Can X4 Be Regarded as Spatial?

[bobc2]

I'm wondering if some of you have ideas or references that would help in answering the question, "Can the 4th Dimension Be Regarded as a Spatial Dimension in the Same Sense That X1, X2, and X3 Are Spatial?" To refine the question I present graphics below that review the manner in which we describe the motion of a projectile in 3-D space using time as a parameter. For example, we can use Y(t) and X(t), two parametric equations to describe the motion as shown below. See sketches (a) and (b) below.

Now, if we consider the path of an object's world line through 4-dimensional space, can we not consider time as a parameter in the same sense as our 3-D case? That is, an observer moves along his world line at the speed of light. See the crude 4-D universe sketch d) below (Note: No significance is intended with my arbitrary choice of a Big Bang to Big Crunch universe model--really just wanted some generic model that exhibited 4 dimensions and a flow of time).

The comparison perhaps becomes a little muddy if we consider the moving object in 4 dimensions as not really exhibiting motion, because its description seems more like a 4-dimensional object. Therefore, not knowing how to describe the entity that is actually moving, I first resort to an analogy, sketch (C) below, for the 3-D case in which we project a beam of light onto a parabolic 3-D bar, moving the beam along the path of the bar at a constant speed (constant speed along the bar, i.e, not constant with respect to X and Y). This is similar to the world line case, where something analogous could be regarded as moving along the 4-dimensional path at light speed.

I realize that X4 is different in significant ways from X1, X2, and X3. But that is largely because the universe is populated with 4-D objects that a billions of miles long along the 4th dimension and are comparatively extremely small in the other 3 dimensions. Also, time plays a unique role with relation to the 4th dimension--but do those factors necesarily take away from the spatial character of the 4th dimension?

 

[atyy]

Yes, in spirit. There is no "dimension" that is purely time, or purely space. The Lorentz transformations mix space and time, so there are many equally good separations of spacetime into space and time. Each such such separation is a "Lorentz inertial frame". Another way of saying this is that there isn't only one direction in spacetime that we can call the direction of time - instead there are many. However, each observer defines only one of those directions to be "his" direction of time.

 

[bobc2]

Yes, in spirit. There is no "dimension" that is purely time, or purely space. The Lorentz transformations mix space and time, so there are many equally good separations of spacetime into space and time. Each such such separation is a "Lorentz inertial frame". ...

But, what if you derive a Lorentz transformation using purely spatial considerations as shown below. We have two 4-D objects (blue and red) oriented symmetrically with respect to reference coordinates, X1 and X4. There is no mixing of time and space. After using the Pythagorean theorem on the purely spatial diagram, we use the parameter, time (X4 = ct) in order to obtain the time transformation. Thus, we only introduce time as a parameter.

 

[DrGreg]

I think it depends how you interpret the word "spatial". For my money I'd interpret it as "spacelike" (as determined by the metric) so in that sense time isn't a spatial dimension. But is a dimension, one of the 4 dimensions of spacetime, and there is no unique direction for time, but a whole lightcone-full of timelike directions to choose from, so some people might describe that as "spatial".

I'm sure you know the time parameter described in the 2nd para of post #1 is called proper time, and is invariant (coordinate-independent), but each worldline has its own proper time; there's no global proper time, only a global coordinate time dimension that depends on your choice of coordinates.

It is tempting to think of particles moving along 4D worldlines at the speed of light, but really you should think of spacetime as just a static collection of worldlines, with no motion or passage of "external time". Time is already embedded in spacetime. The model of particles moving along 4D worldlines, as proper time evolves, starts to fall over if you have more than one worldline -- how do you synchronise the two worldlines' proper times? You can't in any coordinate-independent way.

 

[bobc2]

I think it depends how you interpret the word "spatial". For my money I'd interpret it as "spacelike" (as determined by the metric) so in that sense time isn't a spatial dimension. But is a dimension, one of the 4 dimensions of spacetime, and there is no unique direction for time, but a whole lightcone-full of timelike directions to choose from, so some people might describe that as "spatial".

Thanks for the excellent comments, DrGreg. Yes, I was intending the dictionary "spatial" as having the nature of space. Space on the same footing as X1, X2, and X3. Again, recognizing that the 4th dimension is distinguished by the geometric properties of 4-dimensional objects--that is, 4-dimensional objects extend billions of miles along the 4th dimension. And, again, proper time is associated with the world line of 4-D objects in that some aspect of observers moves along their world lines at light speed, evoking the impression of time passing.

I'm sure you know the time parameter described in the 2nd para of post #1 is called proper time, and is invariant (coordinate-independent), but each worldline has its own proper time; there's no global proper time, only a global coordinate time dimension that depends on your choice of coordinates.

Yes, of course, I certainly agree with your comments.

It is tempting to think of particles moving along 4D worldlines at the speed of light, but really you should think of spacetime as just a static collection of worldlines, with no motion or passage of "external time".

Yes. Your comment here is consistent with the model described in post #1. The universe is described as a 4-dimensional space. And in this model the universe is populated with 4-dimensional objects. And if the objects are indeed 4-D, then they are positioned in the universe as static objects--there can be no motion associated with these objects (other than a continuous sequence of 3-D cross-section views giving the impression that there are 3-D objects in motion).

Time is already embedded in spacetime. The model of particles moving along 4D worldlines, as proper time evolves, starts to fall over if you have more than one worldline -- how do you synchronise the two worldlines' proper times? You can't in any coordinate-independent way.

Good point. And we have a problem here--the definition of an observer is a little slippery. The observer is assumed to be a sentinel being with a consciousness. But we want to avoid getting into philosophy and metaphysics, so we simply recognize first that all objects embedded in the universe are 4-dimensional. This must be since, as shown in the sketch below, observers moving at different velocities "live" in different instantaneous 3-D cross-sections of the universe. Physically (we are assuming the external physical model above), there cannot be multiple different 3-D cross-sections of an object unless that object is 4-dimensional.

So, the physical body (including brain, neurons, etc.) of an observer is a static 4-dimensional structure. Nevertheless we shall still refer to an observer as moving along his own world line at light speed and leave it to metaphysics and philosophy to worry about the nature of the entity that is actually doing the moving. I used a spot of light moving along the parabolic bar at constant tangential speed as the 3-D analog of a parametric proper time for a 4-D object. And the yellow spot on the 4-D object of post #1 sketch d) was intended to represent the light speed motion of the undefined entity.

Time is passing for the "moving observer", but doesn't require time to be part of the spatial dimension. Again, time can only be regarded as a parameter in no more deeper sense than the use of time as a parameter in the 3-D description of projectile motion sketched in post #1 sketch b). Any deeper understanding would necesarily involve an investigation of consciousness and psychological aspects of observers.