Time As A Dimension

[Pure_Argent]

Am I wrong in thinking that Einstein's space-time continuum model includes time as a dimension? Is there any mathematical proof for time as a dimension? (I bring this up because of an argument I'm involved in on another forum; the opposing side wants more proof than I'm able to give) Or is time just a human convention? I don't think so, but I need your help to prove it.

[Reality Check]

Spacetime includes time.
There is no "mathematical proof for time as a dimension". It is just that the universe can be modeled accurately by considering time as a dimension in spacetime.

[Pure_Argent]

:/

What about relativity? The way time expands and contracts with movement? Should that be a motive to consider time as a dimension?

[Dorfl]

Am I wrong in thinking that Einstein's space-time continuum model includes time as a dimension?

No. Space-time consists of one time dimension and three space dimensions.

Is there any mathematical proof for time as a dimension? (I bring this up because of an argument I'm involved in on another forum; the opposing side wants more proof than I'm able to give) Or is time just a human convention? I don't think so, but I need your help to prove it.

You can't mathematically prove anything about the real world. But, as Reality Check said, using a model of the universe which includes time as a dimension works just fine. It gives predictions which match observations to an enormously high degree of accuracy. If you want to take that as a sign that the universe is really a four-dimensional space-time, or just as a sign that the four-dimensional space-time is a very good model, is up to you.

[Ziggurat]

If you want to take that as a sign that the universe is really a four-dimensional space-time, or just as a sign that the four-dimensional space-time is a very good model, is up to you.

Ah, the joys of epistemology applied to physics!

[Just thinking]

The concept that Time by itself doesn't exist has interested me, as well as many in the past, both philosophers and scientists. It's certainly not absolute as we all know, nor are events always in specific order (depending on your frame of reference). If you think about, though ... when you use any device to measure time, all you're really doing is counting events.

Time (http://en.wikipedia.org/wiki/Time):

... time does not refer to any kind of "container" that events and objects "move through", nor to any entity that "flows", but that it is instead part of a fundamental intellectual structure (together with space and number) within which humans sequence and compare events. This second view, in the tradition of Gottfried Leibniz[6] and Immanuel Kant,[7][8] holds that time is neither an event nor a thing, and thus is not itself measurable nor can it be travelled.

[Dorfl]

Ah, the joys of epistemology applied to physics!

I thought it was ontology?

[BoogieWoogieWookie]

Instead of thinking of dimensions as units needed to describe the shape of size of something such as a box with a certain length, width and height, think of them as coordinates needed to locate a particular point.

If I want to know the exact location of an airplane I need to know the compass coordinates and height of the plane (x, y, z) but these are not enough. The time coordinate must also be given. The plane cannot be located precisely unless t is specified with the other three dimensions.

Ask the person to use only the three spacial dimensions to locate a bird in flight.

[Soapy Sam]

If I want to know the exact location of an airplane I need to know the compass coordinates and height of the plane (x, y, z) but these are not enough. The time coordinate must also be given. The plane cannot be located precisely unless t is specified with the other three dimensions.


And not even then with total precision. (If it's somewhere, at a point in time, how can it be moving?)
*Whips tortoise into frenzy and gallops off over horizon in series of convergent bounds*

[HansMustermann]

Am I wrong in thinking that Einstein's space-time continuum model includes time as a dimension? Is there any mathematical proof for time as a dimension? (I bring this up because of an argument I'm involved in on another forum; the opposing side wants more proof than I'm able to give) Or is time just a human convention? I don't think so, but I need your help to prove it.

It depends on what you understand by "dimension". It doesn't say it's equivalent to space.

You can use anything as a dimension. It just means it's a variable in your problem space.

E.g., if I own a factory making Xnorgs and want to figure out how many units should I produce a year, to maximize my profits, well, I might take a piece of graph paper (or a computer program) and plot two curves: price I can sell a Xnorg vs number produced (the demand curve) and price it costs me to produce a Xnorg vs number produced. Or rather the difference between them, times number. So I'll put money on one axis and number produced on the other, and see where it peaks. I just effectively used money as one dimension, and number of units as the other dimension.

Or in an actual problem, at some point in the early 90's I got into 3D computer graphics. Texturing a polygon boiled down to basically drawing a number of horizontal lines, covering its surface. But each pixel on that line had to come from a texture, not be a solid fill. Depth of each pixel is also important, since polygons in the front occlude ones in the back. And then there's lighting, in my case it was just a simple linear interpolation of the brightness values for the corners.

