# Time as the 4th Dimension

Hey Guys,

I, FYI not a physicist, recently got a fascination to dimensions, and for the life in me I cant seem to wrap my head around the fact that time is the 4th dimension.

From what I can understand, the spatial dimensions can completely perceive the dimension below, but perceive an object in the high dimension in a form comprehensible in their dimension. So far so good, that is until we get to time a.k.a 4D. Now for any changes to occur to any object in any of the dimensions, then time is paramount. For example, when tracking a point, it correct to say that it was at point x,y at time t. While observing a 3D object in motion, I believe it correct to say that it's at x,y,z at time t?

Why is the 4th Dimension, time, present in the lower dimensions? The only conclusions that I can make is that time is independent or that time comes way way before, i.e. Time is the 1st dimension. In the beginning was time, then space came later. For any object in any of the spatial dimensions to comprehend changes in their dimensional environment, then time is there.

So shouldn't time be the 1st dimension? Or am i understanding dimensions the wrong way? Thanx

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What is first and last isn't really the point, I think. In fact, in most calculations, time is taken as the zeroth dimension. But this is purely by convention and not at all important.

I guess my point is, you shouldn't understand it as "first time, then one spacial dimension, then another, then another". They're all just there, and we label them as we please.

Taking a simple scalar field $\varphi$, the simplest energy density (Hamiltonian) is
$H = \sum_{a=0}^3 \partial_a \varphi \partial_a \varphi$
Taking its Legendre transform (quantum physicists call it Wick rotation - "rotating to imaginary time"), for a chosen time direction (0-th), we get Lagrangian density:
$L = \partial_0 \varphi \partial_0 \varphi - \sum_{i=1}^3 \partial_i \varphi \partial_i \varphi$
The first term is kinetic energy, the second potential energy of internal stress of the field.

From the point of view of energy density, there is usually no difference between time and spatial directions - complete 4D symmetry.
The difference appears when we are interested in evolution among some chosen direction, which we call time - getting Lagrangian density.
And so as we know from STR, there is some freedom of choosing this time direction - different choices differ by Lorentz transformation/boost.

Taking a simple scalar field $\varphi$, the simplest energy density (Hamiltonian) is
$H = \sum_{a=0}^3 \partial_a \varphi \partial_a \varphi$
Taking its Legendre transform (quantum physicists call it Wick rotation - "rotating to imaginary time"), for a chosen time direction (0-th), we get Lagrangian density:
$L = \partial_0 \varphi \partial_0 \varphi - \sum_{i=1}^3 \partial_i \varphi \partial_i \varphi$
The first term is kinetic energy, the second potential energy of internal stress of the field.

From the point of view of energy density, there is usually no difference between time and spatial directions - complete 4D symmetry.
The difference appears when we are interested in evolution among some chosen direction, which we call time - getting Lagrangian density.
And so as we know from STR, there is some freedom of choosing this time direction - different choices differ by Lorentz transformation/boost.
Wait, wick rotations and legendre transform are two entirely different things. And the hamiltonian is actually the lagrangian of the field (for appropriate metrics and some stuff). I'm not sure I get your point.

From the perspective of field theory, Wick rotation works the same way: while "rotating time to imaginary direction", there would appear i in the partial differentiations, changing sign of $\partial_0 \varphi \partial_0 \varphi$ - getting us from symmetric Euclidean norm, to Minkowski with emphasized time.

The difference between Hamiltonian and Lagrangian is that the former is just energy density, while the latter is used with Euler-Lagrange equations to get evolution in the chosen time direction.
$\pi = \frac{\partial L}{\partial(\partial_0 \varphi)} (=\partial_0 \varphi)\qquad\qquad H= \pi \partial_0 \varphi - L$.

