Space Definition and 1000 Threads

As in title: Plugging in the definition is straight forward, I am too lazy to type, I will just quote the book Fetter 1971: Up to here everything is very straight forward, in particular, since we are working on free electron gas, ##E=\hbar \omega## However, I have no idea how to arrive...

(Not sure about the prefix. Choose intermediate to be sure.) This article Physicists Recreate Classic 'Double-Slit' Experiment Using Time Instead of Space has it's basis in this paper: Double-slit time diffraction at optical frequencies It certainly sounds intriguing but is there actually...

Many explanations out there for gravity. One that I saw a few months ago explained that our gravity hete on earth is space moving toward the center of the earth, and we should be falling to the center of earth at a rate of 9.8 m/ s/s. But the surface of the earth is in the way... What happens to...

Hello, I am solid on the following concepts but less certain on the correct understanding of what a random variable is... Random Experiment: an experiment that has an uncertain outcome. Trials: how many times we sequentially repeat a random experiment. Sample space ##S##: the set of ALL...

In the book "Group theory and it's Applications to the Quantum Mechanics of atomic spectra " by Eugene P. Wigner in chapter 4 The elements of quantum mechanics it is written What does the wave-packet and the refractive index implies here.How to interpret this?

Dear Everybody, I am having trouble with last part of this question. I believe the answer is no. But I have to proof the general case. Here is my work for the problem: Suppose that we have two distinct norms on the same vector space ##X## over complex numbers. Then there exists no ##K## in...

Dear everybody, I am having some trouble proving the implication (or the forward direction.) Here is my work: Suppose that we have an arbitrary linear functional ##l## on a Banach Space ##B## is continuous. Since ##l## is continuous linear functional on B, in other words, we want show that...

Hello :), I am wondering of the right and direct method to calculate the following tangent spaces at ##1##: ##T_ISL_n(R)##, ##T_IU(n)## and ##T_ISU(n)##. Definitions I know: Given a smooth curve ##γ : (− ,) → R^n## with ##γ(0) = x##, a tangent vector ##˙γ(0)## is a vector with components...

How long does it take water ice H20 in space in our solar system to sublimate, say a basic ice cube? It starts as a solid cube at the temperature of whatever space is above Earth and then completely turns to vapor. Just looking for ballpark situation here. Does anyone know of a table or place...

TL;DR Summary: Solar sytem forces on Unity Hello ! For my last year in my school, I've got a project to do, and I wanted to recreate the Solar system with forces on Unity. My forces are Velocity and Acceleration (I'm using the Frenet's formulas). I'm sorry I'm not a physicist and that's why...

I understand people collapse multi dimensional functions to make simpler visualisations, eg if you have a 500 dimension objective function in machine learning you can collapse it to 2D or 3D to get a visual idea of the objective-space.is this why Einstein did it as well? to make simpler...

Homework Statement:: Model for Gravity Relevant Equations:: Rg -Rg = G/(8pi*c^4)T The rubber table model for gravity can't quite translate to reality. One sees that a ball placed on the table distorts it, but this is due to it being in a uniform gravity field. Just what mechanism distorts...

Let ##u## be a generator of ##H^1(\mathbb{R} P^\infty; \mathbb{F}_2)##. Prove the relations $$\text{Sq}^i(u^n) =\binom{n}{i} u^{n+i}$$

I the lambda-CDM model, is the expansion of spacetime uniform around all of spacetime, is there a smooth transition between expanding parts of spacetime (the voids) and non-expanding parts of spacetime, or is there a sharp distinction between expanding and non-expanding parts of spacetime. Is...

I am following [this YouTube lecture by Schuller][1] where he finds the appropriate formalism for the quantum mechanics in the physical curved space. Everything makes sense to me but at the very end I see that we find the pull backed connection one-form on the base manifold. He says to the end...

Can someone please explain to me how can we obtain this integral in eq. 5.27 from eq. 5.26? I quite do not understand how is it possible to make this adjustment and why the (p_(f))^2 appeared there in the numerator and also why a solid angle appeared there suddenly.

First of all, all the physical quantities presented in this topic are unknown variables, and I require a functional relationship between these unknown variables. In a vast space that does not consider gravity , there are many ideal rigid balls moving freely. And in equilibrium. The ball is...