Probably everyone recognized by now that's how 3D games work.

Well, it all boiled down to interpolating a 6 dimensional line: X, Y, Z on the screen, the two coordinates in the texture, and the brightness value as the 6'th dimension. (Various other algorithms use more or less coordinates there.) Literally the Bresenham interpolation algorithm in 6 dimensions.

Ok, really 5 because the Y is constant for each scan line, so it can be optimized out.

I'm geeking out, ain't I? :p

Anyway, the point is that you can use anything as a dimension. If a line in a 6-dimensional space is what models your problem, you use a 6-dimensional space.

And if you think that's crazy, wait until you see how many dimensions the String Theory uses :p

[Towlie]

If time is considered to be a dimension, it's not the fourth dimension. You could call it a fourth dimension, meaning an additional dimension, but it ranks in first place. A point, normally considered to have zero dimensions, could be considered to have a single dimension if you specify when the point began to exist and when it ceased to exist.

Perhaps time should be referred to as the "zeroeth dimension". ;)

[Ziggurat]

I thought it was ontology?

It probably falls under both. But philosophers bug me, so I can't say I actually care.

[Just thinking]

If I want to know the exact location of an airplane I need to know the compass coordinates and height of the plane (x, y, z) but these are not enough. The time coordinate must also be given. The plane cannot be located precisely unless t is specified with the other three dimensions.

Not necessarily.

If you know that the plane is consuming fuel as a function of its position, you can (in theory) determine its location by knowing the 3 spacial dimensions along with the plane's mass.

[Towlie]

Not necessarily.

If you know that the plane is consuming fuel as a function of its position, you can (in theory) determine its location by knowing the 3 spacial dimensions along with the plane's mass.The plane's mass when? See? You still need to know the time, since flying planes don't just hang in permanent locations.

B.W.W. is talking about having to use four numbers to specify (describe) the location of a flying airplane, but you're talking about determining the location. Those are two very different subjects. I don't think you completely understood what he said.

[Just thinking]

The plane's mass when? See? You still need to know the time, since flying planes don't just hang in permanent locations.

You are by default requiring 4 dimensions and then stating that you need them. Well, of course that requires using time (t), by your definition. You set it up that way.

B.W.W. is talking about having to use four numbers to specify (describe) the location of a flying airplane, but you're talking about determining the location. Those are two very different subjects. I don't think you completely understood what he said.

When you say "the location of an airplane", you really only need 3 dimensions, flying or not. Of course, if flying, its geogrid locations are changing, but you can give its actual location with just 3 dimensions. You seem to only require another dimension to describe where the plane either no longer is, or where it will be. In either case, you can re-describe those locations by measuring its mass instead.

[Tumbleweed]

I look at it this way; L, L squared, and L cubed are the first three dimensions. THEN you give that L cubed motion, making T ( velocity), T squared ( acceleration) and T cubed ( jerk) the next three

[Tumbleweed]

If you were to define a motionless, fixed point in space, would not your relationship ( relative motion) to that fixed point be considered a dimension?

[Tumbleweed]

On a somewhat related note:Can gravity waves be of the same nature as the wave you see in sports stadiums as the fans stand up? That is a manifestation of T cubed or jerk. If gravity is akin to and indistinguishable from acceleration, can gravity waves be akin to jerk, making gravity a sort of derivative of jerk

[Dorfl]

It probably falls under both. But philosophers bug me, so I can't say I actually care.
Agreed.

[Reality Check]

I look at it this way; L, L squared, and L cubed are the first three dimensions. THEN you give that L cubed motion, making T ( velocity), T squared ( acceleration) and T cubed ( jerk) the next three
A bit simplistic. Area is not a length squared (otherwise you could only have squares!). It is the addition of another dimension.

A better way to look at is:
A point is a zero-dimensional spacial object.
A line is a 1-dimensional spacial object.
An area is a 2-dimensional spacial object.
A volume is a 3-dimensional spacial object.
Spacetime is a 3-dimensional spacial object with the addition of a time dimension (treated differently from a spacial dimension).

Motion is a change in position (not L cubed or T).
Acceleration is a change in motion (not T squared).
Jerk is a change in acceleration (not T cubed).

