Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework.
Debates concerning the nature, essence and the mode of existence of space date back to antiquity; namely, to treatises like the Timaeus of Plato, or Socrates in his reflections on what the Greeks called khôra (i.e. "space"), or in the Physics of Aristotle (Book IV, Delta) in the definition of topos (i.e. place), or in the later "geometrical conception of place" as "space qua extension" in the Discourse on Place (Qawl fi al-Makan) of the 11th-century Arab polymath Alhazen. Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, particularly during the early development of classical mechanics. In Isaac Newton's view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another. In the 18th century, the philosopher and theologian George Berkeley attempted to refute the "visibility of spatial depth" in his Essay Towards a New Theory of Vision. Later, the metaphysician Immanuel Kant said that the concepts of space and time are not empirical ones derived from experiences of the outside world—they are elements of an already given systematic framework that humans possess and use to structure all experiences. Kant referred to the experience of "space" in his Critique of Pure Reason as being a subjective "pure a priori form of intuition".
In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space.
I was talking to a graduate physics student about the issue of energy conservation in an expanding universe. I paraphrased the argument against energy conservation as follows -
Suppose we have a photon in outer space that is very far from earth. The universe is expanding (by this I meant that in...
Homework Statement
A) Use Gauss's Law to derive the electric field in all space for a non-conducting sphere with volumetric charge distribution ρ=ρ0r3 and radius, R.
B) Repeat when there is a concentric spherical cavity within the non conducting sphere with radius, A.
Homework Equations...
I be grateful for any feedback on this argument:
- First assume space is continuous
- Then there is an actually infinite amount of information in a spatial volume of 10000 cubic units
- There is also an actually infinite amount of information in a spatial volume of 1 cubic unit
- But this is a...
Homework Statement
I am trying to learn how to create state space representations. I am using this link to study.
http://www2.ensc.sfu.ca/people/faculty/saif/ctm/examples/pend/invpen.html
Could someone explain how to get the matrix A from my equations?
Homework EquationsThe Attempt at a...
TL;DR at bottomIt's a somewhat accepted convention that in realistic scifi space fighters should be impossible to use effectively. In general they are regarded as being easy targets that, at interstellar ranges would be unable to survive in a battlefield that employs current plausible scifi...
Hello,
I've a fundamental question that seems to keep myself confused about the mathematics of quantum mechanics. For simplicity sake I'll approach this in the discrete fashion. Consider the countable set of functions of Hilbert space, labeled by i\in \mathbb{N} . This set \left...
nmh{796}
$\textsf{Suppose $Y_1$ and $Y_2$ form a basis for a 2-dimensional vector space $V$ .}\\$
$\textsf{Show that the vectors $Y_1+Y_2$ and $Y_1−Y_2$ are also a basis for $V$.}$
$$Y_1=\begin{bmatrix}a\\b\end{bmatrix}
\textit{ and }Y_2=\begin{bmatrix}c\\d\end{bmatrix}$$
$\textit{ then }$...
At an L1 LaGrangian point between two bodies, one could - materials science notwithstanding - pit two of Newton's Laws (LM3,UG) against each other to provide thruster-free stationkeeping.
Would it be feasible to use that to launch free from the system ? either spit out like a watermelon seed...
I've been trying to wrap my head around the geometry of the expansion of space, from Science Channel shows I vaguely understand the "every point in space is moving away from every other point in space" and iirc this was uniformly so. Is that correct? If not ignore the rest of this post I suppose...
In the 19th century Lord Kelvin made the first numerical calculation of the age of the Earth not based on the Bible.From his initial guess that the Earth started as a molten rock and that today the temperature of the interior increases at a certain rate as you approach the center, he got an age...
To explain the concept of curved space time, we often use analogy of rubber sheet. If we put a heavy ball at the centre of sheet then it creates a depression and now a smaller ball will fall towards that heavy ball because of depression. But in this analogy smaller ball is falling down the slope...
Hello,
I am taking a quantum mechanics course using the Griffiths textbook and encountering some confusion on the definition of inner products on eigenfunctions of hermitian operators. In chapter 3 the definition of inner products is explained as follows: $$ \langle f(x)| g(x) \rangle = \int...
In texts treating Hilbert spaces, it's usually given as an example that "any finite dimensional unitary space is complete", but I've found no proof so far and failed prove it myself.
