Why is Planck time measured differently from other dimensions?

[spizma]

Hi, I'm new here. I've loved all the qualitative aspects of physics for a while now and I'm just starting to get into some of the more quantitative aspects of physics, the actual math. I'm such a complete amateur at all this and I'm only in high school so please excuse my question if it seems ridiculous. I was thinking about how time is a dimension today and this thought occurred to me. From what I understand, the Planck length is the smallest possible length. Now, it being a length, it would measure a 1 dimensional object. If we were to measure a 2 dimensional object, the curvature of 1 dimensional space, we would measure it in Planck lengths squared, the Planck area I suppose. If we were to measure a 3 dimensional object, the curvature of 2 dimensional space, we would measure it in Planck lengths cubed, the Planck volume I'm assuming. So, from this logic, one would assume if we were to measure something in the fourth dimension of time, the curvature of 3 dimensional space, we would measure it in Planck lengths to the 4th power. But we don't, we measure it in a completely separate unit, the Planck time. Why?

 

[Ballon]

Without going very far, my logic will tell me that (perhaps) that it is because 3d are Spaces, and time is Time...

 

[spizma]

Yes but why is there such a separation of space and time?

As far as I can tell, time is a dimension, and it is basically the curvature of the 3rd dimension. For example, if an ant was walking on the edge newspaper, a metaphor for the 2nd dimension, we could fold the newspaper in the third dimension to the other edge (folding the dimension) and put the ant on another position in the 2nd dimension. I figured the same went for the 4th dimension, our 3 dimensional selves are curving in 4 dimensional space, time, which allows us to end up in another position in the 3rd dimension. We can measure all those other curvatures in space using powers of the Planck length, so why can't we measure the curvature in the 4th dimension using a power of the Planck length. I'm probably just completely wrong in what I understand about dimensions, but could someone please tell me where I'm getting it wrong.

 

[Jimmy Snyder]

If we were to measure a 2 dimensional object, the curvature of 1 dimensional space, we would measure it in Planck lengths squared, the Planck area I suppose.

I'm not sure what you mean about curvature. However, your supposition is correct: http://scienceworld.wolfram.com/physics/PlanckArea.html

If we were to measure a 3 dimensional object, the curvature of 2 dimensional space, we would measure it in Planck lengths cubed, the Planck volume I'm assuming.

You are correct again: http://en.wikipedia.org/wiki/Orders_of_magnitude_(volume)

So, from this logic, one would assume if we were to measure something in the fourth dimension of time, the curvature of 3 dimensional space, we would measure it in Planck lengths to the 4th power.

Here you are wrong, you should keep to the pattern you set in the first two suppositions:
If we were to measure a 4 dimensional object, the curvature of 3 dimensional space, we would measure it in Planck lengths to the 4th power, the Planck 4-dimensional volume I'm assuming.
That would be correct.

But we don't, we measure it in a completely separate unit, the Planck time. Why?

Time is 1 dimension and Planck time measures 1 dimensional time. The unit is not completely separate from the Planck length. It is the Planck length divided by c, the speed of light. This is the standard ratio for making time and length commensurate.

 

[spizma]

Ok, I understand most of what you said, but I still have one question.

Time is 1 dimension and Planck time measures 1 dimensional time.

I thought time was the 4th dimension?

 

[RandallB]

spizma

I assume you’re trying to relate to time as “THE FOURTH DIMENSION” as completing the count of dimensions used in General Relativity.
That is not correct.
You do not reach the four dimensions of GR, by simply calling what we observe as time to be “THE FOURTH DIMENSION”.

GR is not that simple, it assumes a four dimensional reality that we can see no part of those dimensions in our experienced reality. That is, using 4 spatial GR dimensions of A, B, C, & D no one of those matching directly to anyone of our 3 spatial dimensions x, y, z or to the parameter we call time.

Some attempts have been made to view the GR four spatial dimensions as needing a parameter like GR TIME to allow for, or measure, change in that 4D world, just as we do in our 3D world. That could be described as a 5 parameter reality, comparable to our 4 parameters of 3D plus time.