[Reality Check]

If you were to define a motionless, fixed point in space, would not your relationship ( relative motion) to that fixed point be considered a dimension?
It would be a velocity vector pointing from your current position to that "motionless, fixed point in space".

[sol invictus]

And not even then with total precision. (If it's somewhere, at a point in time, how can it be moving?)

By having a velocity (and a non-zero duration), of course.

If you just want to know that the plane was somewhere and somewhen, all you need are its 4 spacetime coordinates. But if you also want to know where it was a little later and a little before, you need to specify its velocity also. According to the laws of classical physics that's all you need to specify: position, time, and velocity (but not acceleration, for example).

[Towlie]

Acceleration is a change in motionAcceleration is a change in velocity.

By the way, velocity is the rate of change of position, and is a vector quantity incorporating both speed and direction. For that reason, I cringe when I hear people speak or write of the "velocity of light" as a constant. Perhaps they say "velocity" to sound more scientific and intellectual, but they're wrong. The velocity of a beam of light changes whenever it's reflected from a mirror. What they really mean is the speed of light.

Also, many people speak of mass when they really mean matter. Mass is a property that matter has. Mass is not changed into energy in a nuclear reaction, matter is changed into energy.

[Just thinking]

Mass is not changed into energy in a nuclear reaction, matter is charged into energy.

Is that with Visa or Mastercard?

:D

[Towlie]

Is that with Visa or Mastercard?Thanks, I'm glad you spotted that. ;)

[Tumbleweed]

A bit simplistic. Area is not a length squared (otherwise you could only have squares!). It is the addition of another dimension.

A better way to look at is:
A point is a zero-dimensional spacial object.
A line is a 1-dimensional spacial object.
An area is a 2-dimensional spacial object.
A volume is a 3-dimensional spacial object.
Spacetime is a 3-dimensional spacial object with the addition of a time dimension (treated differently from a spacial dimension).

Motion is a change in position (not L cubed or T).
Acceleration is a change in motion (not T squared).
Jerk is a change in acceleration (not T cubed).
I'm not saying that T squared IS acceleration, Im' just saying that it is necessary to have acceleration. The first three dimensions, up to a literal cube require no time. If you want to add dimensions to that cube you have to change its position or spatial coordinates. Hence time and its manifestations of velocity acceleration and jerk. Jerk by the way is observed by helicopter pilots watching traffic every morning. It manifests itself as a wave brought on by stop and go traffic. A caterpillar wave is one too- I think. All "waves" have a T cubed?

[Tumbleweed]

To me, 3 cubed does not just simplistically mean 3 times 3 times 3. It means a cube, a structure 3 units on a side. And I think that any equation using a unit cubed can geometrically be represented using cubes
Take the Pythagorean Theorem: a* + b* = c*. It means more than just trig to find lengths and angles. It is finding area as well. It literally means that if you have a right angled triangle with sides a b and hypotenuse c, the AREA of the square with one side as a, plus the AREA of the square with one side as b will equal the AREA of the square with side c. If they don't adjust until they do and you will have property lines line up at a right angles and true squares as parcels. 3,4,5 for example, as the Egyptians said. So again, x squared is way more than just x times x

[Dorfl]

To me, 3 cubed does not just simplistically mean 3 times 3 times 3. It means a cube, a structure 3 units on a side. And I think that any equation using a unit cubed can geometrically be represented using cubes. Take the Pythagorean Theorem: a* + b* = c*. It means more than just trig to find lengths and angles. It is finding area as well. It literally means that if you have a right angled triangle with sides a b and hypotenuse c, the AREA of the square with one side as a, plus the AREA of the square with one side as b will equal the AREA of the square with side c. If they don't adjust until they do and you will have property lines line up at a right angles and true squares as parcels. 3,4,5 for example, as the Egyptians said. So again, x squared is way more than just x times x

What geometrical figure does rπ correspond to?

[Towlie]

Take the Pythagorean Theorem: a² + b² = c². It means more than just trig to find lengths and angles. It is finding area as well.Yes, the Pythagorean Theorem can be illustrated geometrically:

http://z.about.com/d/math/1/5/e/D/pythagoreantheorem.gif

But does that mean that any squared term represents something geometric?

In the formula E = MC² does the C² term represent an area?

[ponderingturtle]

In the formula E = MC² does the C² term represent an area?

No. It is a conversion factor.

