/sci/ - Science & Math

[Anonymous No.3207078]

What would happen if space wasn't a linear tessellation?

[Anonymous No.3207084]

Is it a linear tessellation?

What does that mean, exactly?

[Anonymous No.3207090]

Not sure what you mean, but coordinate systems are mathematical tools; there isn't an invisible x, y, and z axis floating about somewhere in space.

[Anonymous No.3207095]

>>3207084
That the planes that are all perpendicular to each other,AKA space, didn't maintain a smooth constant extension.

[Anonymous No.3207121]

You can have different rules for geometry than Euclidean geometry, and we live in a space that follows non-Euclidean rules, but I don't know what specifically you're asking about.

>>3207095
>maintain a smooth constant extension.
What does that part mean?

[Anonymous No.3207123]

>>3207095

some things can be prove a priori. This means they can be proven simply by reasoning about them, without the need for evidence. If it is impossible to imagine an alternative then the assertion is proven a priori.

I say this to pre-empt the inevitable response of "prove it" when I tell you that space literally cannot exist in some sort of wacky funhouse configuration. What you have described is senseless and impossible.

[Anonymous No.3207136]

>>3207123
Is this a troll?

[Anonymous No.3207145]

ITT: we pretend we dont know about general relativity

[Anonymous No.3207157]

>>3207145

The large-scale structure of space is flat. Local warping in the fabric of space is negligible on the largest scale.

[Anonymous No.3207189]

Does liner tessellation involve fractals?

[Anonymous No.3207203]

>>3207189
Black holes with a fractal hologram on the interior wall of the event horizon.

[Anonymous No.3207227]

It isn't. Mass warps space-time.

[Anonymous No.3207234]

>>3207203
If the universe is a 3d hologram on the event horizon of a black hole in a 4d universe, shouldn't the universe be shrinking?

[Anonymous No.3207235]

>>3207095
because that's a geometric interpretation coined by Descartes.

In reality "space" isn't like a x-y-z axis but rather warped due to energy and mass

[Anonymous No.3207246]

>>3207227
Mass is energy.

The observable universe is our time space since the big bang, the beginning of time itself.

Time stops at the event horizon of a black hole.

[Anonymous No.3207262]

>>3207235
Particles represent the wavelike "strings" or just the warpings of space due to waves that resonate and cancel.

Time?

[Anonymous No.3207329]

OP here

do dimensions exist in pieces?

[Anonymous No.3207331]

You're wrong in assuming space "maintains a constant extension" due to the fact that x, y, and z axis do not physically exist. We perceive space as being linear, e.g two points can be connected by a line, when in reality time is a measure of change and space is a measure of time's movement. There is no linearity in the universe, we just perceive it that way because of how we interact with the 4th dimension (time).

[Anonymous No.3207339]

>>3207234
>If the universe is a 3d hologram
well, this my friend, is where you were wrong for the first time

>on the event horizon of a black hole in a 4d universe
this, was the second time

> shouldn't the universe be shrinking?
and finally, this was the third and last time you were wrong

[Anonymous No.3207349]

>>3207331
>in reality time is a measure of change and space is a measure of time's movement.
That's not reality, those are your own personal metaphysical beliefs. And they don't even make any sense. You've defined space and time in a vague, self-referential loop.

[Anonymous No.3207354]

>>3207262
>he believes in string theory

[Anonymous No.3207357]

>>3207331
10/10
would troll again

[Anonymous No.3207363]

>>3207246
>Time stops at the event horizon of a black hole.
no

[Anonymous No.3207371]

>>3207262
both. if you see it from the perspective of string theory, space itself is a 'network' of strings vibrating in different 'resonances' which are characteristic to what we know as a 'particle'.

[A guide for 4-dimensional flatlanders No.3207380]

From time to time there are discussions involving general relativity here, so I thought I'd put together some pictures to try to explain what spacetime curvature is, how I like to picture it, and how exactly it leads to gravity.

It's easiest to start with two dimensions and work our way up. Imagine a two-dimensional creature who lives in the surface of a cube. He has no concept of a third dimension, except maybe in an abstract mathematical sense. He can look up, down, left and right, but never in the direction perpendicular to the surface. And when he moves from one square to another, he can't feel his body bending. How can he find out that he lives in the surface of a cube rather than an infinite plane?

[Anonymous No.3207383]

One way would be to travel all the way around the cube. But if he lives near one of the vertices, he can do it without traveling very far at all. He can draw a triangle around the vertex with three right angles, which would be impossible if he lived on a plane.

You might object that it's not a real triangle; the lines are bent in the third dimension. But the flatlander thinks that all this talk about bending in the third dimension is a lot of hot air. There's no way for him to see or measure this third dimension or how things bend in it. He's perfectly capable of modeling the reality he can observe using only two-dimensional concepts. And if we are to understand our own curved spacetime, we should learn how he does it.

