Questions tagged [geometry]

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

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85 votes
9 answers
139k 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 $\vec{A}...
claws's user avatar
  • 7,235
55 votes
8 answers
14k views

Why do we use cross products in physics?

We can define cross products mathematically like if we take two vectors, we can find another vector with certain properties but why do we use it in physics, if we consider a hypothetical physical ...
hezizzenkins's user avatar
  • 1,414
13 votes
13 answers
6k views

Is it possible for a physical object to have an 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 ...
Nick Anderegg's user avatar
22 votes
7 answers
9k 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 ...
George Edison's user avatar
7 votes
2 answers
13k views

Analytic solution for angle of minimum deviation? [closed]

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 ...
stochastic13's user avatar
  • 3,138
13 votes
9 answers
15k 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 ...
german's user avatar
  • 131
10 votes
3 answers
4k 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 ...
26 votes
4 answers
221k 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 ...
Ben's user avatar
  • 981
169 votes
2 answers
41k 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 gives ...
user avatar
13 votes
5 answers
2k views

Geodesics: Straightest or Shortest? When and Why?

In classical General Relativity (meaning not modified) one can think of geodesics in two ways. One way is to say that a geodesic is the curve which is the straightest (in analogy with the flat case) ...
user avatar
17 votes
4 answers
7k views

How is Chasles' Theorem, that any rigid displacement can be produced by translating along a line and then rotating about the same line, true?

Chasles' Theorem in its strong form says: The most general rigid body displacement can be produced by a translation along a line (called its screw axis) followed (or preceded) by a rotation about ...
shivams's user avatar
  • 651
10 votes
4 answers
18k 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 ...
samthebrand's user avatar
1 vote
2 answers
661 views

Euler Rotations in Ordinary Space

I'm reading LittleJohn's notes on Rotations in Ordinary Space on Quantum Mechanics. Link: http://bohr.physics.berkeley.edu/classes/221/1011/notes/classrot.pdf. I'm trying the last question given in ...
Junaid Aftab's user avatar
7 votes
1 answer
7k 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 ...
user12345's user avatar
  • 2,273
8 votes
1 answer
792 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 ...
pritam's user avatar
  • 269
2 votes
4 answers
12k views

Why railroad tracks seem to converge?

I stand up and I look at two parallel railroad tracks. I find that converge away from me. Why? Can someone explain me why parallel lines seem to converge, please?
Stefan's user avatar
  • 608
97 votes
4 answers
18k views

Seeing something from only one angle means you have only seen (what?)% of its surface area at most?

Is there a logical/mathematical way to derive what the very maximum percentage of surface area you can see from one angle of any physical object? For instance, if I look at the broad side of a piece ...
BarrettNashville's user avatar
14 votes
5 answers
12k views

Why does the light side of the moon appear not to line up correctly with the evening sun?

I live at roughly $(52.4^\circ,-2.1^\circ)$. On sunny evenings I've often looked at the Moon and the Sun and noticed that the light part of the Moon does not appear to line up with the Sun. For ...
d4nt's user avatar
  • 405
7 votes
2 answers
1k views

$N$ point charges on a sphere

When charges are released on sphere, what is the shape made by charges? Two charges are on opposite points of one diameter of the sphere. Three charges make a shape of an equilateral triangle. ...
RAM's user avatar
  • 193
13 votes
2 answers
3k 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 ...
hkBattousai's user avatar
8 votes
2 answers
3k views

Rotation in Higher Dimensions

In a world of three spatial dimensions plus time, every atom rotates around a line, the axis of rotation. In a world of $N$ spatial dimensions where $N$ is greater than 3, must every atom rotate, ...
B. Ferguson's user avatar
7 votes
3 answers
1k 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 ...
thyme's user avatar
  • 1,393
6 votes
2 answers
5k 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 $r=R$...
thyme's user avatar
  • 1,393
5 votes
5 answers
779 views

The effects of Lorentz transformation on shape

Imagine a solid 3D cube. Now imagine that this cube is traveling close to the speed of light. To what degree will the spatial geometric properties of this object (or in general of any 3D object) ...
Maverick's user avatar
  • 411
4 votes
2 answers
913 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. ...
massimo's user avatar
  • 89
1 vote
2 answers
419 views

Angular velocity and banking angle

In a swing ride in an amusement park, the angle and speed of a seat in circular motion can be modelled by the banking angle equation: $$\tan \theta=\frac{r\omega^2}{g}$$ Since tan of $90^{\circ}$ or $\...
Sankalp Kapur's user avatar
1 vote
2 answers
176 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 ...
user12262's user avatar
  • 4,248
1 vote
0 answers
92 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 ...
user12262's user avatar
  • 4,248
64 votes
12 answers
16k views

Mathematically prove that a round wheel roll faster than a square wheel

Let's say I have these equal size objects (for now thinking in 2D) on a flat surface. At the center of those objects I add equal positive angular torque (just enough to make the square tire to move ...
John T's user avatar
  • 853
23 votes
9 answers
20k views

Why can the cross product of two vectors be calculated as the determinant of a matrix?

