within each other’s light cone. But again, this raises two questions. Suppose I have an event that is changing at a certain point, take for example, me bouncing a ball but staying at a certain x, y, z, position. Is this event being affected at all? And if so, what does a light ray have to do with it. Secondly , a squared term, taking a negative value. That’s down right devilish to handle.

Finally, I also read that nothing in this definition depends on a particular inertial frame. Woah! This means that I could chose a certain inertial frame, perform the analysis using the given equations, if I first understand how to do it that is, and some how or rather end up with the same results.

So my journey of learning GR continues. I admit that I am making some progress albeit quite slowly. I took a trip down to the library that day and browse through another book by Sean Carroll, which shed some light on this spacetime diagram. To me, GR is not easy. I can only hope that one day I can comprehend, though not the level of the mind of God, the four dimensions that span spacetime.

I guess that was what Thomas Pynchon meant. The road to learn the inner workings of science may prove very difficult. But isn’t it comforting to know that no matter what car one drives, for a professor maybe a Ferrari and for me a Volkswagen Beatle, each one of us will always move forward.

*I always want to make my blogs as intellectual as possible. If you are a casual reader, I’ll be glad if you can understand the slightest bit of what I say from here on.