[Anonymous No.3207384]

>>3207354
well if you actually go beyond all the pop science books for laymen written by hawking, kaku & co. and 'science' documentaries by bbc and discovery explaining string theory it's a very interesting theory and i honestly don't know why people disregard it (even though they do not fully comprehend it) or should i say 'despise'?

[Anonymous No.3207385]

Here is the flatlander's conception of what his reality is like. He doesn't know or speculate about how it might be bent in the mysterious third dimension. In order to draw the diagram, he has to cut the surface in several places, which he imagines sewing back together in his mind. But the places where he makes the cuts are arbitrary (compare the upper and lower pictures, which are equivalent). The interesting points are the so-called vertices. Around each of these points, 90 degrees of angle has gone missing. He calls this missing angle the "defect."

[Anonymous No.3207388]

You probably remember the rule that the angles of a triangle add up to 180 degrees. That rule and several others can be derived from a more general and also more intuitive rule: In plane geometry, the exterior angles of any polygon always add up to 360 degrees, a full circle. A few examples are shown in the pic.

But around one of the vertices of the cube, a full circle is not 360 degrees but 270 degrees. So if you draw a polygon around one of the vertices, 270 degrees is what the exterior angles sum to. So one way to figure out the defect of a vertex is to draw a polygon around it, add the exterior angles, and subtract the result from 360 degrees.

[Anonymous No.3207389]

The defect at a vertex can be a negative number. Here's an example of a vertex at which the defect is -90 degrees.

[Anonymous No.3207390]

And this is the flatlander's conception of the above geometry. It's a bit tricky to draw because two different parts of the geometry overlap in the diagram.

[Anonymous No.3207393]

What happens if we draw a polygon around two vertices? We simply add the defects from each vertex together to get what we call the "total curvature" inside the polygon. When we subtract the total curvature from 360 degrees, we get the sum of the exterior angles.

In this diagram, the polygon contains a total curvature of 180 degrees, with 90 degrees coming from each vertex.

[Anonymous No.3207397]

To see why, it's useful to move to a more broadly applicable definition of total curvature. The flatlander draws a vector (shown in red) and moves it around the polygon. At each step, he tries to keep the vector pointing in the same direction. But when he gets back to the starting point, he will see that the vector has rotated.

In the figure, the vector starts out pointing along the edge of the polygon. At each angle, the edge of the polygon makes a 90-degree turn relative to the vector. When the vector gets back to its starting point, the polygon has turned 270 degrees with respect to the vector instead of the expected 360 degrees.

The angle by which the vector has rotated after being parallel transported around the polygon is equal to the total curvature inside the polygon.

[Anonymous No.3207401]

Using the parallel transport definition of total curvature, it's easy to see why total curvature is an additive quantity. Parallel transporting a vector around each of the two polygons is equivalent to parallel transporting it around the two polygons joined together.

[Anonymous No.3207403]

In the examples considered so far, the total curvature has been concentrated at discrete points. But in a more realistic geometry, the total curvature will be spread out in a continuous fashion. The simplest example is the surface of a sphere. We can draw a triangle on a sphere with two right angles on the equator and a third angle at the north pole. The total curvature inside this triangle is equal to the angle at the north pole, which is proportional to the area of the triangle.

As you might guess, the proportionality between the area of the shape and the total curvature inside holds for any shape drawn on the sphere. Total curvature per unit area is called "curvature." On a sphere, the curvature is constant, but in a general space, it will vary from place to place.

[Anonymous No.3207405]

The flatlander would have no trouble doing calculations about the curved space of the sphere, but he might have trouble visualizing it. But he can approximate the sphere using polygons, which he already knows how to visualize. The curvature of the sphere gets concentrated into discrete points, but the total curvature inside any large shape is about the same. The more numerous the points and the smaller the defect at each point, the better the approximation.

[Anonymous No.3207408]

Now let's apply the flatlander's approximation to three dimensions. We imagine space as being composed of 3-dimensional polyhedra with the faces attached to each other. Where two faces meet, we will notice nothing unusual. Only along the lines where three or more polyhedra meet will we notice something funny.

This is a picture of a room with one of these lines passing through it. The defect at this line is 90 degrees.

One way three-dimensional curvature is different from two-dimensional curvature is that curvature is no longer a scalar. It has an orientation. We can see this in two ways:

First, if we parallel transport a vector around this line, it will rotate 90 degrees in the plane perpendicular to the line.

[Anonymous No.3207414]

Second, imagine that we're using a large number of these lines to visualize a space with continuous curvature, as illustrated in the figure. A horizontal loop will intercept several of these lines, so it will have some total curvature inside it. But a vertical loop will not intercept any of the lines, so the total curvature inside it will be zero.

