Ask a physicist anything. (7)

[Resha Caner]

Well, yeah, paths are a bit more difficult. The problem is that in an abstract sense, a path is an infinite-dimensional object (since you can have an infinite number of locations among the path, each of which is defined by two values in two dimensions). Dealing with generalized paths can be quite a fun mathematical/computational problem, and commonly appears in General Relativity and Quantum Field Theory.

Hmm. I'm not sure I get why it would be infinite-dimensional, but it's good to know others have thought of this. I mean, it would be really cool if I were on to something truly unique, but I haven't had a good idea yet that someone else hasn't already considered. So, maybe I'm treading on old ground here.

Anyway ...

What I was heading toward is that maybe the proper model should be based on the material one is trying to model, not on an abstract space in which that material resides. It would mean everything depends on the interaction of the material involved - even "dimension", "time", etc. As such, time would be different for every system - and I don't really mean that in the typical relativistic way ... at least I don't think so. It could mean that from the perspective of a linear system, the time in some other system might progress parabolically, or cubically, or logarithmically.

 

[Chalnoth]

Hmm. I'm not sure I get why it would be infinite-dimensional, but it's good to know others have thought of this. I mean, it would be really cool if I were on to something truly unique, but I haven't had a good idea yet that someone else hasn't already considered. So, maybe I'm treading on old ground here.

Well, you have to use an infinite number of variables to describe an arbitrary path. Infinite variables = infinite-dimensional.

What I was heading toward is that maybe the proper model should be based on the material one is trying to model, not on an abstract space in which that material resides. It would mean everything depends on the interaction of the material involved - even "dimension", "time", etc. As such, time would be different for every system - and I don't really mean that in the typical relativistic way ... at least I don't think so. It could mean that from the perspective of a linear system, the time in some other system might progress parabolically, or cubically, or logarithmically.

It is generally the case that if you can find a way to exploit some particular aspect of the problem you're trying to solve, you can improve the problem-solving time. But what you're describing here is a little bit too vague for me to see what you're talking about specifically.

 

[Resha Caner]

Well, you have to use an infinite number of variables to describe an arbitrary path. Infinite variables = infinite-dimensional.

Ah, OK. I missed that you were describing an arbitrary path. My description was not of an arbitrary path, so I think it would be less than infinite-dimensional. Still, it seems others have covered this before.

It is generally the case that if you can find a way to exploit some particular aspect of the problem you're trying to solve, you can improve the problem-solving time. But what you're describing here is a little bit too vague for me to see what you're talking about specifically.

This all started years back when I was working pretty intensely on some nonlinear systems. I had this niggling idea that "nonlinear" was really a term that meant it was nonlinear relative to the mathematics one chose to use. So, if I could find a different math system that matched the behavior of the material, the behavior would be "linear" because of a correspondence between the real system and the model.

It seemed to work pretty well, but there are some details that have held me up and prevented me from fully implementing the idea.

Regardless, the key to the mathematics was to adopt Non-Newtonian calculus. That meant I could define new velocity and acceleration terms (with the implication being that time was also different).

For example, a traditional velocity would be v = (x2-x1) / (t2-t1). A Non-Newtonian version with a consistent "geometric" calculus might be v = (x2/x1) ^ (1/(t2-t1)).

 

[GrowingSmaller]

I dont quite get curved space. In diagrams like this:

It seems always to show objects travelling above the line of the radius, as if travelling above the centre of gravity or flying over the top of the gravity and being sucked in away from the initial trajectory. But what if the object is travelling in line with the raduis, straight for the centre? Then it doesn't seem like you can make curved space into a downhill shape, because that would put the object on an unnatural trajectory away from the radius and centre. IOW a straight line can't be represented as a curved hill. So how is space "curved" in this scenario?

 

[Chalnoth]

I dont quite get curved space. In diagrams like this:

It seems always to show objects travelling above the line of the radius, as if travelling above the centre of gravity or flying over the top of the gravity and being sucked in away from the initial trajectory. But what if the object is travelling in line with the raduis, straight for the centre? Then it doesn't seem like you can make curved space into a downhill shape, because that would put the object on an unnatural trajectory away from the radius and centre. IOW a straight line can't be represented as a curved hill. So how is space "curved" in this scenario?

I don't quite get what you mean. The object traveling inward in the direction of the radius would simply speed up as it moves "downhill".

I should mention, however, that these visualizations of the curvature are just analogies to what is happening, as there isn't any good way to actually map the real geometry of space-time to a simple image for us to observe.

 

[GrowingSmaller]

I don't quite get what you mean. The object traveling inward in the direction of the radius would simply speed up as it moves "downhill".

Yes I get that but then the type of picture above would be inapt wouldn't it, as you seem to have said below.

I should mention, however, that these visualizations of the curvature are just analogies to what is happening, as there isn't any good way to actually map the real geometry of space-time to a simple image for us to observe.

So how is the geometry represented. Is it vector coordinates or field properties or something?

Also whats the difference between Newton's inverse square law and Einstein's curved space, or is it just 2 ways of saying the same thing? In that case why would one be preferred above the other?

 

[Chalnoth]

Yes I get that but then the type of picture above would be inapt wouldn't it, as you seem to have said below.


So how is the geometry represented. Is it vector coordinates or field properties or something?

Also whats the difference between Newton's inverse square law and Einstein's curved space, or is it just 2 ways of saying the same thing? In that case why would one be preferred above the other?

