To be used for questions on geometry closely pertaining to physics. Includes differential geometry and euclidean geometry.

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26
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6answers
64k views

What is the physical significance of dot & cross product of vectors? Why is division not defined for vectors?

I get the physical significance of vector addition & subtraction. But I don't understand what do dot & cross products mean? More specifically, why is it that dot product of vectors ...
17
votes
7answers
5k views

Experimental evidence of a fourth spatial dimension?

As human beings, we observe the world in which we live in three dimensions. However, it is certainly theoretically possible that more dimensions exist. Is there any direct or indirect evidence ...
15
votes
4answers
91k views

Does the rotation of the earth dramatically affect airplane flight time?

Say I'm flying from Sydney, to Los Angeles (S2LA), back to Sydney (LA2S). During S2LA, travelling with the rotation of the earth, would the flight time be longer than LA2S on account of Los Angeles ...
7
votes
11answers
2k views

Is it possible for a physical object to have a irrational length?

Suppose I have a caliper that is infinitely precise. Also suppose that this caliper returns not a number, but rather whether the precise length is rational or irrational. If I were to use this ...
7
votes
4answers
6k views

How far into space does one have to travel to see the entire sphere of earth?

Virgin Galactic will take passengers aboard SpaceShipTwo as high as 65 miles above the surface of the earth. But from this altitude, passengers will only be able to see a certain segment of the ...
5
votes
2answers
624 views

Textbook on the Geometry of Special Relativity

I am looking for a textbook that treats the subject of Special Relativity from a geometric point of view, i.e. a textbook that introduces the theory right from the start in terms of 4-vectors and ...
2
votes
1answer
283 views

Why is physical space equivalent to $\mathbb{R}^3$?

Why is physical space equivalent to $\mathbb{R}^3$, as opposed to e.g. $\mathbb{Q}^3$? I am trying to understand what would be the logical reasons behind our assumption that our physical space is ...
2
votes
0answers
76 views

If two ends were a certain “length” apart were they therefore at rest (or at least rigid) to each other? [closed]

Considering the definition of the SI unit of "length" [1] and [2 (" method a.")] I'm missing any requirements about the two "ends" of the required "path travelled by light" being "at rest to each ...
5
votes
2answers
183 views

What is the notion of a spatial angle in general relativity?

Is there a notion of spatial angles in general relativity? Example: The world line of a photon is given by $x^{\mu}(\lambda)$. Suppose it flies into my lab where I have a mirror. I align the mirror ...
1
vote
2answers
134 views

Are signal fronts in a beam not at rest to each other?

I'd like to investigate how the notion of "mutual rest" might be applied consistently, but distinctively, in the following thought experiment: Consider a light source ("$A$") which directs a beam ...
7
votes
6answers
3k views

Gravitation is not force?

Einstein said that gravity can be looked at as curvature in space- time and not as a force that is acting between bodies. (Actually what Einstein said was that gravity was curvature in space-time and ...
3
votes
2answers
441 views

Shape of electric charges on sphere in equilibrium state

When electric charges of equal magnitude and sign are released on a regular sphere (and assume that they stick to the surface of the sphere, but they are free to move along its surface), what is the ...
5
votes
5answers
2k views

Do perfect spheres exist in nature?

Often in physics, Objects are approximated as spherical. However do any perfectly spherical objects actually exist in nature?
4
votes
2answers
322 views

Can Minkowski spacetime be redefined as a non-flat riemannian manifold?

Minkowski space time is defined in terms of a flat pseudo-Riemannian manifold. I have wondered if it can be redefined as Riamannian manifold and in the case what type of curvature would there appear. ...
5
votes
2answers
134 views

Which causal structures are absent from any “nice” patch of Minkowski space?

Which "causal separation structures" (or "interval structures") can not be found among the events in "any nice patch ($P$) of Minkowski space"?, where "causal separation structure" ($s$) should be ...
1
vote
1answer
677 views

Geometry in diagonal matrix and inertia tensor

For this problem, can anyone explain to me why when $x_1$ axis is aligned with the diagonal of the cube, the resulting inertia tensor will become diagonal? How to interpret this result geometrically? ...
1
vote
1answer
701 views

How to calculate spatial distance in space-time?

Pinning two test particles at two different points in space, how can I calculate their spatial distance, when the geometry is given by the Schwarzschild metric? Let's say particle 1 is pinned at ...
19
votes
3answers
7k views

Why can't a piece of paper (of non-zero thickness) be folded more than $N$ times?

