Lizzie L

Absolutely (ETA and apologies!) The OP said: And I sort of agree, still. Units of time are defined in terms of clocks, and clocks involve oscillating things. A world without change would be a world without oscillators for a start - and I don't see that "time" would exist as a dimension in such a world. Until someone can persuade me that there is a spacetime that is unit free I'm sticking to my guns here But what seems more interesting to me is why time should be unidirectional, and the spatial dimensions bidirectional. And I think that has more to do with the way we think than anything "out there" in the world.

Oh, give over, Schneibster. You were the one who said "Note that Fourier transforms are from the frequency domain to the time domain". I just pointed out that they go the other way. It's true that the inverse function will often take you back (although not in practice in my field). Neither of us were "pwnt". I'm neither playing games nor "trying little digs". I was just trying to show you that "x-vt" doesn't give convert the x dimension into t! Well, it depends entirely on what you are talking about. But when the transform is from one reference frame into another it doesn't change the units. Which was my point. So it's good if you agree. Well, no, I'm not. I don't think that "space and time" are "essentially the same thing". I think they are dimensions of the same thing ("spacetime"), but I think they are in different units. And I don't think that is "denying relativity". But clearly you think I've got that wrong. That's fine. Let's leave it there.

Yes, for some functions you can use the inverse Fourier transform to get your time series back. But this is a bit off topic. The Lorentz operation transforms your data into a different reference frame. It doesn't transform the axes into different units. At least not as far as I can see, and at least some people seem to agree.

Time domain to frequency domain.

What is true, of course, is that you can express distance in terms of time by invoking a velocity. Hence light-years and light seconds. But that doesn't turn them into the same units. It does make for neat equations though, because we can declare the speed of light ot be one, and say that moving one light second along a spatial axis is equivalent to moving one second along the time axis, and your cone will make a neat 45%. But the units are still different. A light second is not a second. And I think when you defined the speed of light as one second per second, you meant one light-second per second. Which is obviously true, by definition.

Well, that's a relief!

No, I didn't leave out time. See the part where it says "one hour"? Anyway, Schneibster, I didn't come here to wrangle with you. I followed your link, and it looked like a cool forum and the nature of time is sort of in my baileywick, after a fashion. So I'll eave the math argument to better mathematicians than I. I'm going to stick to my guns, though, and say that in an unchanging world, time would have no meaning, or, in other words, that time only makes sense in world in which things change. And the fact that our units of time require reference to some oscillator seems to me to demonstrate that quite well. As for the direction of time - I think that is largely an artefact of what we are privy to as observers, and what we can predict from what we are given in the present.

Let's take a simple Galilean transform for two reference frames, and an object travelling at velocity v along the shared x axis: At any given time, the x coordinates of the object in reference frame xyz will be given by x minus the velocity of the object times the time it's been travelling. So if it's been travelling at one mile an hour for one hour, we subtract 1 mile off the x' axis to get the coordinates on the x axis. And the other coordinates will, in this example, be the same in both. So the transform works just fine, and doesn't convert spatial units into time. 3 miles minus the distance travelled by object x in one hour is still a distance.

No, it doesn't. The units in v2/c2 cancel (as both are velocities) leaving us with x-vt which is the same as a galilean transform - you just move the thing along the original x axis by the amount of distance that the thing will have travelled in time t. And a square root, but that's not going to turn space into time.

Because you are also rotating the t axis. You aren't rotating time into space. You end up with a new set of xyz axes and a new t axis, and the units remain distance on the xyz and time on the t. I'm a patient woman

I'm paying attention to the math, but I'm not seeing space and time proved equivalent. I'm not seeing the transform as transforming space into time. I'm seeing them as transforming an event from the reference frame of one observer into the reference frame of another.

No problem It could be wrong anyway. But it does have some evidence and reasoning behind it as I said.