The type of quantity needed to describe curvature in general is not a scalar or a vector, but a fourth-rank tensor. I won't go into the mathematical details in this thread, but if you want to learn more about general relativity, a good place to start is learning about tensors.

[Anonymous No.3207417]

Here's an illustration of negative curvature. The table and chair are the same, but the table is mostly hidden in the extra 90-degree angle you can't see. In the continuous case, you would be able to see the table, but it would appear squeezed into the region where the curvature was.

[Anonymous No.3207421]

Next we add a fourth dimension, time. A point in spacetime is given by the four coordinates x, y, z, and t. For example, such a point might represent the position of a particle at a particular moment in time. Each particle has a one-dimensional path in spacetime called a "worldline" composed of all its spacetime positions.

What we want to do is visualize curved spacetime by breaking it up into flat four-dimensional polytopes. At the three-dimensional faces where two polytopes come together everything will seem normal. Spacetime curvature will be concentrated on the two-dimensional surfaces where three or more polytopes meet.

One type of two-dimensional surface in spacetime is a line which persists over time. This is no different from what we considered in the three-dimensional case, except that now the line may be moving. This kind of curvature is predicted by general relativity, but not in amounts large enough for us to notice in everyday life.

[Anonymous No.3207422]

Another type of two-dimensional surface in spacetime is a plane that exists only for an instant. In this picture, the green dot is a cross-section of such a plane. The diagram is drawn in the two directions perpendicular to the plane, one of which is time. Around the plane there is a small missing angle, represented by the white area; like in the two-dimensional flatlander's diagrams, you should imagine the boundaries of the dark region as being sewn together. The red and orange lines are the worldlines of two spheres.

[Anonymous No.3207423]

This animation is what the spacetime of the previous diagram would look like from the red sphere's perspective. Initially, the orange sphere is at rest relative to the red sphere. When the green plane flashes in and out of existence, each of the spheres will see the other start moving toward it. But neither sphere, from its own perspective, has accelerated.

An object has an acceleration from its own perspective (known technically as "proper acceleration") if and only if a net force is applied to it. There is no force on either sphere, only spacetime curvature between them.

[Anonymous No.3207427]

And this animation is from the orange sphere's perspective.

[Anonymous No.3207428]

By using several such planes, we can approximate a curved spacetime. The larger the frequency and the smaller the spacing between the planes, the better the approximation. Animation is from the red sphere's perspective.

[Anonymous No.3207431]

We can also have spacetime curvature with the opposite sign, which makes the objects on each side begin moving away from each other. This is again animated from the red sphere's perspective.

[Anonymous No.3207434]

In a locally Minkowskian coordinate system with t, x, y, and z coordinates, one of the components (the tt one) of Einstein's field equations can be written as:
<div class="math">{R^x}_{txt} + {R^y}_{tyt} + {R^z}_{tzt} = 4 \pi G (T_{tt} + T_{xx} + T_{yy} + T_{zz})</div>
The left-hand side is a sum over the three possible orientations of attractive curvature. The right hand side is <span class="math">4 \pi[/spoiler] times Newton's gravitational constant times the sum of the energy density and of the pressure as measured in all three directions. To avoid needless complication, we work units where the speed of light is 1.

This diagram is a sketch of the spacetime curvature produced by the earth. The green lines represent planes of attractive curvature, and the red lines repulsive. Inside the earth (at least in the central part where the density is greatest), the curvature is attractive in all directions. Above the earth's surface, the right hand side of the above equation is zero, so the attractive curvature from the vertical planes which extend out of the earth must be balanced by a repulsive curvature from the horizontal planes.

[Anonymous No.3207437]

This animation is from the perspective of someone either at the center of the earth or very far away from it. From the center of the earth's perspective, the apple accelerates toward it, and from the perspective of objects far away from the earth, the apple accelerates away.

[Anonymous No.3207443]

Space is curved.

True story.

[Anonymous No.3207446]

But from the apple's perspective, as shown in this animation, it isn't accelerating at all. There is no force on the apple.

The only thing in the diagram that's accelerating from its own perspective is the tree and the ground beneath it. It can't start moving toward the earth because the part of the earth below it is holding it up with a real force.

[Anonymous No.3207467]

One small question: Is this space-time curvature possible with other fundamental interactions, such as electromagnetism?

[Anonymous No.3207494]

>>3207467
Do you mean do they cause spacetime curvature, or are they spacetime curvature?

Electromagnetic energy does cause spacetime curvature, as does the energy from all the other interactions.

There are also theories in which electromagnetism and other forces are the result of spacetime curvature. To do this, they have to add extra dimensions to spacetime, generally curled in a tiny loop so we don't detect them. None of these theories has been confirmed experimentally, though.