I've plotted out the trajectory of an imaginary electron in 3D; next I represent it's points with the matrix A(x1 y1 z1) "throughout it's orbit":
( -1/2 -1 1
( -2 -1.5 2
(-1/2 2 3...
Homework Statement
"Do there exist any event spaces with just six elements?"
Homework EquationsThe Attempt at a Solution
Suppose ##F_1## is an event space with a non-trivial event ##A##. Then ##F_1=\{∅,A,A^c,Ω\}##. So ##inf(|F_1|) = 4##, since if you remove any of these events, ##F_1## is no...
Homework Statement
If ##\lim_{n \rightarrow \infty} x_n = L## then ##\lim_{n\rightarrow\infty}cx_n = cL## where ##x_n## is a sequence in ##\mathbb{C}## and ##L, c \epsilon \mathbb{C}##.
Homework Equations
##\lim_{n\rightarrow\infty} cx_n = cL## iff for all ##\varepsilon > 0##, there exists...
Lagrangian Mechanics uses generalized coordinates and generalized velocities in configuration space.
Hamiltonian Mechanics uses coordinates and corresponding momenta in phase space.
Could anyone please explain the difference between configuration space and phase space.
Thank you in advance for...
Can string theory be made without time equations?
According to Carlo Rovelli in his latest book "The Order of Time"
https://www.amazon.com/dp/073521610X/?tag=pfamazon01-20
:
"The equations of loop quantum gravity on which I work are a modern version of the theory of Wheeler and DeWitt. There...
Hey! :o
For a field $K$ and $1<n\in \mathbb{N}$ let $A\in K^{(n-1)\times n}$ aa matrix with rank $n-1$. For a row vector $z\in K^{1\times n}$ let $\left (\frac{A}{z}\right )\in K^{n\times n}$ be the matrix that we get if we add as the $n$-th row of the matrix $A$ the vector $z$.
To show that...
The permittivity of free space, ε0, is usually given without any derivation or historical context as to how it was experimentally determined.
Could you explain to me how the value of ε0 was first determined experimentally or provide a resource that gives such a derivation?
Thanks!
Hi everyone, I need a little help understanding how periodic reciprocal space applies to the Debye model for solids. Many thanks in advance!
If we start with the general derivation of a dispersion relation for a 1D system, with atoms coupled by springs, one gets the following relation
$$\omega...
Hello! I am a bit confused about the sign in space and time translation operators acting on a state. I found it with both plus and minus sign and I am not sure which one to use when. The equations I am talking about are: $$U(t)=e^{\pm iHt/\hbar}$$ and $$T(x)=e^{\pm ixp/\hbar}$$. Is it a plus or...
So, let me preface by saying I’m neither a scientist nor a mathematician, so am requesting some talented help here checking the accuracy of my source information and math.
Regarding star formation, I got curious about how much volume of space in the interstellar medium is actually required to...
Homework Statement
"If ##A_1,...,A_m\in O## and ##k\in ℕ##, show that the set of points in ##Ω## (the sample space) which belong to exactly ##k## of the ##A_i## belongs to ##O## (the previous exercise is the case when ##m=2## and ##k=1##)."
Homework Equations
Event space: O
##O\neq ∅##...
1) space is expanding at an increasing rate, therefore things are getting farther from each other and therefore increasing in velocity. 2) the faster an object moves relative to another, the more mass it has 3) supermassive objects can turn into neutron stars, black holes, etc. Therefore, will...
Homework Statement
I am asked to write an expression for the length of a vector V in terms of its dot product in an arbitrary system in Euclidean space.
Homework EquationsThe Attempt at a Solution
The dot product of a vector a with itself can be given by I a I2. Does that expression only apply...
When we see with open eyes or in visible range of wavelength of e.m. wave we see black space containing some star...but if we see the space in other range of wavelength of the e.m. wave what will we see?? Also why the space is black?? Which matel we find in space mostly? Can we extract those...
In the field of medical physics, specifically in monte carlo simulation of radiation beams produced by electron accelerators, people call ‘phase space’ to a file that contains the data of a large number of particles when they traverse a reference surface in the machine (usually a plane), i.e...
Homework Statement
1.1.3
1) Do functions that vanish at the endpoints x=0 and L=0 form a vector space?
