Caruthers

Before asking additional questions I wanted to see, and hopefully understand, your full explanation and I am still working through bit by bit. I also got sidetracked getting my sailboat ready for winter. As you are aware my focus is on how time is represented. What has become clear is that in these diagrams the slope becomes a factor of c, and that makes sense to me. I am now thinking about what it means to rotate the "time" axis. I get the math, but what does it mean for the direction of time to point in a different direction? Anyway, still studying your excellent presentation of the subject.

Got it. Very clear and helpful. If is not in a book it should be! Eagerly awaiting the next part.

As a mechanical Engineer I have worked in aerospace, oil & gas and software simulation.. Should the right hand side of the equation also contain i ? Evaluating the equation assuming 1. The object is traveling only in the x direction and y and z = 0 2. Using units of million meters per second 3. The object travels 300 units in one second (i.e. approximately the speed of light, c) Then we get either Root(300^2 + i.300^2) = root(300^2 - i.300^2) and I don't understand what i is or where it came from ,or if the i is not present in rhe right hand we get Root(300^2 + i.300^2) = root(300^2 - 300^2) and I still don't understand i I am confused 😢

Not really a problem, I am just asking about something I don't, yet, understand. Yes, you can plot anything against anything and some times it is meaningful, and sometimes it isn't. In this case I am learning about space-time diagrams. I had assumed that one of the axes had to be time. I understand that in order to simplify the diagram one dimension of space is often shown. In that case I expected the other would be time. Apparently that is not the case as, in order to create the equations, all axes must be in the same units and are expressed as distances. The x axis is the distance an object moves and the y axis is the value of ct, which is also a distance. I suppose that the value of t is the time that the object takes to reach the distance x, however multiplying by c, which has units of distance/time, results in a distance. This seems to me to result in determining - what is the distance an object travels when it travels a distance x, which obviously doesn't make sense, so I am asking what is the right way to understand the use of ct as an axis. My question then is how is time represented in a space-time diagram. When you say "related by a constant of proportionality" taking a simple case such as a constant of, say, 300 then should we say something like y (meters) = 300x (meters) ? Taking an object moving along the x axis 300 units in 2 seconds then the value of y (ct) would be 600, i.e the point plotted would be at 300 units in one direction and 600 in the other. This has been explained to me as not being a "regular graph" but should be thought of as a map. I guess my basic question is what is the meaning of this type of diagram and where is time represented in a space-time diagram.

Does i denote an imaginary number?

Yes, I do recognize that you are trying to help me, and I thank you for that. I am a retired engineer with, finally, a little time to study subjects like this solely for pleasure. I am ok at calculus, and have read through several papers on relativity, but at this moment I am focused on understanding the progression from Einstein's work to the graphical representation. You are also correct, my last post contained a silly error. Speed/velocity is not represented by a point and I do understand that. I guess my fingers were ahead of my brain. Thank you for pointing that out. Coming back to the topic, just to be sure we agree on terminology, a map is a graphical representation of the location (e.g. latitude, longitude, elevation) of various features. I understand your analogy of a space-time diagram to a map as both axes are distances, however that brings me back to my original question - if it is a "map" of distances how is time represented? If we consider our diagram to be scaled in units of, say, million meters/second then an object travelling a c will reach 300 (approx) Mm/s after 1 second. Now we need to decide where it will be on the y axis. Given that the y axis is ct then we can say it is at c X 1 second = 300 and we can locate that point. Now lets consider an object travelling a 0.5 c. After 1 second it will be at 150 M m/s. As before we determine the value on the y axis which would again be 300 X 1 second. Now we have an object "mapped" to 300 on one axis and 150 on the other and we can plot that, however we have no representation of time and cannot determine the value of time at that point simply be looking at the diagram. The diagram is representing space-space not space-time. I hope I have explained this properly and that my terminology works for you, and please remember this is a subject where I am in the learning phase.

Yes, surely it is supposed to be a plot of distance against time. I thought the essence of a space-time diagram is to represent a one dimensional space against time. If that is correct then any point plotted does represent a velocity. I am not sure I understand the analogy to the surveyor.. The surveyor plotting 2 or 3 dimensions in space in order to create a map is not considering time. If the diagram plots distances against distances where does time appear? and how can we call it space-time?

Thank you for your help.. Why can't I pick seconds for the time axis and light-seconds for the distance axis? That way the plot would be velocity. "Consistent units" of plotting distance against distance seems non-intuitive. If we take an object moving at c then at one second it will be at a distance of one light second. This is on a 45 deg line as expected. If an object travels at 0.5c then after 1 second it has traveled 0.5 light seconds. Since ct = x then we must plot ct at 0.5 which is on the same 45 deg line. Any object traveling at any speed seems to lie on the same 45 deg line. I don't quite understand. It only makes sense to plot different units. We do this all the time - distance v time, acceleration v time, height v age, etc. We do not invalidate a plot of height against age because they have different units. Measuring along a curve of height against age is pretty meaningless. A "composite" as you describe it of (say) meters v meters is like saying - how many meters are there per meter. The answer in every case is 1.

When constructing a space-time diagram using Minkowski's methodology the time axis is defined as ct. C × t = distance This is akin to saying - how far does light travel when it travels one meter. The value of c is large, therefore to create a meaningful diagram the values of x must be correspondingly large. Perhaps a suitable scale would be light-seconds. In the case of an object traveling at c, then in one second it will travel one light-second. The time axis in a space-time diagram is scaled also in light-seconds (ct for 1 second). Doesn't this simply mean that the time taken to travel one light-second is one second, or in reverse the distance light travels in one second is one light second. Why aren't the axes, for example, t (time in seconds) and x (distance in light-seconds).