Time and space are functionally identical and can be meaningfully described in the same units; and of this I have found the most marvellous proof -
Wait one, someone’s just come up the road from Porlock and I can’t post while he’s banging at the door. :mad:
What is the sound of one hand clapping……you across the face for asking such a question!?
You have to understand what is meant by “dimension.”
Simply put, a dimension is a number required in order to locate a point, given a starting point.
Thus, in a one-dimensional space, you only need one number to locate the point. In two-dimensional space, you need two numbers. In three-dimensional space, three, etc.
But these numbers need not be linear measurements. For instance, in a two-dimensional space, you can locate the point given an angle and distance. Thus the point at 2, 2 on a plane can also be located with an angle of 45 degrees and a distance of the square root of eight (assuming my math is correct – but you get the idea).
Now, lets assume in our two dimensional world that you are trying to hit a moving target. Point 2,2 is fine if the object is stationary, but to hit the moving target, you need to include the time where the target is at point 2,2. Time thus becomes a dimension.
And we use time as a dimension all the time. “It’s a one-hour drive,” for instance, is one way of locating a position. (Note that we often leave out dimensions for convenience sake. You would say, “It’s five miles to the gas station,” and ignore the depth and breadth, since they don’t matter and a road can be considered to be a line, even though, technically, it’s three dimensions, not one.)
[QUOTE=Frylock]
How many seconds are there in a meter?
[/QUOTE]
The # of seconds in a meter varies, based on the ammount of energy expended in traversing the meter.
[QUOTE=RealityChuck]
And we use time as a dimension all the time. “It’s a one-hour drive,” for instance, is one way of locating a position. (Note that we often leave out dimensions for convenience sake. You would say, “It’s five miles to the gas station,” and ignore the depth and breadth, since they don’t matter and a road can be considered to be a line, even though, technically, it’s three dimensions, not one.)
[/QUOTE]
Do I understand correctly that you’re point here is to offer a case in which we use a “time measure” in order to indicate a “space distance?” So for example, in this case, the answer to my OP would be something like “In this case there are five miles in an hour”?
-FrL-
[QUOTE=Pasta]
I grant your point. The OP’s question can only be addressed if one is willing to place an artificial constraint on the form of the bilinear. The answer, then, is as artificial as the constraint. That is, the answer is not a physical law but a metric-dependent statement. A relevant statement, given the ubiquitousness and Occam-pleasingness of the “+/-1” metric, but yes, a non-fundamental one.
[/QUOTE]
Thanks for all the discussion.
It looks like you guys are zeroing in on saying the answer to the question “How many meters in a second” could be anything, depending on what arbitrary equivalence you’ve agreed to use beforehand.
If that’s right, then just to make this more clear to me, could you give me an example as to how I could, so to speak, “make” one second equal one meter, and then an example as to how I could, so to speak, “make” one second equal two meters?
-FrL-
[QUOTE=FRDE]
There is a straightforward answer to this
[/QUOTE]
this was the best answer by far. Until you clarified it was an error. 
[QUOTE=Frylock]
If that’s right, then just to make this more clear to me, could you give me an example as to how I could, so to speak, “make” one second equal one meter, and then an example as to how I could, so to speak, “make” one second equal two meters?
[/QUOTE]
The numerical answers given above all set 1 time unit equal to the distance a massless particle travels in that time unit. That happens to be the velocity c that shows up in the equations as written typically. But, if your equations had a bunch of twos all over the place, then you might have set c’=c/2, and you might then have decided it sensible to equate 1 time unit with half the distance a massless particle travels in that time unit. 2 and c’ always show up multiplied together, so it’s an obvious simplification to choose a unit equivalence that casts away the product 2c’ rather that one which casts away only the c’ (leaving twos everywhere), but Orbifold points out that there’s no fundamental reason to choose one over the other. Nature doesn’t care how we choose to relate the meter and the second.
So, if you want to equate 1 s = 1 m, the bilinear (and everything else) needs to reflect the choice:
1 s = 1 m
ds[sup]2[/sup] = (299,792,458*dt)[sup]2[/sup]-dx[sup]2[/sup]-dy[sup]2[/sup]-dz[sup]2[/sup]
1 s = 2 m
ds[sup]2[/sup] = (149,896,229*dt)[sup]2[/sup]-dx[sup]2[/sup]-dy[sup]2[/sup]-dz[sup]2[/sup]
1 s = 299,792,458 m
ds[sup]2[/sup] = dt[sup]2[/sup]-dx[sup]2[/sup]-dy[sup]2[/sup]-dz[sup]2[/sup]
One second is equal to the number of meters the universe has expanded in that second.