2) How about periodic functions? obeying f(0)=f(L) ?
3) How about functions that obey f(0)=4 ?
If the functions do not qualify, list what go wrong.Homework Equations
The Attempt at a...
Homework Statement
Homework Equations
definition of null vector,
[/B]
The Attempt at a Solution
null vector : ## |0 \rangle = (0,0,0) ##
inverse of (a,b,c) = ( - a, -b, -c)
vector sum of the two vectors of the same form e.g. (c,d,1) + ( e,f,1) = ( c+e, d+f, 2) does not have the same...
For the vector valud function F in the image, the three components of the output vector at a point are functions of (x,y,z)the three coordinates of the point.But while calculating divergence, why is the rate of change of x component of the output along x direction alone is accounted(similarly...
To specify, by fuel I mean fossil, nuclear, chemical and pressurized air. Want to build and design a satellite for my daughter with a camera in it (goPro) with an antenna to be able to view anytime she likes. Want to do a hexagonal approach with solar cells only for power. Want to power the...
I am looking at a couple of very interesting papers, published in MNRAS, that deduce, that the accelerated expansion of the Universe we observe can be attributed to gravitational waves, produced by a very distant merger of two or more universe-mass-scale black holes. The last one is on the...
Homework Statement
The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong.
Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...
I was looking at NASA saying they might not have enough funds to put humans on Mars by 2040. OK, zoom forward maybe 200 years and the cost might be, maybe, just buy a ticket. Or people might already be working there; they will be paid to go there. So, here's the question. What is the least cost...
Hello! I am a returning student, 19YO, and will be starting back in community college in a month retaking classes I failed the first go-around with college. This is hopefully less of a "tell me what to do" thread, and more of a "clarify my misconceptions."
Basically, I am very interested in a...
as we know light has momentum so theoretically we can use it but is it practical?
(also this is it that light only exerts force if incident on something?)
If I consider a tetrahedron of four densely packed spheres of unit radius, what it the radius of the largest sized sphere that can fit in the space in between?
How do you show that there can be only one tangent space at a given point of a manifold? Geometrically it's pretty obvious in 3 dimensions, as one notices that there can be only one tangent plane at a point. But how could we show that using equations?
It seems to me that the concept of a space elevator does not take Coriolis force into account. If the elevator were in built with a space station in geosynchronous orbit and counterweight then there is more to reaching the space station than just climbing the rope. The rope would have to be...
I hear that deepest void of intergalactic space may contain say one particle per cubic cm. I don't want to quibble the amount but let's take that as close enough for my purposes.
Now is this figure a statistical average so that if it were correct that each cubic kilometre of deep space would...
Here is a rather interesting NY Times obituary of Constance Adams, space habitat architect.
I never heard of her, she died young (53), but sounds quite interesting and did some fun design stuff.
What is the largest possible Rotating wheel space station possible to be constructed with current materials? and what would be the population it would support. also formulas used for calculation.would be useful.
Could constructing cylindrical space elevators support more population,
Could someone tell me in what sense the following photo of Hilbert is a infinite dimensional Hilbert Space?
It's shown in a pdf I'm reading.
Perhaps I'm putting the chariot in front of the horses as one would say here in our country, by considering infinite as infinite dimensional?
Homework Statement
From Classical Mechanics, Gregory, in the chapter on Hamilton's equations of motion:
14.13: Decide if the energy surfaces in phase space are bounded for the following cases:
i.) The two-body gravitation problem with E<0
ii.) The two-body gravitation problem viewed from the...
So, I'm investigating a certain way of steering a rocket in space for my first undergraduate research project. Essentially, the idea is to control the location of some mass located on a horizontal track perpendicular to a rocket, so that when the mass is moved, the center of mass of the rocket...
I am reading Feynman's book on QED and something struck me about light. I know that we can only calculate the probability of where a photon goes. After that I came across how a partial reflection affects light. My question is, is there a place in the universe where there is a great thickness of...
Wolfram says that an example of an inner product space is the vector space of real functions whose domain is an closed interval [a,b] with inner product ##\langle f, g\rangle = \int_a^b f(x) g(x) dx##. But ##1/x## is a real function, and ##\langle 1/x, 1/x\rangle## does not converge... So how is...