>*Gravitational waves cause space itself to stretch*
Please ELI5 specifically and empiracally what "Space Itself" actually is.
Would it be possible to build a 'galactic clock & Compass' - a "clock" to the regular pulses of a pulsar and the galactic direction the pulsar is in relation to the terrestrial compass (magnetic) on earth...?
What is the pulsar with the most reliable timings?
Space itself means the metric of spacetime: how you measure distances. Locally, if there was no gravity, you could think of space as a plain reticule where distances would be calculated through eucledian geometry: that means cartesian distances, parallel lines would be parallel and the angles of a triangle would sum 180, and the shortest path between two points is the straight line.
General Relativity successfully establishes that the presence of mass distorts this, so it defines a mathematical object (the Einstein tensor) that reacts to the distribution of mass and energy and precisely describes the changes to the metric. For example it can model how the mass of the sun distorts the space so that light from distant stars appear to follow a curved path because very close to the sun a curved path is now the shortest path.
The Einstein tensor defines how distances --and time-- are measured and it's the best mathematical model that we have about what "space itself" is. Future theoretical advances could take us forward and demonstrate that space itself emerges from other more fundamental elements, but this needs bridging quantum mechanics and gravity. We don't really know what space is made of, but scientists have precisely modelled how it reacts to mass (and energy) with utmost precision.
NB: At cosmic scales the exercise becomes more difficult, as there is an expansion of the metric of spacetime that is not due to the presence of mass, in fact it is caused by the _absence_ of mass as it seems to be due the energy of empty space: the phenomenon called Dark Energy.
If this is ELI5, we hang around very different kinds of five year olds :-)
Joking aside, thank you for the deeper explanation. The idea that spacetime isn’t fundamental is a very non-intuitive concept given how we’ve evolved to interact with the world. Any suggested reading on this topic for laypeople?
> Any suggested reading on this topic for laypeople?
Carlo Rovelli is a bona fide theoretical physicist, very involved in the development of Quantum Loop Gravity (one of the attempted approaches to bridge GR and QM). Turns out he is also a good pop-sci writer, so I would begin there. His book "The Order of Time" deals with the nature of time, which is not about the nature of space, but then again reading it you see that the mental gimnastics are similar.
I also found useful contributions from regular contributors at /r/cosmology (thank you /u/jazzwhizz) but it's less straightforward and alas, Reddit has its own issues.
I find /r/cosmotology a hairy subject hard to swallow...
I think they push string theory too much, and try too hard to braid it into the fabric of our societies, with their little shops and what not... It gets everywhere, and tomorrow is their favorite day "Friday"!
let's think about space like a giant, invisible playground. Normally, it's flat like your bedroom floor, where you can measure how far your toys are from each other with a ruler straight across. That's like when there's no gravity in space.
But guess what happens when something really heavy, like a big bowling ball (that's like a star or planet) comes into your playground? It makes everything around it bend and curve. So, the distance between your toys is no longer a straight line. It's like when you throw a ball, it doesn't go straight, it goes in a curve.
This bending is what a really smart guy named Einstein explained in a thing called General Relativity. He came up with a way to measure how space bends around heavy stuff.
And you know what else? There's this really weird stuff called Dark Energy that's everywhere but we can't see it. It makes space grow bigger and bigger, not because of heavy stuff, but because there's a lot of empty room. Scientists are still trying to understand this, but it's like blowing up a balloon: even though there's no heavy stuff inside, the balloon still gets bigger!
Out of curiosity, from someone who has never worked with non-euclidean geometry, what does it mean for a path to be curved in non-Euclidean space? My outsider understanding of curvature is that the inside of a curve is shorter than the outside of the curve, whereas a line has the same length on either side (assuming we give these curves and lines some thickness). But, if the shortest path can be curved, what do we mean by curved?
I've found it's easier to think about this stuff in two dimensions. The surface of a sphere (or the Earth!) has non-Euclidean geometry.
Imagine two people standing some distance apart from each other at the equator. They both begin walking in straight-line paths due south. At first, their paths are parallel. But as they move toward the south pole, they begin to drift closer to each other, as though their paths were curving towards each other. When they reach the south pole, they bump into each other. But they were both walking straight forward following the shortest path to the south pole the whole time. The curvature of the surface causes their initially-parallel paths to converge.[1]
On a plane (which has Euclidean geometry), initially-parallel paths never converge.
[1] Don't take this too literally; the real planet Earth is three-dimensional, and its gravity keeps us on the surface. But mathematically, it's possible to describe a curved two-dimensional space without referring to any higher dimensions. When I talk about "the surface of a sphere", that's what I mean -- the surface is the entire 2D space.
Maybe it's worth adding that in this way of thinking (intrinsic geometry of the surface), great-circle paths have exactly the property the GP brought up about straight lines: neighboring paths aren't shorter on one side and longer on the other. (If you think of them as 3-d paths then there's a shorter path below vs. longer above, but that's not part of the intrinsic geometry.)
If two people are in parallel, they will make two parallel circles. If two people aimed at a singe point, they are not in parallel.
