Tensors and Time

[foodchain]

Yes, I don’t have a high level of understanding in regards to tensor stuff so please bear with me.

 

In regards to tensors can time be applied in varying degrees successfully, such as in the billionth of a second to maybe an hour? What I mean is basically can you use time in a sort of fashion to dissect actions/reactions for example in tensor terms? When I think about it such seems very rigid, or how could it adequately describe the motion in the ocean for at least one minute of time, or how each wave on a shore might look exactly for the period of ten minutes or some other increment of time.

 

Such as basically taking matter and energy, along with the various physical functions used to describe them currently into tensor along with time to explain in more real time sense a particular action again for example.

 

Also I was thinking that if you start to take time into such small increments that it might sort of disintegrate or time no longer being applicable to physical reality, such as cutting a second down in time I imagine can be done sort of infinitely, but of course you cant reach infinity so that time would still be occurring, just at a pass maybe, or is there some inverse of C or something really in regards to time and action. I think this still applies to the topic just not sure really.

 

My math skills are not to a level to be able to digest simply seeing an equation about such either, so you will have to use words, and thank you in advance for whomever posts or even reads this:D

[Bignose]

Rumplefloople?!? I am exceptionally confused about what you mean by "...describe them currently into tensor..." and oceans and etc.

 

But, I'm going to at least take a stab. The fluid mechanics equation describe the motion of the fluid in the ocean. They can be written like:

 

[math]\frac{D \mathbf{v}}{D t} = \nabla \cdot \mathbf{T} [/math] where [math]\mathbf{T}[/math] is the stress tensor of the fluid. The terms that make up the components of the stress tensor can be functions of time, as well as position. But, any time later, the components can change. There are times when you can approximate the stress tensor as unchanging in time -- that depends on the situation, but in general is can be a function of time and position.

 

A tensor is a mathematical construct so that the equations can be written like the one above rather than write out each term explictly. The above equations in spherical coordinates take an entire page to write out, but in vector/tensor notations it is one short line. Time may or may not be a coordinate to be included, but either way, the tensor notation is just a convenience not whatever you were trying to say above.

[foodchain]

Rumplefloople?!? I am exceptionally confused about what you mean by "...describe them currently into tensor..." and oceans and etc.

 

But, I'm going to at least take a stab. The fluid mechanics equation describe the motion of the fluid in the ocean. They can be written like:

 

[math]\frac{D \mathbf{v}}{D t} = \nabla \cdot \mathbf{T} [/math] where [math]\mathbf{T}[/math] is the stress tensor of the fluid. The terms that make up the components of the stress tensor can be functions of time, as well as position. But, any time later, the components can change. There are times when you can approximate the stress tensor as unchanging in time -- that depends on the situation, but in general is can be a function of time and position.

 

A tensor is a mathematical construct so that the equations can be written like the one above rather than write out each term explictly. The above equations in spherical coordinates take an entire page to write out, but in vector/tensor notations it is one short line. Time may or may not be a coordinate to be included, but either way, the tensor notation is just a convenience not whatever you were trying to say above.

 

So a tensor is a snapshot then, or can only view physical phenomena in the form of a snapshot in regards to time? How small of an increment of time is that? So for instance you took a series of these based on something in motion, say light hitting a rotating object, say a sphere of h20, would it be possible to describe such in tensor notation then? Basically I have somewhat of an idea. Basically when instruments are used to identify a piece of material, say carbon in some molecule, its able to do such in the form of a constant overall right? So if you could hook some apparatus up overall to discern this in the form of a tensor equation on some read out, would it be possible to study such then over a period of time. Such as the tensor equation is based in real time application and refreshes every hundredth of a second for example, and be able to form an equation that would register any sort of change.

[Bignose]

let me try to separate the two issues here. One the one hand, a tensor is a mathematical construct whose real motivation is for ease of notation. It is far nicer to write a bold T rather than writing out all the components. For example, in the mometum equation I wrote above, these is a velocity component in each direction. That means there are 3 separate velocity equations, one for each component (the velocity in the x direction, the velocity in the y direction, the velocity in the z direction). The stress tensor, T, has 9 components (xx,xy,xz,yx,yy,yz,zx,zy,zz). I could write each of them out for each equation, but, it is far easier to write out that one line above.

 

Secondly, in tensor formation, the equations are indifferent about what coordinate system used to describe the situation. This is desirable, since nature is indifferent about what coordinate system we as humans choose to impart. I.e. instead of the x,y,z components of velocity, maybe it would be easier to study the system in the cylindrical coordinates (r, theta, z), or spherical (r,theta,phi), or any other ones. The rules of tensor calculus set up exactly how to translate from one coordinate system to another, and the really nice thing is that the basic equation remains the same. The momentum equation I wrote above is correct for al systems of coordinates. This is the real benefit of tensor notation.

 

Now, on the issue of time. Let me write a very basic equation:

 

[math]\frac{df(t)}{dt} = g(t)[/math]

 

The time derivative, [math]\frac{df(t)}{dt}[/math], is the limit as the time increment goes to zero (delta t). This represents the instantaneous rate of change of the function f at a given time t. It has no other information at all. The same statement can be said about the right hand side, which the rate of change is equal to. It is a function of t, but the evaluation of that function happens only at one specific point in time, t. Not a range.

 

Assuming the equation can be solved (which it can for a lot of interesting phenomena), it is solved for all t. It is not solved for a period of time that you specify, you get a solution f(t)= something. That function again only gives the value at one specified t and does not necessarily say anythng about the times around it.

 

Now, that is the mathematical world's ideal. Obviously, in the real world, we never have the entire continuous data. We never know g(t) for all t. We take readings in time, whether by hand (looking at an instument) or by computer (which samples at a fixed rate, cannot be infinite). In fact, in solving differential equations numerically, we assume g(t) is not constantly changing, but is fixed for a short time. But, and here is the really important part, this is only an approximation. It does not describe the curves perfectly. One of the major issues with computational solutions is to choose small enough periods of time that g(t) would not vary too much over that period, yet large enough periods of time so the computer program can end. I.e. if g(t) changes slowly, say 0.1% every second, there is not much advantage to simulating on the microsecond scale. But, the physics of the equations determine this. Again, I cannot emphasize this enough, they are only approximations.

 

So, I hope I have explained well enough that time is not trapped in a tensor formulation. On the one hand you have a mathematical construct that has some very nice properties to make the math easier to handle. On the other, you have phenomena that vary in time. The two are actually very different subjects. All the way back to the fluid mechanic equation, the velocity field, v, changes in time and those changes in time are written using a tensor. But, I could have also just written out each component term by term. Time is not wrapped up in the tensor forumation.