Lecture on Newtonian spacetime

[Dale]

Note: this thread is a fork from https://www.physicsforums.com/threa...s-nobel-laureate-stanford-university.1067929/

I saw a rather interesting lecture recently that showed how Newtonian spacetime followed from taking Newton’s first law seriously (ie as something applicable to the real world and not as a special case of the 2nd law). In the resulting Newtonian spacetime there was a concept of absolute time and absolute space, but even so there was not a notion of the velocity of space.

 

[gmax137]

I saw a rather interesting lecture recently that showed how Newtonian spacetime followed from taking Newton’s first law seriously (ie as something applicable to the real world and not as a special case of the 2nd law). In the resulting Newtonian spacetime there was a concept of absolute time and absolute space, but even so there was not a notion of the velocity of space.

Sounds interesting, @Dale . Is the lecture available online or ?? Thanks!

 

[Dale]

Sounds interesting, @Dale . Is the lecture available online or ?? Thanks!

Yes, this is lecture 9 by Dr Schuller at the Heraeus International Winter School on Gravity and Light. It is all quite good, but lecture 9 was the one I was specifically referring to.

 

[gmax137]

Thank you @Dale . Parts of that lecture were over my head but the professor is doing a very good job. I think I will start with Lecture 1 and see what I can learn.

 

[Dale]

I did the same for the same reason!

 

[cianfa72]

@Dale Lecture 9 referenced in your post #10 is basically the Newton-Cartan theory ?

ps. very interesting his point of view about Newton I law/axiom: basically it defines what physically/geometrically an uniform straight line is.

 

[Dale]

@Dale Lecture 9 referenced in your post #10 is basically the Newton-Cartan theory ?

I would say it is the spacetime of Newton-Cartan theory, but in that lecture he didn't develop the theory further. I.e. no description on how matter curves Newton-Cartan spacetime other than the restrictions on the connection that he mentions.

ps. very interesting his point of view about Newton I law/axiom: basically it defines what physically/geometrically an uniform straight line is.

I agree. The other alternatives I have seen are:

1) Treat it as a special case of the second law
2) Treat it as the definition of an inertial frame

His third approach

3) Treat it as the definition of a straight line in spacetime

Has some benefits. I think the other two are still viable, but I see the appeal of the third way.

 

[cianfa72]

1) Treat it as a special case of the second law
2) Treat it as the definition of an inertial frame

His third approach

3) Treat it as the definition of a straight line in spacetime

I think 1) isn't useful since it is contained in the Newton's second law.

Btw, as he pointed out, to be applicable the first law actually requires knowing when/in which circumstances there is "no-force" acting on a particle (viewing/including gravity as a force).

As far as I understand, what he says about Laplace's idea was to "overcome" this difficulty (basically a "circular" argument) by attempting to treat gravity as curvature of just space (gravity is no longer a force) turning the second law in an autoparallel equation. It turns out that in space alone isn't possible, yet in spacetime it is.

 

[Dale]

Btw, as he pointed out, to be applicable the first law actually requires knowing when/in which circumstances there is "no-force" acting on a particle (viewing/including gravity as a force).

Yes, and that is why making gravity not a force is so useful. Then you can just attach an accelerometer. If it reads 0 then the particle is free. It makes the theory empirically clear.

Laplace's idea was to "overcome" this difficulty (basically a "circular" argument) by attempting to treat gravity as curvature of just space (gravity is no longer a force) turning the second law in an autoparallel equation. It turns out that in space alone isn't possible, yet in spacetime it is

I wonder historically why that didn’t occur to Laplace.

 

[pines-demon]

I agree. The other alternatives I have seen are:

1) Treat it as a special case of the second law
2) Treat it as the definition of an inertial frame

His third approach

3) Treat it as the definition of a straight line in spacetime

I got a different approach from my advanced mechanics teacher. He said that Newton's 1st law postulated the existence of inertial frames.

I wonder historically why that didn’t occur to Laplace.

He was close, he came up with the idea of light being deviated by gravity.

 

[cianfa72]

I got a different approach from my advanced mechanics teacher. He said that Newton's 1st law postulated the existence of inertial frames.

This point of view assumes knowledge of what an uniform straight line is (i.e. what means straight in space and uniform in time). Basically it assumes un underlying Euclidean geometry for the space.

 

[Dale]

I got a different approach from my advanced mechanics teacher. He said that Newton's 1st law postulated the existence of inertial frames.

Yes, that was 2) in my list.