(yet another misinterpretation of your question)
Minutes and seconds are just a way of dividing units into smaller units based on the sexagecimal system (base 60). An hour is divided in to 60 minutes, which is subdivided into 60 seconds. There are also 3600 seconds in a degree of latitude. So there are 3600 seconds in a meter in exactly the same way there are 1000 millifurlongs in a furlong.
I dunno, but there are usually two in a duel.
Regardless of the above, all relativists set c equal to 1. Imagine a graph with the vertical axis as time and calibrated in years, and the horizontal axis as distance with units of lightyears.
The Plot of something moving at c would be a line at 45 degrees with a slope of 1. So 1 year would equal 1 lightyear. From this it is very simple to figure out how many meters are in a second.
But this is all just because relativists are lazy.
Actually, in the above I should have said in the world where c =1,1 year can equal 1 lightyear.
But this is all just because relativists are lazy. I.e.
E[sup]2[/sup] = m[sup]2[/sup]c[sup]4[/sup] + p[sup]2[/sup]c[sup]2[/sup]
Becomes E[sup]2[/sup] = m[sup]2[/sup] +p[sup]2[/sup]
or if and only if p =0
E = mc[sup]2[/sup]
Becomes
E =m
This is probably entirely incorrect for reasons that I can’t being to comprehend. But it sounds really nice so I’ll propose it anyway and legitimize it under the terms of my artistic license.
The smallest measurable length (Planck length) is 1.6162412 x 10^-35 meters and the smallest measurable time (Planck time) is 5.3912140 x 10^-44 seconds. Assuming they are equal, it works out to about
299791698 meters = 1 second
(which is interestingly similar to the speed of light at 299792458 m/s)
or
3.33564 x 10^-9 seconds = 1 meter
and in fact, if you go back to how Planck length and time are derived, if you were to divide them the terms would indeed cancel to C, the speed of light. any apparent disparity is due to rounding errors.
[QUOTE=tomndebb]
I suppose that one could identify one second as equal to 299,792,458 meters, (the number of meters that light travels in 1 second in a vacuum).
[/QUOTE]
Not to go off on a Tangent, or anything.
[QUOTE=Contrapuntal]
Not to go off on a Tangent, or anything.
[/QUOTE]
[QUOTE=H. G. Wells]
‘You must follow me carefully. I shall have to controvert one or two ideas that are almost universally accepted. The geometry, for instance, they taught you at school is founded on a misconception.’
‘Is not that rather a large thing to expect us to begin upon?’ said Filby, an argumentative person with red hair.
‘I do not mean to ask you to accept anything without reasonable ground for it. You will soon admit as much as I need from you. You know of course that a mathematical line, a line of thickness nil, has no real existence. They taught you that? Neither has a mathematical plane. These things are mere abstractions.’
‘That is all right,’ said the Psychologist.
‘Nor, having only length, breadth, and thickness, can a cube have a real existence.’
‘There I object,’ said Filby. ‘Of course a solid body may exist. All real things–’
‘So most people think. But wait a moment. Can an instantaneous cube exist?’
‘Don’t follow you,’ said Filby.
‘Can a cube that does not last for any time at all, have a real existence?’
Filby became pensive. ‘Clearly,’ the Time Traveller proceeded, ‘any real body must have extension in four directions: it must have Length, Breadth, Thickness, and – Duration. But through a natural infirmity of the flesh, which I will explain to you in a moment, we incline to overlook this fact. There are really four dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter, because it happens that our consciousness moves intermittently in one direction along the latter from the beginning to the end of our lives.’
[/QUOTE]
from Chapter 1 of The Time Machine. U.S. Copyright has expired; full text available online at Project Gutenberg
[QUOTE=Keeve]
from Chapter 1 of The Time Machine. U.S. Copyright has expired; full text available online at Project Gutenberg
[/QUOTE]
Is there some significance to the quote that I’m missing? I’m not catching the relevance to my OP except that both mention that time is (or can be considered as) a dimension.
-FrL-
People tend to equate “dimension” with “spatial distance”. I’ve always thought this portion of that book does a nice job of illustrating how important and real Duration is, despite its not being spatial. That’s all.
I went a totally different direction with this, based on just the OP, without reading all the other responses.