Updated: In order to fold anything in half, it must be $\pi$ times longer than its thickness, and that depending on how something is folded, the amount its length decreases with each fold differs. ...
5
votes
1answer
1k views

How is the equation of motion on an ellipse derived?

I would like to show that a particle orbiting another will follow the trajectory \begin{equation} r = \frac{a(1-e^2)}{1 + e \cos(\theta)}. \end{equation} I would like to do this with minimal ...
1
vote
4answers
358 views

Why drops form spheres?

Consider a drop of water floating in an inertial frame in STP air (e.g., the ISS). Intuitively, the equilibrium shape of the drop is a sphere. How would one prove that? Is it equivalent to showing ...
3
votes
1answer
277 views

Can a very small portion of an ellipse be a parabola?

We consider that when a body is projected from any height from the earth surface with a speed lesser than the orbital speed ( tangentially to the earth surface at that point.) it follows an elliptical ...
7
votes
2answers
658 views

Is Dyson Sphere a stable construction?

Suppose that a star is encompassed by a Dyson Sphere. Do we need a position control system for the Dyson Sphere to keep its origin always aligned with the center of the star? Will it stay aligned ...
4
votes
0answers
150 views

Can space move? [closed]

I have some questions; I hope you don't mind: $\bullet$ If the space between two distance galaxies is increasing, then is the volumes of space in which the galaxies find themselves also moving apart? ...
3
votes
1answer
6k views

Analytic solution for angle of minimum deviation?

Consider a simple prism with a prism angle $A$, angle of incidence $\theta_1$, angle of emergence $\theta_4$ and the first and second angle of refraction as $\theta_2,\theta_3$. the refractive index ...
2
votes
2answers
350 views

How far would I have to go to see a fully rounded Earth? [duplicate]

How far would I have to go to see a fully rounded Earth? Recently, I saw a video on Youtube in which a sky diver called Felix Baumgartner ascends to $120,000$ feet (= $39$ miles) in a stratospheric ...
1
vote
1answer
16 views

How to express in terms of spacetime intervals whether two participants in a flat region were at rest to each other

Given a flat region of spacetime as set $\mathcal S$ of events together with values of spacetime intervals (up to a common non-zero constant) for each pair of events, $s^2 : \mathcal S \times \mathcal ...
1
vote
1answer
202 views

Rømer's determination of the speed of light

I am trying to understand Rømer's determination of the speed of light ($c$). The geometry of the situation is shown in the image below. The determination involves measuring apparent fluctuations in ...
1
vote
2answers
1k views

What is the physical meaning of a dot product and a cross product of vectors? [duplicate]

My teacher told me that Vectors are quantities that behave like Displacements. Seen this way, the triangle law of vector addition simply means that to reach point C from point A, going from A to B ...
0
votes
1answer
40 views

New direction vector after collision of spheres [closed]

I have a volume in 3-space in which random spheres are spawned in motion. They have the following attributes to them: position known (in three axes) a direction vector (in three axes) a scalar speed ...
-3
votes
1answer
55 views

3D donuts and the centre of a black hole [closed]

I have been trying to imagine ways to visualise the gravitational field of the earth... It's near zero at the outer edges of the atmosphere, strong at sea level, strongest at the quarter point, then ...
121
votes
2answers
8k views

Why do sunbeams diverge even though the sun is much more than a few kilometers away?

Consider this picture of sun beams streaming onto the valley through the clouds. Given that the valley is only (at a guess) 3km wide, with simple trigonometry and the angles of the beams, this ...
16
votes
9answers
4k views

Why do objects appear smaller when viewed from a distance? [duplicate]

Yes, I know all about perspective (I'm an artist). I even have some basic knowledge of descriptive geometry. I know how it works. My question is more about why it works. I have a sneaking suspicion ...
15
votes
2answers
589 views

Generalized Complex Geometry and Theoretical Physics

I have been wondering about some of the different uses of Generalized Complex Geometry (GCG) in Physics. Without going into mathematical detail (see Gualtieri's thesis for reference), a Generalized ...
11
votes
2answers
860 views

The Reeh-Schlieder theorem and quantum geometry

There have been some very nice discussions recently centered around the question of whether gravity and the geometry and topology of the classical world we see about us, could be phenomena which ...
-2
votes
3answers
2k views

