|MadSci Network: Physics|
Dear Chris, I’m afraid I must disagree with the author of the article you sited. I don’t agree that space-time and curved space-time are purely “mathematical constructs”, rather they seem to be the most natural way of describing the phenomena we see around us, like gravity. I wouldn’t say that they are the only way of describing gravity, but we seem to be led to them by the simplest assumptions that we can make. Before we talk about curves, let’s think about straight. What is a straight line, and how might we define it? Keep in mind that in general we must consider points that are not only at different places in space, but at different times, too! But let’s start simple. If we have two points in space at the same time, what is a straight line that connects them? Well, why not the line with the smallest distance, such as that formed by a “taut rope” as in the article. That certainly seems reasonable, and we can accept that for the time being. Now what if we consider two points (typically they are called “events”) that are at different places in space and at different times, too. What is a straight line that connects these two events? You can’t stretch a rope between two different times! Let’s pick two events within a coordinate system, for example the event at the origin (t, x, y, z) = (0, 0, 0, 0) and the event 10 second later and 10 meters along the x-axis, or (t, x, y, z) = (10s, 10m, 0, 0). What is the straight line that connects these two? You might guess the line that changes its “t” and “x” values in a uniform way, as (0, 0, 0, 0), (1s, 1m, 0, 0), (2s, 2m, 0, 0), (3s, 3m, 0, 0), (4s, 4m, 0, 0), (5s, 5m, 0, 0), (6s, 6m, 0, 0), (7s, 7m, 0, 0), (8s, 8m, 0, 0), (9s, 9m, 0, 0), (10s, 10m, 0, 0), might seem reasonable. What is this line? You might say it is the line taken by a body moving through space and time with a constant speed, 1m/s in the positive x direction. That is, the straight line corresponds to that taken by a body moving with no acceleration between the initial and final events. (Keep in mind that this idea would not work if the events are too far apart for a given time, this is because a body would have to travel faster than the speed of light to get from one to the other.) A general statement would then be that straight lines in space-time correspond to the motion taken by a freely moving body between the initial and final events. Does that sound acceptable? I hope so. We call these lines “geodesics”. Now, what is the “distance” between these two events? That’s another problem. The spatial concept of distance will not work, because if we look at these two events from a different reference frame, the measured distance will be different. For example, the body that moves between them will be at each event, and so thinks that the events occur at the same spatial position. The moving object will measure the spatial separation of the events to be zero! This means that ordinary distance is not very useful when dealing with events since its value depends on who is measuring it. I won’t prove this (you can find it in just about any introductory text on special relativity) but a useful invariant “distance” (it doesn’t change with different reference frames) is the “interval”. As defined in your cited article, in flat space-time the interval is defined as ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 (it can be defined like this or with a relative minus sign, it doesn’t really matter). The presence of the “c” just converts the units of time to that of space, that is, seconds to meters. Typically we can measure time and space with the same units by setting “c=1” so that 1 second = 3x10^8m. This is nothing more that unit conversion. Now, back to the article. The author claims that by manipulating the variables in the interval, the “purely time-like” value is made to look “length-like” by multiplying by c. I’m afraid this doesn’t make sense. An interval is either time-like or space-like depending on whether it is positive or negative. Given our definition, ds2 = -c^2 dt^2 + dx^2 + dy^2 + dz2, a positive interval is space-like, a negative interval is time-like, a zero interval is called “null” or “light-like”. You cannot make an interval “look” like something else by multiplying it by c. The author also suggests stretching a rope between two points on the Earth’s orbit and states that this distance is less than that moved by the Earth in the orbit, and therefore geodesics do not minimize distances. Geodesics do not minimize distances, they minimize intervals from event to event along an objects motion. The “rope distance” from A to B in the article is certainly less than the spatial distance the Earth travels, but the Earth also has a coordinate time change of three months, or so, which is not taken into account in the “rope distance”. As we mentioned before, the rope is not stretched between two events that are three months apart, and so cannot be used to compare the interval moved through by the Earth. So is space-time curved? Well, what is flat space-time? One in which parallel lines never intersect; one in which coordinates can be defined in a Euclidean fashion and the interval always has the form ds2 = -c^2 dt^2 + dx^2 + dy^2 + dz2. Unfortunately, or fortunately if you like this sort of thing, in the presence of gravitating bodies, initially parallel lines can intersect and there is no way to have an invariant interval with the flat space-time form. It turns out that it is even impossible to have a series of clocks that can be maintained to run at the same rate. Therefore, we seem to be led to the idea of curved space-time. Using whatever reasonable way we want to define straight, space-time doesn't seem to be flat! There are many great books that introduce Special and General Relativity. For General Relativity I would suggest “A First Course in General Relativity” by Bernard F. Schutz. Well, I hope I have answered your question, Chris. If you would like some more information, please let us know. Sincerely, Jim Guinn
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