Actually, the geometry is represented by a distance measure, called a metric. By integrating the metric across a path, you can obtain the distance along that path. Objects in General Relativity always take the shortest possible space-time path between two space-time points.

This description is rather abstract and often difficult to work with, but it allows you to describe curvature without any extra dimensions. For example, it allows you to describe everything about how the surface of a sphere is curved without any reference to a third dimension.

As far as the relationship with Newton's inverse square law, well, that can be thought of as an approximation to General Relativity. You can correct Newton's inverse square law with additional terms (1/r^3, 1/r^4, etc.) which are needed at short distances to get the correct behavior (such as the orbit of Mercury). It's also worth mentioning that there are some other rather distinct differences, such as the fact that light responds to gravity twice as strongly as you'd expect from Newtonian gravity alone.

 

[Lillen]

May i ask, why aren't all matter dated to big bang when everything supposingly were created?

For instance messuering the date of when a comet had a impact on earth, you will need to messure how old the stones are. But the comet has been there since the birth of universe, or atleast since the collapse of a star, has it not?

 

[Chalnoth]

May i ask, why aren't all matter dated to big bang when everything supposingly were created?

Radiometeric dating dates matter to the time it last solidified, not to the time it was first created. So when a volcano erupts and spews lava, for example, that lava can later be dated back to the date of the eruption.

 

[Lillen]

And also as I see the above questions, curved space diagrams neglect that the space is threedimentional. Explaining gravity that way, with a two-dimentional paper is the only way to explain it. But it neglect that the universe is three-dimentinal. How would the moon travell around the earth using a three-dimentional diagram?

 

[Lillen]

Oh! But the comet has been solidified longer then its impact on earth, has it not! ?

 

[Chalnoth]

Oh! But the comet has been solidified longer then its impact on earth, has it not! ?

Well, sure. But how much of the comet survives the impact intact? Instead, we date the rocks and other things around the impact zone that were melted and then re-solidified. Obviously it has to be a pretty major impact for this to work.

Alternatively, smaller meteors that survive impact can be dated to find the time when they first formed (and they all date to right around 4.5 billion years ago).

 

[Lillen]

Oh! I see...

 

[mzungu]

I dont quite get curved space. In diagrams like this:

It seems always to show objects travelling above the line of the radius, as if travelling above the centre of gravity or flying over the top of the gravity and being sucked in away from the initial trajectory. But what if the object is travelling in line with the raduis, straight for the centre? Then it doesn't seem like you can make curved space into a downhill shape, because that would put the object on an unnatural trajectory away from the radius and centre. IOW a straight line can't be represented as a curved hill. So how is space "curved" in this scenario?

Actually in a physical world where gravity plays a role or is in existence then suffice it to say that Linearity is not possible. Also there is no way we can depict in 2D or even in 3D the curvature of spacetime. The only way we can "see it" is through mathematics.

 

[GrowingSmaller]

Actually in a physical world where gravity plays a role or is in existence then suffice it to say that Linearity is not possible. Also there is no way we can depict in 2D or even in 3D the curvature of spacetime. The only way we can "see it" is through mathematics.

So is "curvature" actually the wrong word?

 

[Chalnoth]

So is "curvature" actually the wrong word?

Not at all. It's exactly the same sort of math you would use to describe the curvature of the Earth, in particular how the Earth's curvature impacts the distances between various points. It's just that space-time has too many dimensions for us to visualize the curvature: our brains are incapable of imagining four-dimensional objects.

 

[mzungu]

Two documentaries to keep you all occupied. Enjoy:

The Fabric of the Cosmos | Watch Free Documentary Online

The Truth About Christmas Carols | Watch Free Documentary Online

 

[GrowingSmaller]

Thanks for the comments.

Now on th entropy.

Does it have units?

Also, I understansd that at the "beginning" (BB singularity etc) there was higher entropy, which decreases overall over time (except in open sysrtems IIRC where things like life might develop).

So is the human brain, or maybe a platonic solid, or another artefact, anywhere near the low level of entropy at the beginning. Is it possible for that to happen, or can the original low level not be recreated at all no matter what, except parhaps in a black hole?

 

[Chalnoth]

Thanks for the comments.

Now on th entropy.

Does it have units?

Sort of. The fundamental definition of entropy is:

S = k log(Omega)

Here k is Boltzmann's constant, and it has units of energy over temperature. But log(Omega) is the logarithm of the number of ways the system can be reconfigured and still have the same appearance. The units are just something that humans put in there as a convention. The logarithm of configurations is not.

So I would say that in practice, yes, entropy has units. But it doesn't need them. Those units are not fundamental to the definition, they're just there due to convention.

Also, I understansd that at the "beginning" (BB singularity etc) there was higher entropy, which decreases overall over time (except in open sysrtems IIRC where things like life might develop).

It's the other way around. Entropy has been increasing steadily. Though given your next statement I assume you just misspoke.

So is the human brain, or maybe a platonic solid, or another artefact, anywhere near the low level of entropy at the beginning. Is it possible for that to happen, or can the original low level not be recreated at all no matter what, except parhaps in a black hole?

Well, it happened once, so there's no reason to believe it can't happen again. However, to us the start of a new universe like our own would just look like a microscopic black hole that pops into existence then rapidly evaporates into nothing.

 

[GrowingSmaller]

Thanks again. Is there such a thing as low and high entrophy behavior. For example I might draw an intricate pattern on a piece of paper, or just crumple it up and throw it in the bin. Would these count as examples of low and high entropy activities respectively?