Should linear algebra and vector calculus from traditional courses be replaced with `geometric algebra`? [closed]

geometric algebra gives geometric meaning to linear algebra and much more. it can provide a coordinate free geometric interpretation of spaces. those who learn of ...
9
votes
1answer
237 views

Cutting a circle and moving endpoints

A metal (or otherwise, suitably elastic) circle is cut and the points are slid up and down a vertical axis as shown: How would one describe the resultant curves mathematically?
3
votes
0answers
392 views

Geodesic distance in de Sitter space

Consider $N$ dimensional de Sitter space embedded in $N+1$ dimensional Minkowski space: $$\eta_{\mu\nu}X^\mu X^\nu=1, \hspace{1cm}\eta_{\mu\nu}=\text{diag}(-1,1,\dots,1)$$ where I set $H=1$ for ...
5
votes
4answers
743 views

How did “no prior geometry” father 50 years of confusion?

I've come across this quote attributed to Misner, Thorne & Wheeler from their book, Gravitation: Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior ...
3
votes
2answers
232 views

What is the connection between geometry of physical space and Hilbert space?

In Quantum Mechanis (QM), the dynamical variables are the (quantized) coordinates $x_j$ and their canonical conjugate $p_j = -i\partial_j$ with the commutation relation $[x_j,p_k]=i\delta_{jk}$ ...
3
votes
4answers
435 views

Formulation of general relativity

EDIT: I think I can pinpoint my confusion a bit better. Here comes my updated question (I'm not sure what the standard way of doing things is - please let me know if I should delete the old version). ...
9
votes
3answers
517 views

What is the exact meaning of homogeneity in cosmology?

I understand that, in general, homogeneity is the physical attribute of being uniform in composition (" of the same form at every point"), but I'm slightly confused when it is used in cosmology as ...
7
votes
1answer
5k views

How to calculate roll, yaw and pitch angles from 3D co-ordinates (Euler Angles)

I have digitized a video of a flying fly in a 3-dimensional space. At all instants I know the x, y, and z co-oridinates of the following points on the fly's body --- The points are my choice, and ...
3
votes
2answers
7k views

Why do far away objects appear to move slowly in comparison to nearby objects?

When we are in a moving train, nearby stationary objects appear to go backwards. In Physics, relative velocity can be employed to explain the phenomenon: velocity of object w.r.t train = velocity ...
2
votes
0answers
65 views

Is there a theory that physically explains the origin equivalence principle / explains gravity via acceleration?

I've been thinking about how gravity could arise from a 4th dimensional spinning cylinder with space-time that has pliability like rubber (which I think is a generally accepted analogy). The ...
2
votes
0answers
43 views

Can a value of “length, in meters” be attributed to a pair of ends which are rigid (but not at rest) to each other? [duplicate]

The definition of the SI base unit "metre" [1] doesn't seem to rule out explicitly that a certain value of "length, in meters" could be attributed to a pair of ends which are rigid to each other, but ...
0
votes
3answers
216 views

What is the meaning of “a straight line”? [duplicate]

The book "The Dancing Wu Li Masters" page 189 talking about General Relativity says "A geodesic is not always a straight line". Is that true? What is a definition of "straight line" that makes sense ...
0
votes
1answer
134 views

Do lower dimensional objects actually exist? [duplicate]

Take for instance a arbitrarily chosen 2-dimensional rectangle, having a length of $x$, a width of $y$ and height of 0. (Or take for instance a 1-dimensional line. Or take for instance a 0-dimensional ...
6
votes
1answer
517 views

Why are fractal geometries useful for compact antenna design?

While most of what I've read about fractals has been dubious in nature, over the years, I keep hearing that these sorts of self-similar (or approximately self-similar) geometries are useful in the ...
4
votes
1answer
155 views

Flat space metrics

This question concerns the metric of a flat space: $$ds^2=dr^2+cr^2\,\,d\theta^2$$ where $c$ is a constant. Why is it necessary to set $c=1$ to avoid singularities and to restrict $r\ge 0$? Thanks.
3
votes
1answer
1k views

Nanotube chiral angle as a function of $n$ and $m$

I'm looking into nanotubes and I thought I'd assure myself that the basic geometry equations are indeed correct. No problems for the radius, I quickly found the known formula: $$R = ...