 

[pines-demon]

Yes, that was 2) in my list.

Not exactly. (2) is more like taking the law as a definition, you could define something without postulating that it exists.

Edit: for example, a magnetic monopole is that which makes $\nabla \cdot \mathbf B \neq0$. This is just a definition and not a postulate that magnetic monopoles exist.

 

[Dale]

Not exactly. (2) is more like taking the law as a definition, you could define something without postulating that it exists.

Ah, I see the distinction you are making.

The issue then is that as far as I know such frames don’t exist. So it is probably better to just define them rather than assert their existence. They are a useful approximation even if they don’t exist.

 

[pines-demon]

The issue then is that as far as I know such frames don’t exist. So it is probably better to just define them rather than assert their existence. They are a useful approximation even if they don’t exist.

From a modern physics perspective, sure, inertial frames do not exist, but before 1915 that might not be at all clear.

 

[weirdoguy]

I got a different approach from my advanced mechanics teacher. He said that Newton's 1st law postulated the existence of inertial frames.

That's how I do it with my students Although it's not that advanced, just high-school physics. And I got it from an old problem book for high schools. Those old books are such a gems sometimes.

 

[gmax137]

Dr Schuller

I watched Lecture 1 earlier today. I have no previous exposure to the subject (topology). I have to say, Schuller is a master lecturer, and an expert in the use of the chalk board. So clear, and no wasted words. I liked the parts where he admitted the subject seems quite abstract, but "trust me, if you stick with it, it's worthwhile." I'm looking forward to the next lecture.

Too bad so much on the internet is crap, it should all be like this. I'd be interested in hearing what more mathematically sophisticated PFers think of these lectures. It is an investment, each lecture runs about 90 minutes.

 

[Dale]

an expert in the use of the chalk board

I have noticed that he has little marks on the wall to indicate how high the chalk board should be so he can exactly reach the top.

I'd be interested in hearing what more mathematically sophisticated PFers think of these lectures.

I am far from the most mathematical here. I need to go back again and take notes. Just the definitions, I think.

 

[cianfa72]

I'd be interested in hearing what more mathematically sophisticated PFers think of these lectures.

To me in some point they aren't very accurate. For instance in lecture 2 in the definition of topological manifold he doesn't esplicitely demand that the image under chart map $x(U)$ of an open set $U \in M$ is supposed to be open in $\mathbb R^d$ (w.r.t. its standard Euclidean topology).

Next, starting from lecture 4, he defines differential manifold as a triple $(M, \mathcal O, \mathcal A)$. Actually assigning an atlas $\mathcal A$ automatically defines the (unique) topology of $M$ (i.e. a set $V$ is defined open in $M$ iff the images $x_i(V \cap U_i)$ under the chart maps $(x_i, U_i)$ in the atlas $\mathcal A$ are all open).

 

[pervect]

I'm slowly working my way through the lecture by Dr. Schuller, but I think there is an interesting point where I disagree. At one point, around an hour and ten minutes into the lecture, he states that it's gravity itself, not relativity, that determines space-time is curved. But I think that it really IS relativity - not special relativity, but Galilean relativity - that makes the space-time curvature idea work. If objects of different composition fell at different rates, then the curved space-time picture would fail. It is exactly Galilean relativity that motivates the "curved Newtonian space-time" idea, in my opinion.

However, from context, I am sure that he meant "special relativity" in his lecture, even though he didn't use those exact words. Still, I thought it was an interesting point.

 

[Dale]

If objects of different composition fell at different rates, then the curved space-time picture would fail.

But that is gravity, not relativity (Galilean or otherwise).

 

[Sagittarius A-Star]

But that is gravity, not relativity (Galilean or otherwise).

I think it is the Galilean equivalence principle.

 

[Dale]

I think it is the Galilean equivalence principle.

OK, but that is still gravity, not relativity.

 

[cianfa72]

In lecture 9 he defines Newton spacetime as the 5-tuple $(M,\mathcal O, \mathcal A, \nabla, t)$. $\nabla$ is a torsion-free affine connection defining the notion of "uniform straight lines" in spacetime, i.e. inertial worldlines.

Next, the notion of Newton's absolute time is encoded by the global smooth function $t$. I think what he claims at 01:05:05 is no sense (since just preimages of different values of function $t$ are by definition disjoint sets). The point to demand $df \neq 0$ everywhere should be to make sure $M$ is foliated from regular submanifold (by virtue of regular values theorem).