"Time is Not the Fourth Dimension"

[Mobyseven]

Well for sure time is not geometric 4th dimension. Because by no operation in time you can make left boot from right boot .. in space with 4 geometric dimensions all you need to do is rotate the boot around arbitrary plane.
On the other hand, the convention is useful, mostly in relativity problems .. so what the heck ..

No, it's still a 'geometric' dimension. It's just that the geometry of space-time is non-Euclidean and unfamiliar to our everyday notion of geometry.

 

[Molinaro]

Well for sure time is not geometric 4th dimension. Because by no operation in time you can make left boot from right boot .. in space with 4 geometric dimensions all you need to do is rotate the boot around arbitrary plane.
On the other hand, the convention is useful, mostly in relativity problems .. so what the heck ..

It's not a matter of usefulness. It is a necessity. You cannot describe anything by only using a combination of space measurements.

 

[ynot]

In what way is time an actual thing? Time is a measurement between events and therefore can’t exist only in the present event. How can something that is purely a conceptual measurement with no actual existence be a dimension? (IMO)

 

[Illustronic]

A point is not an actual thing, it is a vector coordinate of position. A 2-dimensional plane is not an actual entity you can touch, its also a vector coordinate you can only breach. One and two dimensional space is not an entity, its a mathematical coordinate of position in 3-dimensional space, which is meaningless unless you get there at the right time.

 

[Fnord]

All that I know is that time only flows in one direction noitcerid eno ni swolf ylno emit taht si wonk I taht llA

 

[W.D.Clinger]

It might be best to read my last two paragraphs first. (I started with general relativity and ended up with special relativity. It didn't occur to me that the question was probably intended to be about special relativity until I had written the whole thing.)

In what way is time an actual thing? Time is a measurement between events and therefore can’t exist only in the present event. How can something that is purely a conceptual measurement with no actual existence be a dimension? (IMO)

First, let's note that what you've said there would make just as much (or little) sense if "space" were substituted for "time" throughout.

Second, let's note that the four dimensions we're talking about here correspond to the four coordinate axes of a locally Lorentzian manifold. What that means, roughly, is that we can map the immediate neighborhood of every event (or point) into a small region of Minkowski spacetime with minimal distortion that goes to zero in the limit as we approach the event (point). What's more, these event-centered maps change smoothly as we go from one event (point) to a nearby event (point); that means some of the concepts familiar from freshman calculus carry over in a suitably generalized form.

In particular, we still have a suitably generalized notion of the metric distance between two nearby events (points). Roughly speaking, we can define the distance between two sufficiently nearby events (points) by mapping them into Minkowski spacetime and using the Lorentz metric there as an approximation to the metric distance between them. That approximation gets worse as events become more widely separated, but there are mathematically rigorous ways to deal with that using a suitable generalization of freshman calculus.

The generalization from freshman calculus to differential geometry is not easy, but it was done more than a hundred years ago and the theory has been simplified and extended since then. To a great extent, therefore, your question reduces to a question about the flat Minkowski spacetime used in special relativity.

In Minkowski spacetime, we can choose a single coordinate system in which every event (point) is assigned 4 coordinates of the form (t, x, y, z). The square of the metric distance between two points is the sum of the squares of the differences between their spatial coordinates x, y, and z minus the speed of light squared times the square of the difference between their time coordinates t. The special treatment of the time coordinate in that calculation of the metric distance is what makes Minkowski spacetime non-Euclidean and hard to get used to. It really is a different kind of geometry; Euclid's fifth axiom does not hold.

If you leave out the time coordinate t, then the three remaining coordinates (x, y, z) approximate a conventional Euclidean space. That Newtonian approximation works fine so long as nothing is moving very fast. When things are moving at relativistic velocities, however, then you have to start thinking of time as a 4th coordinate and use the spacetime metric and geometry instead of using just the spatial metric and geometry; otherwise your calculations will yield results that disagree with experiment and other manifestations of what we believe to be reality.

 

[ynot]

First, let's note that what you've said there would make just as much (or little) sense if "space" were substituted for "time" throughout.

In what way is space an actual thing? Space is a measurement between actual things. Give an example of space existing without referring to actual things (non-mathematical).

 

[W.D.Clinger]

In what way is time an actual thing? Time is a measurement between events and therefore can’t exist only in the present event. How can something that is purely a conceptual measurement with no actual existence be a dimension? (IMO)

First, let's note that what you've said there would make just as much (or little) sense if "space" were substituted for "time" throughout.

In what way is space an actual thing? Space is a measurement between actual things. Give an example of space existing without referring to actual things (non-mathematical).

I don't know that space (or time) is an actual thing.

I'm inclined to regard both time and space as mathematical concepts, and to regard actual things/events as physical measurements of those mathematical concepts.

But that's just me. Your mileage (whether it be an actual thing or a mathematical concept) may vary (and not just because your frame of reference is different from mine).

 

[lionking]

"Time is Not the Fourth Dimension" would be a good song title.

 

[ynot]

I don't know that space (or time) is an actual thing.

I'm inclined to regard both time and space as mathematical concepts, and to regard actual things/events as physical measurements of those mathematical concepts.

But that's just me. Your mileage (whether it be an actual thing or a mathematical concept) may vary (and not just because your frame of reference is different from mine).

My mileage does vary and IMO you put the cart befiore the horse as math favoured people do (I blame all that book learnin ;-).

So how can a mathematical concept be a dimension?