Can we say, (y + z ) x1 = (y1 + z1) x is also an equation of a straight line in 3 dimensional space, where (x1,y1,z1) and (x,y,z) are the coordinates of a given point and a variable point respectively on a 3D line that passes through the origin, have seen equation of a straight line in 3...

Since space is expanding, and that expansion moves things, then those moved things must be moved by something. If moved by space itself, then space must be providing the energy. And if gravity is a function of space and time interacting, then what is the energy or mass of a volume of space...

Hi, I am currently preparing for my exam and have just watched a video about motion in phase space. From minute 4 a quadratic potential is introduced and then from minute 6 minute the phase trajectory. Here are the pictures quadratic potential phase trajectory Regarding phase...

First I found work: W=(3.85x10^5)(2.45x10^8) W= 9.43x10^13 Then used that for difference of kinetic energy: 9.43x10^13 = (1/2) (4.55x10^4)v2^2 - (1/2)(4.55x10^4)(1.22x10^4)^2 9.43x10^13 = (22750)v2^2 - 3.386x10^12 9.43x10^13 + 3.386x10^12 = (22750)v2^2 9.77x10^13 = 22750v2^2 9.77x10^13/22750...

Trying to make sense of small and large extra dimension(s) of phyiscal space in a simple intuitive example. Consider a two dimensional manifold like R2 and we are trying to add a small and a large extra dimension. Do we mean by smale extra dimension in this case something like (0,1)×R (the...

I'm reading the Feynman lectures chapter on "Curved Space", section 6-2. Say we're trying to figure out a way to measure average curvature on Earth. We know that 3d space is curved if Euclidean geometry rules don't work - e.g. the ratio of circumference and radius of a circle isn't ##2\pi##, or...

I was reading a recent physics article on the google home page that stated that the current theory of mass is that it is a photon moving at light speed but stationary in space. My analogy of this was like a photon moving in a circle but it is not moving. I wish I could sight the article but...

Just FYI to anyone who is interested or has a friend or relative who might be interested, Kerbal Space Program is free on the Epic Games store right now through January 12th at 10AM (don't know the time zone). Kerbal Space Program is THE game if you are interested in rocketry, realistic space...

This is probably a dumb question. I'm not a physicist and took basic physics a very long time ago. If an object was in deep space, a long way away from gravitational fields and was subjected to a constant 1g acceleration in a straight line what prevents it from eventually exceeding light speed...

It looks to me like no more than a huge ego trip for rich people. How much of the money spent will contribute to our knowledge of Science or Space. One image from JWST probably has more worth than the whole of this fun project. They might just as well send their money to Yemen, Ukraine or Haiti...

Relatively new area to me; will solve one -at- time as i enjoy the weekend with coffee. 1. Unit tangent ##r=xi+yj+zk## ##r=(t-\dfrac{t^3}{3})i+t^2j+(t+\dfrac{t^3}{3})k## ##T=\dfrac{dr}{dt} ⋅\dfrac{dt}{ds}## ##\dfrac{dr}{dt}=(1-t^2)i+2tj+(1+t^2)k##...

Hi, I'm having trouble understanding that for time-like events where the order of time is absolute, these events can be in different spatial orders depending on reference frame. Can someone provide an example please? Thank you!

Quote from NASA: My understanding of dark energy is based on NASA's report: https://science.nasa.gov/astrophysics/focus-areas/what-is-dark-energy; were NASA state as follows: "It turns out that roughly 68% of the universe is dark energy. Dark matter makes up about 27%. The rest - everything on...

Is it possible to have a universe with only space and time but no mass ? I ask this question because a friend told me that time is an illusion. In fact, time does not exist. Because of the existence of matter, time can be felt through the movement of matter. If matter does not exist, time does...

Hello, I'm wondering about the practicality and feasibility of building large power tower structures in space. I could imagine something like follows: Take a long structural beam which is positioned in alignment to the sun rays. At the end closest to the sun a receiver is placed. Consisting...

Hello everyone, Concerning the separation axioms in topology. Our topology professor introduced the equivalent definition for a topological space to be a ##T_{o}-space## as: $$ (X,\tau)\ is\ a\ T_{o}-space\ iff\ \forall\ x\ \in X,\ \{x\}^{\prime}\ is\ a\ union\ of\ closed\ sets. $$ The direction...