No, it didn't. I'm finding the "speed of time" concept incoherent, and the definition of the speed of light as second/second equally so, and I don't see how it flows from anything you've posted or my admittedly crude understanding of relativity. And you haven't persuaded me that the units on the x, y and z units are the same as the units on the t axis. But my ear is open like a greedy shark, to catch the tunings of a voice divine. (Keats, heh).

Well, the light cone is 45% if you choose light-[insert time unit here] for the spatial axes and [insert the same time unit here] for the time axis. It's not a postulate so much as a convenience. And it doesn't put the spatial dimensions into the same units as the time axis. The spatial dimensions have units of distance and the time axis has units of time. You seem to me to be contradicting yourself, but I'll leave it there until we get some more input. ETA: these self-aggregating posts are rather cool, but I got confused as to who this response was from! OK, so I agree with this. Schneibster seems to disagree.

The speed at which future seconds become past seconds. It's one second per second. Which is c. Really?

Well, if we are going to measure time in seconds, that's fine by me, and by using the speed of light we can readily get a distance measure from that e.g. light-seconds (we don't need metres as well). But in that case I rest my case: time is measured in terms of change - if you don't have a changing thing, you can't measure time. And if you can't measure time, you don't have a way of putting anything on a t axis. QED *harrumph*

What do you mean by "the speed of time"?

Well, no, you can't define the time by the speed of light. You can express distance and time in terms of the speed of light, so that distance is measured in, say light-seconds, as you suggested, and if you want to keep your conic angle at pi/2 then you can measure your time in seconds. But that won't help you define the seconds.

I get that. My point is really not very profound and it is simply that along the spatial axes, relative or not, we have units of light-seconds (or whatever time unit you want to use) and on the t axis you will have units of seconds (ditto, if you are setting c at 1). So we need a definition of seconds (or years or whatever). For which we need to refer to an oscillator, no? Sure. I get that. I didn't not Absolutely. But it's orthogonal to my point, as it were It's a beautiful diagram.

Well, no I hadn't, but I appreciate the respect That's fine. But we still have spatial axes and time axes, don't we? Yes indeed. So explain to me the units on the two types of axes, or explain to me why there aren't two types of axes, or why we don't need units for them. Well, I don't understand his claim that I don't see how we can move forward in time at the speed of light i.e. at distance per time interval

In "the math" we have 4 axes - x, y, z and t. As John Baez says in Schneibster's link, we can set the speed of light to 1. That means that if the spatial axes are measured off in light-seconds, the time axis will be measured off in seconds, and the cone will have an angle of pi/4. And my fairly simple point is that those time units are defined in terms of some kind of clock, or oscillator, whether it's the orbit of the earth round the sun, the rotation of the earth, or some property of cesium.

I guess you could if you had a mirror far enough away.

Just to push things a little further - you could argue that "physical reality" - is inside both cones. And we can infer what that reality is by fitting models to data. It won't be perfect in either case, but depending in the kind of data we are talking about, some future models will be more accurate than past ones. Some past histories will have no single solution, given the data, whereas some present states will have a single future. That would be true of some kinds of deterministic worlds, wouldn't it? Yes. You are preaching to the choir here I actually teach quantitative methodology! And I never said "time is change". That's something of a straw woman. Well, I'm not sure that it's their mathyness that makes them practically useful. But I agree (well I would assert!) that the criterion by which we evaluate a model is its usefulness, and for scientific models, that's often quantitative accuracy. General Relativity. Isn't c in there somewhere? I mean that the parameters in the equations include time in units - e.g. seconds.

Fair enough. But it does, doesn't it? I mean it shows quite clearly, I'd say, that "the math" isn't unit-free - it's based on an intrinsic definition of time that relates to some kind of clock. Because you are standing in your own shadow

OK, cool. So if we have time in seconds on the vertical axis we have space in light-seconds on the horizontal axes, right? So how are we defining "seconds"? 9,192,631,770 periods of radiation between the hyperfine levels of cesium 133?