Minkowski space

ghwellsjr
Gold Member
Hi ash, you seem to be doing pretty well with this topic. At the risk of being too simplistic, the thing to understand about "minkowski space" is that, like euclidean space where you have x, y and z coordinates and triangles obey the pythagorean theorem, minkowski space is the same, that is, everything within the space is related to everything else in the same "orthogonal" way as in euclidian space, except that there is this additional axis "t" to account for time.
It's not that you are being too simplistic, but you are teaching wrong concepts in what follows. You have a lot to learn yourself.

In the three dimensional euclidean space, we are used to thinking about a "point" in space which can be identified by the unique x,y and z coordinates that identify that location. In Minkowski space, everything that relates to a unique locus of x,y,z and t coordinates is considered to be not just a "point" location (a place), but rather an "event", because the coordinates include a "place" for time. The difference between the two is that if you want to describe the motion of the earth around the sun in euclidian space, then the path of the orbit is described by an elipse and the earth just keeps going around and around on the path traced by the elipse. In spacetime, the path is described by a helix. This is because you are describing the orbit where the earth's location at any point in time as time progresses. Thus, in spacetime, the orbit of the earth will never retrace itself because, as it orbits the sun, it continues goes around in space in an eliptical rotation that moves along the "time axis". So, what looks like an elipse in euclidian space looks like a helix in Minkowski space.
This is not the significant difference between Euclidean and Minkowski space. You should also add the time coordinate in Euclidean space to show the orbit of earth as a helix that never traces out that same points.

Special relativity describes how different observers at different places (in space and time), relate to each other and to other "events" that they may observe that happen at other places in space and time, and especially where they are in relative motion with respect to each other and/or to such events. So, when people ask about what is the meaning of Minkowski space, the key thing to understand is that everything located in Minkowski space is an event, not just a place. This characteristic has lead to the use of the term "space-time" to describe what Minkowski realized about the meaning of Einstein's theory of Special Relativity and lead him to develop "Minkowski space".
The significant difference between Euclidean and Minkowski space is in the transforms that get you from one frame to another. In Euclidean space, time is not in the transform whereas in Minkowski space, it is.

Others here have posted adequate explanations of the questions you had about what the terms "space like", "time-like" and "light like" refer to. In thinking about these terms, the most important thing to remember is that the speed of light is constant....everywhere, for everybody no matter how fast they are moving toward or away from the source of the light.
That is the significant difference between Minkowski space which is flat and curved space where the speed of light is not the same everywhere.

This fact has important consequences for any set of observers of events happening in space-time (Minkowski space). First, depending on how far away you are from an event....let say, and exploding star, you could observe that event way before someone else who is located much further away from the event than you are. But, on the other hand, he could be much closer to another exploding star than you are. Now if you think about it, if both exploding stars blew up at exactly the same time, but you see the explosion of Star A before you see the explosion of Star B (but you know they exploded at the same time), you can conclude that you are closer to Star A than to Star B. But, if someone else sees Star B explode before Star A, and they also know that both stars exploded at the same time, then they will know that they are closer to Star B than Star A. But, if neither of you know when either star exploded, and the other guy tells you he just saw Star B explode, and you tell him , heck, I just saw Star A explode, and he looks up in the sky and says...Star A is still there, and you look up in the sky and say, well, Star B is still there, you can say that there is a problem that you both are not able to simultaneous observe those two events at the same time from different locations even though both events actually happened simultaneously.
Your last comment, "actually happened simultaneously", is a strong indication that you do not understand relativity. In fact, all these same characteristics that you describe apply in Euclicean space with a finite speed of light.

Another thing that reveals that you do not understand relativity is your statement, "the other guy tells you he just saw Star B explode", as if instant communication over a long distance is possible.

The ability to observe an event in space-time is intimately bound up with the time it takes for light coming from the source of the event to reach the observer. It is this fact that gives rise to the concept of the light cone in space-time. In order to understand it, you have to understand that in Minkowski space, the time axis, "t", is to be treated conceptually as orthogonal to each of the other space axes.
There's more to it than that. This would also be true in Euclidean space with a finite speed of light.