 

[ynot]

The current standard model requires that there be a fourth dimension, but try as they may no one could find a one so one needed to be created. It was thought that something that was nothing would be very good for the purpose and time was selected. Besides, time had a bit of an attitude problem (with all that not waiting for no man and never being enough of it stuff) and needed to be taught a lesson by having to do some real work for a change.

 

[Dr.Sid]

By geometric dimension (or maybe 'spatial' would be better word) I mean this: when you go from 1D to 2D, or from 2D to 3D, there is some regularity in this. Flipping orientation is of them.
In 1D, you can have 2 same objects (short line), with different orientation. One pointing in one direction, second pointing in other direction. If it was object with fixed size, you would not be able to change the orientation in 1D space. But if you would extend the space into 2D, it would be simple. You would rotate the object around point.
In 2D, you can have different objects too. Take letter R, and it's mirror image. Again, in 2D you are not able to change one such object into another. But again, if you extend the space into 3D, you can change one into another by rotating it around axis.
Also if you have some system of coordinates, let's say XYZ, then Z extends XY in this was, same as Y extends XZ and X extends YZ. Also you can define any 2D subspace in 3D space, and all those subspaces would have same properties.
With time it's different. There are spatial operations which would be possible in 4D space. Rotation around plane, which would make left boot from right boot is one of them. Also 3D boot should be definable (or moved into) in any 3D subspace of 4D space. like XYT.
So yes, you can make any system of coordinates if you want to examine some number of values in some system. But time is nothing special here. It can be XYZ and weight, speed, color, or presidential election statistics. You can have 3D space of time, weight and price .. and it clearly is not geometric system in which object can have length or volume.
Spatial coordinates are tied together in special manner, and no other quantity in universe behaves like this, much less time.

 

[sphenisc]

You need 3 numbers to specify a location in space (plus 1 to specify an event in spacetime), just as you need two (latitude and longitude) to specify a point on the (2D) surface of the earth.

That's the definition of the number of dimensions - it's got nothing to do with the human mind.

I wonder if a single number/point on a space-filling fractal would work?

 

[W.D.Clinger]

You need 3 numbers to specify a location in space (plus 1 to specify an event in spacetime), just as you need two (latitude and longitude) to specify a point on the (2D) surface of the earth.

That's the definition of the number of dimensions - it's got nothing to do with the human mind.

I wonder if a single number/point on a space-filling fractal would work?

Well, there's a little more to it than sol invictus said.

According to ZFC, you could take any well-ordering of spacetime events and specify each event by a single ordinal. The problem with that is that the ordinal indexes wouldn't have any kind of smooth relationship to spacetime, so you couldn't do calculus-like things with them.

If you take sol invictus's definition and add the requirement that nearby events should have nearby coordinates, then you need 4 separate coordinates to specify an event in spacetime.

 

[sol invictus]

In what way is space an actual thing? Space is a measurement between actual things. Give an example of space existing without referring to actual things (non-mathematical).

I can't make any sense of your posts on this topic.

Suppose I have a meter stick and a stopwatch. There's a car driving down a road. When the car passes me, I start my watch. When it passes you down the road, I stop my watch and measure the distance between us with the meter stick.

I've just measured the time and distance between two events (car passes me and car passes you). There's not much difference between those two measurements.

 

[sol invictus]

In what way is space an actual thing? Space is a measurement between actual things. Give an example of space existing without referring to actual things (non-mathematical).

I wonder if a single number/point on a space-filling fractal would work?

Well, there's a little more to it than sol invictus said.

According to ZFC, you could take any well-ordering of spacetime events and specify each event by a single ordinal. The problem with that is that the ordinal indexes wouldn't have any kind of smooth relationship to spacetime, so you couldn't do calculus-like things with them.

If you take sol invictus's definition and add the requirement that nearby events should have nearby coordinates, then you need 4 separate coordinates to specify an event in spacetime.

There are attempts to do such things floating around in physics. One fairly trivial example is to put spacetime on a lattice, as you must do if you want to simulate physics on a computer. In the limit the points get very fine, this usually reproduces continuum physics pretty well. Since there are a finite number of points, so if you wanted to you could label each with a single integer.

But labeling all points by a single integer (instead of a vector of 3+1 integers) would make your life much harder than it needs to be. The laws of physics are local, which means they're pretty simple expressed in terms of those 3+1 vectors. But they'd look horrendously complicated written in terms of that single integer.

 

[gnome]

but we have little or no idea how to make sense of theories with more than one time dimension.

Ask a Dr. Who fan

 

[blobru]

... One interesting implication of the differentness of time is that we know perfectly well how to formulate physics in n space dimensions and 1 time (and some such theories with n>3 are even completely consistent with all our experimental data), but we have little or no idea how to make sense of theories with more than one time dimension.

Though hasn't Hawking proposed a second time dimension (perpendicular to real time: imaginary time^WP) as a way for cosmologists to frame what might have happened before the Big Bang? Which seems (in my layman's understanding, at least) to make sense with Many-Worlds Theory (I think Hawking posits a wave-function for all worlds in imaginary time, with ours [this universe] at a probability spike).

 

[sol invictus]

That's actually a space dimension, not an additional time dimension - space is imaginary time.

 

[Roboramma]

That's actually a space dimension, not an additional time dimension - space is imaginary time.

That makes me wonder, is there a way to tell the difference between a universe with 1 time dimension and 3 space dimensions and one with 3 time dimensions and 1 space dimension?