Special relativity says that there are different choices of time direction, differing by boost - but we usually have no problem to choose the most appropriate reference frame.
If fundamental equations (like energy density) are symmetric, this symmetry is usually broken by concrete solutions. Like dropping a stone to infinite flat water surface - there appears asymmetry and propagates from the center. The "stone drop" of our Universe - the event breaking 4D symmetry, was our Big Bang.
It was also spatially localized - low entropic event: creating the entropy gradient - our 2nd law of thermodynamics.

ps. To see how nonintuitive this symmetry can be, there are GRT solutions (as wormholes or slowly rotating black holes), where spacetime is not time-orientable (here are some links: http://iopscience.iop.org/0264-9381/19/17/308/ ). I completely don't see them realistic, but it would mean that there would exist a loop with configuration of light cones like here:
http://dl.dropboxusercontent.com/u/12405967/loop.jpg [Broken]
So if a rocket would make such a trip and returned back to Earth, e.g. a mug inside would still break to increase entropy ... but from our time perspective it would be pieces getting into a mug ...
Our mugs "break forward", because we can only make mugs from in Big Bang->..->humans-> ..->mug time direction.

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So shouldn't time be the 1st dimension? Or am i understanding dimensions the wrong way? Thanx
As was mentioned, dont call it "the" fourth dimension. Its "a" dimension, one component of four that make up space-time. Dimension is much more general than that though. Maybe these might illuminate a little;
http://en.wikipedia.org/wiki/Dimension_(mathematics_and_physics)
http://en.wikipedia.org/wiki/Dimension_of_a_physical_quantity

Also, this isnt really a quantum physics thread. Maybe this makes more sense to be in relativity or just plain ol' physics.

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This thread should go to the SR forum.

Now for any changes to occur to any object in any of the dimensions, then time is paramount. For example, when tracking a point, it correct to say that it was at point x,y at time t. While observing a 3D object in motion, I believe it correct to say that it's at x,y,z at time t?
First of all, when considering the spacetime as a whole, nothing "changes" and nothing "happens". Everything just "is". It's like looking at a celuloid tape with a recorded movie. The spacetime is a still eternity.

Now every physical object of some dimension gets one additional dimension. Spatial points (0d) become so called world lines (1d). Every spatial surface (2d) becomes a space-time volume (3d) and so on. The only zero-dimensional thing in the spacetime is a so-called event, which is something different than a spatial point.

Now, the fact that the time dimension is really something we could call "time" comes from the fact that spatial position of any spatial point (which is a spacetime line!) is a function of time. Put in other words, for every chosen moment of time, there is only one spatial position of a spatial point. This in turn comes from the fact that the metric of the spacetime is hyperbolic.

I like to distinguish the ideas of "microscopic time" (various physical features regarding the 4th dimension, casuality and so on) and "macroscopic time" (the mental experience we humans percieve). These two are not completely equivalent and in fact many conditions must be met for the "macroscopic time" to exist as we understand it. The existence of the 4th dimension is only one of them.

HallsofIvy
Homework Helper
Hey Guys,

I, FYI not a physicist, recently got a fascination to dimensions, and for the life in me I cant seem to wrap my head around the fact that time is the 4th dimension.

From what I can understand, the spatial dimensions can completely perceive the dimension below, but perceive an object in the high dimension in a form comprehensible in their dimension.
I'm not sure what you mean by this. "Dimensions" don't perceive anything! I suspect you mean that a three dimensional being can perceive two dimensions, a two dimensional being could perceive one dimension. But that's not true- as far as we know, there are no "two dimensional beings". In fact, there are no three dimensional beings! We and everything we know exist in four dimensions because we perceive things changing with time.

So far so good, that is until we get to time a.k.a 4D. Now for any changes to occur to any object in any of the dimensions, then time is paramount. For example, when tracking a point, it correct to say that it was at point x,y at time t. While observing a 3D object in motion, I believe it correct to say that it's at x,y,z at time t?
Yes, that's the point. Physics studies "events"- things that happen at a particular location and particular time. It takes three numbers (latitude, longitude, altitude perhaps) to designate the location and one number to designate the time. That's all "dimension" means.

Why is the 4th Dimension, time, present in the lower dimensions?
I don't know what you mean by "present in the lower dimensions" or, for that matter by "lower" dimensions.

The only conclusions that I can make is that time is independent or that time comes way way before, i.e. Time is the 1st dimension.
Yes, time is independent which is why it is a separate dimension. And calling it the "fourth dimension" is not intended to say is is "higher" or "lower", more or less important, than the space dimensions. We just need to put them in some order to write them and "x, y, z, t" is typical.

