Do we have "Newtonian space-time" in classical physics?

[etotheipi]

I've managed to thoroughly confuse myself. Before Minkowski came along and combined 3-dimensional Euclidian space and time into Minkowski spacetime, I was under the impression that we only dealt with three dimensions and that time was just a universal parameter. Thorne and Blandford write

...[Newtonian Physics'] arena is flat, 3-dimensional Euclidian space with time separated off and made universal...

Though evidently to specify an event (a concept which exists in all flavours of physics, not just relativistic), we need four pieces of information: $(x,y,z,t)$. But in the realm of classical physics, writing an event in this manner - with $t$ as a coordinate - looks odd because in order for it to be a coordinate, it must also be a dimension (I could be wrong about this...).

Then to add to the confusion, I came across this description by V.I. Arnold,

The Galilean space-time structure consists of ... the universe, a four-dimensional affine space $A^{4}$. The points of $A^{4}$ are called world points or events. The parallel displacements of the universe $A^{4}$ constitute a vector space $\mathbb{R}^{4}$

So now I'm completely lost, since he seems to be including time as a dimension. But I thought Newtonian physics operated under the assumption of motion in a Euclidian space, only parameterised by time!

So I wondered whether someone could clarify whether or not, in classical physics, time is a dimension - since these two sources seem to completely contradict.

Thank you!

 

[PeroK]

I've managed to thoroughly confuse myself. Before Minkowski came along and combined 3-dimensional Euclidian space and time, I was under the impression that we only dealt with three dimensions and that time was just a universal parameter. Thorne and Blandford write

Though evidently to specify an event (a concept which exists in all flavours of physics, not just relativistic), we need four pieces of information: $(x,y,z,t)$. But in the realm of classical physics, writing an event in this manner - with $t$ as a coordinate - looks odd because in order for it to be a coordinate, it must also be a dimension (I could be wrong about this...).

Then to add to the confusion, I came across this description by V.I. Arnold,So now I'm completely lost, since he seems to be including time as a dimension. But I thought Newtonian physics operated under the assumption of motion in a Euclidian space, only parameterised by time!

So I wondered whether someone could clarify whether or not, in classical physics, time is a dimension - since these two sources seem to completely contradict.

Thank you!

It works both ways. If you apply the principles of symmetry and homogeneity to space and time you can derive the following for your transformation between frames:
$$x' = \gamma(x - vt), \ \ t' = \gamma(t - kvx)$$
Where $k$ is a constant, and $\gamma = \frac 1 {\sqrt{1- kv^2}}$

If $k = 0$, then you get:
$$x' = x - vt, \ \ t' = t$$
Otherwise, we find that $k = \frac 1 {c^2}$, where $c$ is an invariant speed.

With hindsight, therefore, we can look at Newtonian mechanics two ways. With time as a parameter or time as a dimension of Galilean spacetime.

 

[Dale]

They are mathematically equivalent. I.e. there is a one to one mapping between $(x(t),y(t),z(t))$ and $(t,x(t),y(t),z(t))$

 

[etotheipi]

$$x' = \gamma(x - vt), \ \ t' = \gamma(t - kvx)$$

If $k = 0$, then you get:
$$x' = x - vt, \ \ t' = t$$
Otherwise, we find that $k = \frac 1 {c^2}$, where $c$ is an invariant speed.

That's quite a neat way of looking at it!

They are mathematically equivalent. I.e. there is a one to one mapping between $(x(t),y(t),z(t))$ and $(t,x(t),y(t),z(t))$

Right, I see what you mean. Thank you!

 

[anorlunda]

It was Einstein, in his famous 1905 paper, who taught us that time and space are not totally independent. In particular, he taught us that simultaneity is frame dependent, and that was the insight that led him to special relativity.

https://en.wikipedia.org/wiki/Relativity_of_simultaneity
http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf

 

[vanhees71]

Another mathematical interpretation of Newtonian physics (and I think that's just the modern way to express Newton's original ideas of "absolute space" and "absolute time") is that of a fibre bundle. You have an continuum of copies of the 3D Euclidean affine space along a time axis.

In some sense the spacetime of SRT (Minkowski space) is more elegant and simple: It's a 4D affine pseudo-Euclidean space with the fundamental form having the signature (1,3) (or equivalently (3,1) if you prefer the mostly-plus-convention for the pseudometric).

 

[etotheipi]

Another mathematical interpretation of Newtonian physics (and I think that's just the modern way to express Newton's original ideas of "absolute space" and "absolute time") is that of a fibre bundle. You have an continuum of copies of the 3D Euclidean affine space along a time axis.

Interesting - I've seen the term "fibre bundle" written but the mathematics appears a little too complex for me to understand right now. Though your description provides a good conceptual way of viewing it, thanks!

 

[PeroK]

Interesting - I've seen the term "fibre bundle" written but the mathematics appears a little too complex for me to understand right now. Though your description provides a good conceptual way of viewing it, thanks!

At least you know now that it isn't a TV + Internet broadband package!

 

[etotheipi]

At least you know now that it isn't a TV + Internet broadband package!

It would be quite the subliminal marketing campaign...