[Beerina]

Instead of thinking of dimensions as units needed to describe the shape of size of something such as a box with a certain length, width and height, think of them as coordinates needed to locate a particular point.

If I want to know the exact location of an airplane I need to know the compass coordinates and height of the plane (x, y, z) but these are not enough. The time coordinate must also be given. The plane cannot be located precisely unless t is specified with the other three dimensions.

Ask the person to use only the three spacial dimensions to locate a bird in flight.

The point with Einstein, though, that this 4-dimensional spacetime thing, whatever it is, physically deforms somehow (whatever physically means) which is what gives rise to gravity. Since you always travel at c through the spacetime continuum, and gravity warps that, you must accelerate without moving through the 3 normal dimensions. Hence time slows and you feel gravity, aka acceleration through the 4-d spacetime continuum.

How well that interpretation maps to actual reality, I have no idea. But resistance to acceleration and gravity aren't just similar phenomena, but are the exact same phenomena.

[Reality Check]

I'm not saying that T squared IS acceleration, Im' just saying that it is necessary to have acceleration. The first three dimensions, up to a literal cube require no time. If you want to add dimensions to that cube you have to change its position or spatial coordinates. Hence time and its manifestations of velocity acceleration and jerk. Jerk by the way is observed by helicopter pilots watching traffic every morning. It manifests itself as a wave brought on by stop and go traffic. A caterpillar wave is one too- I think. All "waves" have a T cubed?
A change in velocity is all that is necessary to have acceleration.
To measure velocity you need a coordinate system.
This coordinate system used is 3 dimensions of space and one of time in our universe (or you can add more space dimensions for string theory).

There is no such thing as your "T", "T squared" or "T cubed" in physics. The problem is that you defined

L cubed as a 3-D space with the same values for all coordinates, e.g. (L,L,L). This is not a general 3-D space which can have different values for each coordinate, e.g. (x,y,z)
And then T as "THEN you give that L cubed motion, making T ( velocity)". But spacetime does not move. Things move in spacetime.
The alternative is that you have your own personal definition of "squared" and "cubed", e.g. squared = take L, add another coordinate K to make the coordinate pair (L,K) and cubed = square L, add another coordinate N to make the coordinate triplet (L,K,N). That is just a 3-D space so you may as well use the standard terms.

[Skamandros]

Ther is 3 space demensions and three time demensions unlocked by big bang. All univers is made of three every thing when four is reeched then flageled the end. So 3 is sustaning capitular thing for all becaus two is no enougf.
If this prinsipal is learned science are easy.

[Towlie]

A good caption for this pic:

http://farm2.static.flickr.com/1062/787494068_a17c9b2a98.jpg?v=0

[Tumbleweed]

Yes, the Pythagorean Theorem can be illustrated geometrically:

http://z.about.com/d/math/1/5/e/D/pythagoreantheorem.gif

But does that mean that any squared term represents something geometric?

In the formula E = MC² does the C² term represent an area?
Well, we've gone over what a second squared is already a while back. Just replace the LWH (xyz) coordinates with T, T squared and T cubed. Merge the time graph with the 3 dimennsional space LWH (xyz) graph and you have a space time graph

[Reality Check]

Well, we've gone over what a second squared is already a while back. Just replace the LWH (xyz) coordinates with T, T squared and T cubed. Merge the time graph with the 3 dimennsional space LWH (xyz) graph and you have a space time graph
T, T squared and T cubed are not dimensions. They are velocity, acceleration and jerk, i.e. vectors. Replacing the LWH (xyz) coordinates will give you a 9 space + 1 time dimensional space time graph.

[Tumbleweed]

What geometrical figure does rπ correspond to?
Beats me, but it isn't a square/circle/ area or sphere/ cube/volume unless n is 2 or 3. Beyond 3 escapes me as far as a stationary geometrical shape goes. I can describe that 3 dimensional shape's motion in relation to coordinate 0,0,0 three ways as well, but that's about it

[Dorfl]

Beats me, but it isn't a square/circle/ area or sphere/ cube/volume unless n is 2 or 3. Beyond 3 escapes me as far as a stationary geometrical shape goes. I can describe that 3 dimensional shape's motion in relation to coordinate 0,0,0 three ways as well, but that's about it

Fair enough. But is there any geometrical shape corresponding to r-2? Because time squared isn't anywhere in the formula for acceleration, time raised to minus two is.