So, when you look at an image of a light cone, you are actually looking at a "surface" which relates the speed of light traveling through space to locations where that light can be observed. If you are located in the "path" of light coming from the source at the origin of the light cone, you have to be located somewhere on that surface. But, you have to always remember, that every observable event that happens in space time has its own unique light cone. The only events you can observe happening are those events where the surface of the light cone intersects the space-time coordinates of where you are "located" in space-time...that is...the "event" of "where/when" you are making your observation. Now, if you are in motion with respect to events that are happening (or did happen), you are changing the where/when of your observation with respect to the light cones of those events. Depending on how your motion relates to the light cones of those events, you may not be able to observe those events where if you were not moving, you would have been able to observe them.
No, events do not have unique light cones. And your statement that you can only observe certain events is totally wrong. Everyone can observe all events no matter where they are if they just wait long enough.

With this understanding, it is possible to gain some insight into what is meant by space-like, time-like and light like events. Lets go back to the situation of two observers, only, instead of thinking about two different exploding stars, lets think about two comets. On Sunday night, at 8:00 pm, you look up in the sky and you note that you see Comet A passing exactly across the path of the North Star, or better, at celestial coordinates x(a), y(a), and z(a), and they see Comet B, way over on the horizon. Someone else located far away (like, on a planet in orbit around another star far away), walks out on the very same Sunday at 8:00 pm and looks at the exact same celestial location (x(a), y(a), and z(a) where you are looking, but they see Comet B at the same place where you are seeing Comet A! When they look to find Comet A, it is way off at a completely different place in the sky!!
Again, a total lack of understanding of relativity. Your idea that there is a "very same Sunday" betrays your lack of understanding.

Now the fact is, that Comet A and Comet B never come close to one another even though they trace paths through Space which intersect at x(a), y(a), and z(a) at completely different times. So, how is it possible that both you and the other guy look up at exactly the same time, and see two different comets at exactly the same place?? Well, clearly, one of you is watching something that the other one won't see for a long time...that is, they are watching an event that the other one will only be able to observe in their spacetime future, while the other one is watching an event that happened long ago in the other one's spacetime past, even though you both are making your observations at exactly the same time and observing completely different events happening at exactly the same place. But, at the time that both of you are making your observations, the fact is that neither comet is actually at x(a), y(a), and z(a). We'll call this situation the Comet example and come back to it later.
The fact that different observers see events in different orders is also true in Euclidean space with a finite speed of light but does not justify your continued statement "at exactly the same time".

Now, lets consider a different example of two exploding stars. In this example, the two exploding stars are exactly the same distance from you, but, there is a gigantic gas cloud that lies closer to exploding star A than exploding star B and lies between the two stars such that it would take 3 months for light to get from star B to the gas cloud, but only about 1/3 that time for the light from star A to get there. However, the light from both exploding stars will take at least a year to reach you. Now, if you observe the gas cloud lighting up exactly one month after you saw both exploding stars blow up, can you determine which exploding star caused the gas cloud to light up?
This has nothing to do with relativity. If one exploding star is responsible for lighting up gas cloud, then all explanations would have to conform to this fact.

I have to run now, but will come back to continue this if it seems to be helpful, but think about which situations described in these examples might be space like, time like or light like.
Please, study Special Relativity yourself. You are in no condition to be teaching it.

@ghwells; Good grief! I appreciate the energy you devoted to evaluating my post. The idea of over simplifying the concepts ash was wrestling with was concededly probably ill-conceived and, as you so thoroughly pointed out, probably more troublesome than helpful. I deleted the post in its entirety and leave it to you guys to help him gain an exacting appreciation of the subject matter, which is no doubt what he was looking for. BTW, I do accept private messages if you have any concerns about my posts.

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Dale
Mentor
2021 Award
The equation that you said is still a Pythagorean theorem!!
\Delta{s}^2 + c^2\Delta{t}^2 = \Delta{x}^2
Doesn't this look like Pythagorean theorem?
It isn't very useful to think of it as a triangle, it is better to think of it as the radius of a circle. If you start at the origin of a Euclidean space and trace out all of the points that are a distance ds away then you use the usual formula and the result is a circle. If you start at the origin of a Minkowski space and trace out all of the events that are a timelike interval ds away then you use the relativistic formula and the result is a hyperbola.

ghwellsjr