Space-time is 4d array: array of framebuffers. You can stretch your mathematical model all day long, but you knowledge must be mapped to reality somehow. In model we have space-time, while in real world we have "physical vaccum" ("something nothing" or "phaccuum", for short). I prefer to name that thing "ether", because I like that word.
> If two people are in parallel, they will make two parallel circles. If two people aimed at a singe point, they are not in parallel.
In spherical geometry, the equivalent of a straight line is a great circle. There are no parallel great circles. That's why I used the phrase "initially parallel" -- at the starting point, both people's paths are at a 90-degree angle to the great circle connecting their locations.
I didn't want to get into "locally flat" vs. "globally curved" in something that started as an ELI5 thread.
> In spherical geometry, the equivalent of a straight line is a great circle.
Yes, of course. If we substitute parallel lines with straight lines in spherical geometry and mix 2D and 3D spaces, then our mental model will be nonsensical but cute.
We found no evidence of fourth dimension in the real world, so we cannot map this cute mathemagical model to reality.
Picture curves on the surface of the earth. They seem flat locally, but if you go a mile north, a mile east, a mile south, and a mile west, you don't end up _exactly_ where you start. (In the northern hemisphere you end up a little east of where you start; in the southern, a little west.)
Same thing in general relativity: the metric tensor measures the failure of closed loops on each axis to not close perfectly, the way they would in Euclidean space.
Basically even as a small creature on earth you can 'figure out' about the curvature by carefully measuring small-ish loops. The same is true for spacetime, but the loops' deformities are even smaller.
It's not too complicated. Get a round ball of some sort. When you draw on the surface, that's a "non-Euclidean space".
Take a straight line down from the "north pole" of your ball to its equator. Draw another straight line around a quarter of the equator. Draw a third line back to the pole. You've just drawn a triangle with 3 straight lines and the angles add to 270 degrees.
A non straight line is just not the shortest distance between two points on that surface.
Distance from center as described by the path - so if the distance to center is held, but the trajectory is changed, the line will be expressed as a curve around a tehtered point to the center.
If the 'tether' is a gravitational link (meaning that the teather, is a constant pull against the trajectory, regardless of the trajectory, the object will continue to curve around center.
I cannot be of help here but I'd say that your concept of curvature is too informal, but more formal maths can deal with it, take a look at the wikipedia page for "Geodesic", the maths are way above my head but the diagrams are cool :D
Super helps! thanks - while I knew a bunch of pieces and tidbits, you cemented it for me - thank you.
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TensorFlow
>*General Relativity successfully establishes that the presence of mass distorts this, so it defines a mathematical object (the Einstein tensor) that reacts to the distribution of mass and energy and precisely describes the changes to the metric.*
This leads me to think that TensorFlow was attempting to map the 'weight' among topics of intersecting interests, sciences, etc... and seeing who the "tensor warping" was most strong with and adding higher eval weights to things that "gravitated" to one another based on the informational difference in distance?
(I dont know the nomenclature, but is that were using 'tensors' in AI/ML/whatever 'weights' come from?
A tensor is a mathematical object that has a lot of uses beyond general relativity, of course. It's a little tricky to visualize (I think there are good YouTube videos) but tensors can be thought of as multidimensional arrays, and also the whole tensor algebra can be done in terms of multidimensional array operations.
So reasoning about a neural network weights and operations in terms of tensors makes sense and I guess that's what the name Tensorflow comes from.
Yes, but your explaination helped me to better understand the other types of tensors in a way I can 'grok' it - as opposed to trying to be at the level of some google quant or whatever with ML understandings of weights.
It helped my put my own internal visualizations to the understanding.
and I had a weird peripheral memory on this from a thought I had whilst driving in 1999 where I was thinking of tensors in this way, but I didnt know what I was just daydreaming about... but apparently, it was einsteins tensors coupled with information theory - and while to me it was a pedestrians take on the premise - it turns out that that day dream was correct!
And it all ties back to when I was ten years old and meditating on the Mind of God -- It all tied into one another - and you gave me some cord to pull these experiences together with understanding which I havent had in 40 years... so that was nice. Thanks.
Anyone else recall daydreams from their past where their later understanding was confurmed, even though you were just "daydreaming at the time"?
Take a sheet of elastic material, like a balloon, but square. Suspend the corners in clamps on posts so that it’s “perfectly” flat. Draw a line that is the shortest distance between two points on the material: it will be a “straight” line like what you learnt in geometry class. Now place a marble in the center of the sheet and let it stretch the sheet. Now draw the shortest possible line connecting these two points: it will curve (and unless you chose by coincidence) it will likely not overlap with the first line: the “shortest distance” between the points has changed.
There’s another aspect to this: the expansion coefficient. One such model is that the coefficient depends only on time: as time passes, distances increase. To model this, draw the same line on the flat sheet of material, then expand it uniformly in all directions. The distance is still a straight line, but the line is longer after the expansion.
Please ELI5 specifically and empiracally what "Space Itself" actually is.
Would it be possible to build a 'galactic clock & Compass' - a "clock" to the regular pulses of a pulsar and the galactic direction the pulsar is in relation to the terrestrial compass (magnetic) on earth...?
What is the pulsar with the most reliable timings?