However if true, since our concept of time is so dependent on GR 4D curves and movements, it is hard to imagine any way mathematically obtaining a reference on, or measure of such a fifth parameter like GR TIME.
GR doesn’t even make any thing like a direct measure of anyone of the A, B, C, or D dimensions of 4D available to us. Using instead parameters that represent curves and warps in that 4D space useable into use mathematically in our reality.

Defining time as a fourth dimension is a sloppy method for explaining General Relativity, used by too many scientists that in effect are talking down to the public when they do.

 

[Jimmy Snyder]

I thought time was the 4th dimension?

Forget relativity for a moment and think about ordinary space. What is the 3rd dimension?

 

[pervect]

Hi, I'm new here. I've loved all the qualitative aspects of physics for a while now and I'm just starting to get into some of the more quantitative aspects of physics, the actual math. I'm such a complete amateur at all this and I'm only in high school so please excuse my question if it seems ridiculous. I was thinking about how time is a dimension today and this thought occurred to me. From what I understand, the Planck length is the smallest possible length. Now, it being a length, it would measure a 1 dimensional object. If we were to measure a 2 dimensional object, the curvature of 1 dimensional space, we would measure it in Planck lengths squared, the Planck area I suppose.

One could measure areas in Planck units squared. The Planck system doesn't actually have units, though, so one would interpret this as "just a number". To recover SI units from Plank units, though, it would be necessary to know that the number represented an area (length^2).

Your statement about curvature is puzzling. There are different sorts of curvature. Sectional curvature would be measured in units of 1/length^2, i.e. the inverse of what you wrote. Path curvature would be measured in units of 1/length Offhand I can't think of any sort of curvature measure that's measured in units of length^2.

See for instance http://en.wikipedia.org/w/index.php?title=Geometrized_unit_system&oldid=119291125

and some of the associated references.

Example: the acceleration of a body would be an example of a path curvature. This has SI units of meters/sec^2. In geometric and also Planck units, this is equivalent to 1/meter^2, because meters and seconds share the same unit.

Example: components of the Riemann curvature tensor (which can be physically interpreted as the tidal force on a non-rotating body following a geodesic) would have units of acceleration / length. This would be 1/sec^2 from our previous argument, equivalent also to 1/meter^2.

Note that as has been mentioned in other threads, the reason for giving space and time the same units is that they can "mix together". For instance, observer A might regard two events as being simultaneous, so that the time interval between them is zero, and there is only a space interval between them. Observer B, moving relative to observer A, does not see the events as being simultaneous, and measures a separation in both space and time between the two events. Thus one observers spatial separation is seen as both a space and a time separation (shorthand: a spacetime separation). Thus space and time are often treated as a unified entity in relativity, rather than as totally separate things.

 

[spizma]

Alright, I understand. It seems like the sources I've gotten information on physics from have been dumbing it down too much. Thanks for the info!

 

[MeJennifer]

Now, it being a length, it would measure a 1 dimensional object. If we were to measure a 2 dimensional object, the curvature of 1 dimensional space, we would measure it in Planck lengths squared, the Planck area I suppose. If we were to measure a 3 dimensional object, the curvature of 2 dimensional space, we would measure it in Planck lengths cubed, the Planck volume I'm assuming. So, from this logic, one would assume if we were to measure something in the fourth dimension of time, the curvature of 3 dimensional space, we would measure it in Planck lengths to the 4th power. But we don't, we measure it in a completely separate unit, the Planck time. Why?

Not quite.

You cannot generally say that any n dimensional manifold can be embedded in a n+1 flat dimensional space. You can "flex" the manifold in many different directions.

Take for instance a Lorentzian manifold, if you take all the possible metrics on this manifold you can only embed it into a 91 dimensional flat space. However that does not automatically mean that you would need 91 dimensions to represent all valid metrics of general relativity. And that does not automatically mean that all those metrics of that particular subset are physically valid solutions.

But there are simple cases, for instance the Schwarzschild solution can easily be embedded in 5 dimensions.

I don't know the minimum number of dimensions required to embed all valid metrics of GR.
Let me know if anyone does find out.