The cross product $\vec{a} \times \vec{b}$ can be written as the determinant of the matrix: $$\left| \begin{matrix} \vec{i} & \vec{j} & \vec{k} \\ a_i & a_j & a_k \\ b_i & b_j &...
agha rehan abbas's user avatar
12 votes
2 answers
27k views

Why sphere minimizes surface area for a given volume?

I was studying surface tension recently. Rain drops or bubbles of any kind which form are always of a spherical shape. This is because the liquid tries to minimize the surface area as the molecules ...
Ishita Gupta's user avatar
9 votes
4 answers
1k views

Is there a geometric interpretation of the spacetime interval?

In Euclidean space, the invariant $s^2 = x^2+ y^2+ z^2$ is equal to the length square of the position vector $r$. This is easily understood and can be represented geometrically in a graph. On the ...
Remi's user avatar
  • 93
6 votes
5 answers
9k 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?
Cactus BAMF's user avatar
5 votes
2 answers
198 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 ...
user12262's user avatar
  • 4,248
5 votes
4 answers
3k views

Why is the Pythagorean Theorem used for error calculation? [duplicate]

They say that if $A = X \times Y$, with $X$ statistically independent of $Y$, then $$\frac{\Delta{A}}{A}=\sqrt{ \left(\frac{\Delta{X}}{X}\right)^2 + \left(\frac{\Delta{Y}}{Y}\right)^2 }$$ I can't ...
Leonardo Castro's user avatar
5 votes
2 answers
8k views

Differentials in Spherical Shell - Maxwell Distribution

In explaining the Maxwell distribution of molecular speeds, my pchem textbook uses the following figure: We are basically trying to find the probability of having a particle with a speed $u$ between ...
David's user avatar
  • 283
1 vote
1 answer
34 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 ...
user12262's user avatar
  • 4,248
1 vote
1 answer
3k 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? ...
Lawerance's user avatar
  • 259
1 vote
5 answers
752 views

Why do floating water 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 ...
Simon S's user avatar
  • 885
1 vote
1 answer
1k views

Understanding Euler's rotation theorem

According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible. If two rotations are forced at the same time, a new axis of rotation will ...
Subhranil Sinha's user avatar
0 votes
2 answers
522 views

If we shift vectors any which way in a Cartesian coordinate system, isn't the meaning of the vector changed?

According to 3Blue1Brown, as long as the magnitude and direction of a given vector are the same, even if we move it around in the Cartesian coordinate system, it is still the same vector. Here, we ...
tryingtobeastoic's user avatar
28 votes
2 answers
7k views

Experiment to show that we need two eyes to determine depth

I'm trying to understand Manishearth's experiment in the answer here, To try this out, close one eye. Now hold your arms straight out, nearly stretched, but not completely stretched. Now extend your ...
tryst with freedom's user avatar
21 votes
3 answers
10k 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. –...
iamsid's user avatar
  • 313
20 votes
9 answers
16k 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 ...
Ricky's user avatar
  • 641
17 votes
2 answers
2k 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 ...
user avatar
11 votes
2 answers
4k views

Can an angle be defined as a vector?

In Classical Mechanics angular velocity, angular acceleration, torque and angular momentum can be defined as vectors with clear advantages such as the possibility to use vector product to simplify ...
sebgiles's user avatar
  • 231
8 votes
3 answers
26k 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 ...
Manisha's user avatar
  • 883
7 votes
2 answers
686 views

Is Newton-Cartan theory really equivalent to Newton's theory of gravity?

It is often said that Newton-Cartan theory is a reformulation or perhaps a generalization of Newton's theory of gravity, and it is said that (given certain conditions/assumptions) the two theories are ...
Maximal Ideal's user avatar
7 votes
9 answers
562 views

Understanding Parallelogram Law of Vector Addition?

Recently I've been adding vectors using the Parallelogram Law and the maths is trivial. However, I can't understand the underlying principals. What allows us to move a vector such that the tail meets ...
James Chadwick's user avatar
7 votes
3 answers
6k views

Can I calculate the size of a real object by just looking at the picture taken by a Camera?

Can I calculate the size of a real object by just looking at the picture taken by a Camera? (I think people do that) i dont understand how? (from physics point of view)
Paul's user avatar
  • 3,455