Hi, a question regarding something I could not really understand The question is: Let V be a space with Norm $||*||$ Prove if $v_n$ converges to vector $v$. and if $v_n$ converges to vector $w$ so $v=w$ and show it by defintion. The question is simple, the thing I dont understand, what...

Hi, mathematically in the F = GMm/r^2 equation r can be very close to infinity (or the size of the universe), but gravitational force always will be some number. But how is that in the real world? Let's say we have a perfectly empty universe but only with two sun-like stars. If they are away...

https://www.cnn.com/2022/11/10/world/shuttle-challenger-nasa-discovered-documentary-crew-scn

I’m in need of recommendation of a general science book (‘general’ means just a bit of introduction and its application, not going into its detailed theoretical and technical workings) which contains the following topics (though not exhaustive) Space Technology: the basic concept of launching...

Suppose M is a manifold and $$T_{p}M$$ is the tangent space at a point $$p \in M$$. How do i prove that it is indeed a vector space using the axioms: Suppose that u,v, w $$\in V$$. where u,v, w are vectors and $$\V$$ is a vector space $$u + v \in V \tag{Closure under addition}$$ $$u + v = v +...

I was reading an article by Edward Harrison, which tackles the problems of conservation of energy at cosmological scales. At some part (point 2.4) he cites several article, including one by Rees and Gott, which he says indicates that the internal energy of a comoving volume (e.g. a cosmic...

Suppose you have the map $$\pi : \mathbb{R}^{n+1}-\{0\} \longrightarrow \mathbb{P}^n$$. I need to prove that the map is differentiable. But this map is a chart of $$\mathbb{P}^n$$ so by definition is differentiable? MENTOR NOTE: fixed Latex mistakes double $ signs and backslashes needed for math

Hello everyone, I am struggling to get insight into a certain set in 4D space. Given is a closed path in 4D-space with constant Euclidean norm $$\vec{\gamma} (\theta):[0,2\pi]\to\mathbb{R}^4, \ \ \vec{\gamma}(0)=\vec{\gamma}(2\pi), \ \ ||\vec{\gamma}(\theta)||_2 = \mathrm{const.}$$ I am looking...

Prove that every surjective local homeomorphism ##\pi : \tilde{X} \to X## from a compact Hausdorff space ##\tilde{X}## to a Hausdorff space ##X## is a covering space.

In special relativity we've the invariant ##\begin{aligned} d s^2=&-d t^2 \\ &+d x^2 \\+d y^2+d z \end{aligned}##. For a clock moving along a worldline the above equation reduces to ##\begin{aligned} d s^2=&-d t^2\end{aligned}## , hence we can say that the time measured by the clock moving...

I was thinking the other day about green houses and how they would act in space or on another planet without an atmosphere. I know that green houses work on Earth by stopping convection but could they theoretically trap heat in a non atmospheric environment? I am imagining a material that allows...

so from Fourier transform we know that Ψ(r)=1/2πℏ∫φ(p)exp(ipr/ℏ)dp I proved that <p>= ∫φ(p)*pφ(p)dp from <p>=∫Ψ(r)*pΨ(r)dr so will the same hold any operator??

Are there any kind of observed and experimentally verified processes or mechanisms where photon emission occurs and which are directly cause by spacetime expansion in some way?

Can the magnetic and electric potentials (A and φ) be fixed to zero, or at least some constant value in a region of space? Naively, I'd think something like this might work (a hollow conducting sphere connected to a voltage source connected to ground, would the potentials inside the sphere be...

I'm looking to check my understanding of the information below and ultimately get a better understanding of it. Is spectral decomposition a mathematical procedure? Does "the state space of the measured system" refer to the possible values that the system could take, when measured?

Interested in Banks' HST these days, for discussion,here are some abstract and an email from L.Smolin. A recent conclusive paper as well https://arxiv.org/abs/2201.06988 The formalism of Holographic Space-time (HST) is a translation of the principles of Lorentzian geometry into the language...

In relativity, no signal travels faster than light, and hence if something happened away from me, I will only know about it after some time. This means I cannot measure instantly the position and time of something as it happens; this would violate special relativity. I however imagine that I...