 

[intel]

Forget relativity for a moment and think about ordinary space. What is the 3rd dimension?

I would say z, the 3rd spatial dimension

If that is the case - can I call time the 1st dimension and assume it acts on all ordinary space?

 

[pmb_phy]

Hi, I'm new here. I've loved all the qualitative aspects of physics for a while now and I'm just starting to get into some of the more quantitative aspects of physics, the actual math. I'm such a complete amateur at all this and I'm only in high school so please excuse my question if it seems ridiculous.

Welcome to the forum spizma. Glad to see you here. Don't worry about seeming ridiculous. We can all make statements that we could later come to regret. Besides, if your new here and only have a high school education than there is no reason for anyone to be a jerk to you. As a matter of fact there is never a good reason for that if the person you're speaking with is a good human being.

I was thinking about how time is a dimension today and this thought occurred to me. From what I understand, the Planck length is the smallest possible length. Now, it being a length, it would measure a 1 dimensional object. If we were to measure a 2 dimensional object, the curvature of 1 dimensional space, ...

A 1d space cannot have curvature in the sense of Riemann curvature. The dimension of the space must be two or more for there to be Riemann curvature present.

If we were to measure a 3 dimensional object, the curvature of 2 dimensional space, we would measure it in Planck lengths cubed, the Planck volume I'm assuming.

What is it you'd be measuring? As of yet it doesn't sound like your taking measurements so as to determine curvature of the space.

So, from this logic, one would assume if we were to measure something in the fourth dimension of time, the curvature of 3 dimensional space, we would measure it in Planck lengths to the 4th power. But we don't, we measure it in a completely separate unit, the Planck time. Why?

The units of time are adjusted so that they have the same dimension of space. E.g. the position 4-vector is defined as X = (ct, x, y, z). So it is "ct" that is the 4th dimension. We can think of it as having units of space but its physical significance is still that of time.

Pete

 

[Jimmy Snyder]

I would say z, the 3rd spatial dimension

Now point your finger in the z direction.

 

[intel]

Now point your finger in the z direction.

Relative to where one is in the classical sense - what am I missing?

Can I say without going into GR that time necessarily acts on all space (except a black hole)?

 

[Jimmy Snyder]

Relative to where one is in the classical sense - what am I missing?

Can I say without going into GR that time necessarily acts on all space (except a black hole)?

What I am getting at is the following. The world doesn't have coordinate lines painted on it. There is no absolute meaning to the z dimension. Four dimensional spacetime works the same way. There is no absolute meaning to the time dimension.

 

[intel]

What I am getting at is the following. The world doesn't have coordinate lines painted on it. There is no absolute meaning to the z dimension. Four dimensional spacetime works the same way. There is no absolute meaning to the time dimension.

I do not understand but do appreciate the complexities of time as a dimension. I am asking if it is technically correct to say given a 3d space, time necessarily acts on all space?

If so - can I say that time is a necessary condition for space?

 

[Jimmy Snyder]

I do not understand but do appreciate the complexities of time as a dimension. I am asking if it is technically correct to say given a 3d space, time necessarily acts on all space?

If so - can I say that time is a necessary condition for space?

Dunno. However a better thing would be to say that spacetime is 4 dimensional and that concepts of pure space and pure time are frame dependent. Take away the frame and you take away pure space and pure time and are left with nothing but spacetime.

Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

 

[intel]

Dunno. However a better thing would be to say that spacetime is 4 dimensional and that concepts of pure space and pure time are frame dependent. Take away the frame and you take away pure space and pure time and are left with nothing but spacetime.

Not sure, but that sounds like a yes to me?

Taking it a step further - is there a reason I can not say time is a necessary and sufficient condition for space?

The next step would be that space is a necessary and sufficient condition for time, but find this intuitively harder to suppose.

 

[Jimmy Snyder]

I'm not sure what you are driving at intel. To my ears it sounds the same as if you said that the x-coordinate is a necessary condition for the y-coordinate. I don't even know what that means.

 

[intel]

If they are a necessary and suficient condition for each other, it gives an immense continuity that all and everything goes through - the scale of it blows my mind