In the beginning was time, then space came later. For any object in any of the spatial dimensions to comprehend changes in their dimensional environment, then time is there.

So shouldn't time be the 1st dimension? Or am i understanding dimensions the wrong way? Thanx
Yes, you are misunderstanding. The "fourth" does not imply any "ordering" other than purely conventional.

meBigGuy
Gold Member
Spatial dimensions don't extend to time in a way I can understand.
0 spatial dimensions = point
1 spatial dimension = line (extends from point)
2 spatial dimensions = plane (2nd dimension orthogonal to 1st)
3 spatial dimensions = cube (3rd dimension orthogonal to 1st and 2nd)
4 spatial dimensions = tesseract (well, what the tesseract represents) (4th spatial dimension orthogonal to 1st, 2nd, 3rd)

It is hard for me to consider time as the dimension that is orthogonal to the 1st three. I can kind of visualize it, but then that means a tesseract is a fantasy. Or, does a tesseract somehow represent time?

As far as the math goes, 4 variables represent 4 dimensions, 5 represent 5, and so on. x,y,z,and t can represent motion in 3 dimensional space. x,y,z,q and t can represent motion in 4 dimensional space, it seems.
So I don't think time is really the "fourth" dimension except in the sense that it exists in addition to and independent of the 3 we generally perceive.

Drakkith
Staff Emeritus
Spatial dimensions don't extend to time in a way I can understand.
Of course they don't. They are spatial dimensions.

It is hard for me to consider time as the dimension that is orthogonal to the 1st three. I can kind of visualize it, but then that means a tesseract is a fantasy. Or, does a tesseract somehow represent time?
No, a tesseract's extra dimension is a spatial one, not a time one.

Hey Guys,

I, FYI not a physicist, recently got a fascination to dimensions, and for the life in me I cant seem to wrap my head around the fact that time is the 4th dimension.

From what I can understand, the spatial dimensions can completely perceive the dimension below, but perceive an object in the high dimension in a form comprehensible in their dimension. So far so good, that is until we get to time a.k.a 4D. Now for any changes to occur to any object in any of the dimensions, then time is paramount. For example, when tracking a point, it correct to say that it was at point x,y at time t. While observing a 3D object in motion, I believe it correct to say that it's at x,y,z at time t?

Why is the 4th Dimension, time, present in the lower dimensions? The only conclusions that I can make is that time is independent or that time comes way way before, i.e. Time is the 1st dimension. In the beginning was time, then space came later. For any object in any of the spatial dimensions to comprehend changes in their dimensional environment, then time is there.

So shouldn't time be the 1st dimension? Or am i understanding dimensions the wrong way? Thanx
Spacetime is a single continuum that emerged from vibrant nothingness according to the Lambda-CDM cosmological model. The event in x, y, z coordinate at time t is correct but you must consider that it is unique as well as every each event in the universe. Example, every anniversary of your birthday the star constellations in the night sky is the same, it's not only time t is different but also the x, y, z coordinate of Earth.

"In Relativity the world has four dimensions: three space dimensions and one dimension that is not exactly time but related to time. In fact, it is time multiplied by the square root of -1." That make sense to physicists and mathematicians especially when the equations could be used in a predictive manner... but difficult to layman. Before Minkowski, Einstein et al there was sci-fi writer who explained the spacetime continuum to layman.. her is an excerpt.

**'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?'

'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.'**

meBigGuy
Gold Member
I really like that, but ....

I think the cube "essence" exists at each and every moment in time. We experience it as it changes from time1 to time2.

I think time and spatial dimensions are different. There can be an arbitrary number of orthogonal spatial dimensions and that doesn't change the nature of time.

Time is the fourth dimension only in the sense that it is another dimension.

I'm curious; Is there any meaningful concept of multiple orthogonal time dimensions?

I'm curious; Is there any meaningful concept of multiple orthogonal time dimensions?
Yes it is. String theorists do it quite often.

Note however, that a dimension may be compact (curled into a loop). Compact time dimensions usually don't change anything. In a space with two noncompact time dimensions, there would probably no macroscopic time in our usual sense.