Acceleration

[pioneer]

This idea popped into my head in the forum topic; dimensions. I went for my nightly swim and further ideas about this appeared in my mind. I thought I would put this out there, as food for thought.

 

If you look at a point, that is moving in a line, say along the x-axis, it can be expressed with only two dimensions, one in distance and one in time or (x,t) which amounts to velocity. If we look at an acceleration of the same point, it is (x,t,t). Does acceleration imply 2 time dimensions?

 

In classical physics the answer would seem to say no. But if we look at this in terms of relativity, the answer appears to be maybe. If you were at relativistic velocity, time would dilate and we would have a certain time reference. If we accelerated through a range of relativistic references, we would be in odd reference where time is changing with time. Our reference would never be in any one time reference at a time. Would this acceleration be implicit of the second dimension of time? One wpuld not be able to express it mathematically with only one time variable.

 

Acceleration has always used two time dimensions in the math, that act independantly but remained connected. Maybe the classical physicists were onto something, but may it wasn't yet time for 2-D time. Maybe the accelerated expansion could be explained with a type of 2-D time affect.

[timo]

The definition of acceleration [math] \vec a = \frac{d}{dt} \frac{d}{dt} \vec x(t)[/math] usually assumes all appearing t's to be the same coordinate.

[someguy]

you can fake a 4 dimensional graph on a piece of paper using the same logic as a topographical map faking 3 dimensions. computers are capable of graphing a 4d object, just a moving 3d object. i've often wanted to see a simple example of the faking of a 5 dimensional graph in the same way but i have never been able to find one. if the fifth dimension was acceleration, the faking would look like many of the same 3d object following the same trajectory, or moving the same way, but each would be doing it at a different speed. i think this would be the best way to test your idea. but i don't have the knowhow to create such a graph.

 

i think i just understood what you're getting at. the rate at which objects move inside a body like the gears inside a clock, change velocity in relation to the velocity of the whole system, the whole clock. therefore you're saying that time changes in relation to time, kind of like a double derivative, thus you would require an extra dimension.

 

but i'm not sure you couldn't express it with one variable, since the rate of change of the objects inside the moving body are related to the whole body's motion, like parts of a mechanical system. therefore, you could always describe mathematically the movement of all parts inside the body as proportional to the speed of the body, so then i think that it would be possible to use the same variables. but you would have a formula inside a formula and that would be your double derivative kind of part i think.

[pioneer]

Thanks for the feedback. I presented this to get the mind thinking. I don't really like the idea of two dimensions of time. I prefer the idea of three dimensions of time. What this amounts to is acceleration for acceleration. For example, if we look at gravitational force, if this is constant, we get a constant acceleration. But since the magnitude of the force, is a function of distance, the force increases with decreasing distance. The result is an acceleration that is also accelerating as distance gets closer.

 

These results imply (r,t,t,t) or one distance dimension and three time dimensions. If position in a volume of space is important, than 6-D space-time allows us to know its vector position in both space and time, simultaneously. In this system, we currently lump the three dimensions of time, into the four time vectors which we call the four forces.

 

I can sort of follow this logic, but where to go from here, I am not sure. My gut tells me that maybe it can help quantum theory integrate gravity. It might give a backdoor way to see what is being overlooked.

[Bignose]

But, I still fail to see anywhere that an extra time dimension (or more) is needed. The traveling of a body toward a gravity source can easily be handled in just 1 time dimension. It may not necessarily be solvable directly, but it's description using 1 time dimension is pretty easy.

 

There are situations when using multiple time dimensions is appropriate. When describing a population of cells, one can use the cell's age as one of many descriptors. Then, there are two times -- the regular time that we all know, call it t, and the age of the cell, call it t'. The only difference between the two is that t' resets to zero every time a cell divides. But, the derivative between them is still 1 = dt'/dt They both pass at the same rate.

 

But, I don't see how a derivative in time (which is a velocity, or an acceleration) "implies" another time dimension. If we take the a derivative in x, df(x)/dx, we don't say that that implies another dimension, y? So why would derivatives in time imply that? Especially since if we change the variables, instead of a stationary observer and observe with the moving particle, a derivative in space (d/dx) can be transformed into a derivative in time (d/dt). I don't see any need to invoke another dimension.

[pioneer]

You are sort of correct. The analogy can be seen if we compare Newtonian gravity to relativity. The relativity addendum is not needed for most situations one will encounter. But as velocity gets extreme, or close to the speed of light, then relativistic affects become more pronounced.

 

A good way to look at it is assume you are traveling near C. Ones velocity will determine the time reference. If we accelerate from there, there is no longer a constant velocity such that time reference is also accelerating. If this acceleration was due to a gravity field that was getting stronger as we approach it, then even our rate of acceleration is also accelerating. The time vector would be a way to calculate changes we would experience.

 

I don't know if this is the correct analogy, but here goes. We often use a position vector for a force applied to a gear. We have our torque, with the direction of force at some odd angle in 3-D space, such that all the force is not being used for mechanical advantage. At V near C, we have steady state. But during the transitions of double acceleration there is a time distortion where steady state is not able to form with finite matter systems. The system will take time to stabilize, but may not under certain conditions. It is sort of like the force vector shearing the gear, alterring the ideal force impulse ,that would be transmitted if the shear was not affecting the torque. Or the affective force gets alterred to create a nonideal force, more in line with another force?