Space Definition and 1000 Threads

Homework Statement Find the complement C' of the set C with respect to the space U if: 1. U = {(x,y,z) : x^2+y^2+z^2 ≤1}, C = {(x,y,z) : x^2+y^2+z^2 = 1} 2. U = {(x,y) : |x| + |y| ≤ 2}, C = {(x,y) : x^2 + y^2 < 2} Homework Equations Definition of complement: The elements of the space that...

Heisenberg's uncertainty relation says: $$\Delta x \Delta p \ge \hbar$$ If we assume a massless quantum object then we have the relationship ##\Delta E = c\Delta p## so that the above uncertainty relationship becomes $$\Delta E \ge \frac{\hbar c}{\Delta x}.\tag{1}$$ I understand that if we have...

It is known that vectors change them sing under the influence of parity when ##(x,z,y)## change into ##(-x,-z,-y)## $$P: y_{i} \rightarrow -y_{i}$$ where ##i=1,2,3## But what about vectors in Minkowski space? Is it true that $$P: y_{\mu} \rightarrow -y_{\mu}$$ where ##\mu=0,1,2,3##. If yes how...

Hi, I have the impression that the special thing about Hilbert space for Quantum Mechanics is that it is simply an infinite space, which allows for infinitively integration and derivation of its elements, f(x), g(x), their linear combination, or any other complex function, given that the main...

My query is really to find out what everybody thinks about 'empty space'. Does dark matter/energy exist and if so, it fills all voids in the universe? Where does it come from? Is it produced by some force? Without DM/DE, does space even exist? Say the supernovas, the collision of neutron stars...

Hi, So the geodesic equation is saying in my frame of reference I may see acceleration and then in your frame of reference you may see gravity? So by just changing coordinates you can create a "force" ? And also is this relevant to the Minkowski space or do I need to be in GR to be able to get...

Hi everyone, I'm reading about the solution of the wave equation in free space on Stratton - Electromagnetic Theory and Snider - PDE and I got a little confused. The wave equation in 3D (plus time) is the following $$\frac{\partial^{2} \Psi (x,y,z,t)} {\partial t^{2}}=\nabla ^{2}\Psi...

Einstein's theory states that curvature of space (created by a celestial body around itself) determines the orbital path of other celestial bodies around it within that curved space by a constant lateral force acting towards the centre upon that revolving body. Then why is that a similar force...

Some books argue that typical coordinate transformations such as space translations and rotations are represented in quantum mechanics by unitary operators because the Wigner's theorem. However I do not find any clear proof of this. For instance, suppose 1D for the sake of simplicity, by...

Let's say I have a 10 cm long stick in space. If I want it's right side to rotate as much as possible (even go backwards) by pushing on it's left end, where would the ideal position of the center of mass of the stick be? Would it be the middle, or closer to the right or left? And by how many...

Take a wavefunction ##\psi## and let this wavefunction be a solution of Schroedinger equation,such that: ##i \hbar \frac{\partial \psi}{\partial t}=H\psi## The complex conjugate of this wavefunction will satisfy the "wrong-sign Schrodinger equation" and not the schrodinger equation,such that ##i...

I need some help in checking my work, especially #4. Problem: You have a set of 10 cards numbered 1-10. You choose a card at random. Event A is choosing a number less than 8. Event B is choosing an even number. Draw a Venn Diagram and calculate each of the following probabilities: 1) P(A)...

Greg Bernhardt submitted a new PF Insights post Mathematical Quantum Field Theory - Reduced Phase Space Continue reading the Original PF Insights Post.

How much energy needs to be be expended to throw a non operating satellite (or other space trash) out of orbit (L4 or L5) or L2 into deeper space... thus minimizing space debris in Earth orbit. I could only find formulas for getting an object INTO orbit... Background of question: we were...

Temperature is atomic/molecular motion. However, the temperature of planetary surfaces without an atmosphere are often quoted. For example, the temperature of the lunar surface ranges from about -200 F to +200 F depending on whether or not the surface is exposed to the sun or is in shadow...

Hi, I have an operator given by the expression: L = (d/dx +ia) where a is some constant. Applying this on x, gives a result in the subspace C and R. Can I safely conclude that the operator L can be given as: \begin{equation} L: \mathcal{H} \rightarrow \mathcal{H} \end{equation} where H is...

If me and my pet duck, who weighs next to nothing, were placed far away from any gravity source and in such a way that our mutual attraction was negligible, would I see my duck accelerate away from me? If so, how fast?

How would I go about calculating the ambient temperature in low Earth orbit (LEO) at approximately 400 km? What equations should I be referencing to determine the temperature of sun light side and/or eclipse side? Thanks!

The metric expansion of the Universe does not seem to cause some of the mechanical effects typically associated with motion (commute) of massive objects through space. For example, distant galaxies can be "flying away" from each other at speeds exceeding the speed of light. It seems that space...

I could not find an easy or believable source on this question on the internet. Anyway, if I had a box (volume V) in space full of gas (density N) and a hole suddenly appeared, how long would it take all of the air to completely leave the box out into space? I would assume the air would be at...

Hello, how can you imagine the geometrically meaning of the minus sign in ds2=-dx02+dx12+dx22+dx32, maybe similar to ds2=x12+dx22 is the length in a triangle with the Pythagoras theorem?

If space time (the universe) is infinitely expanding what happens when it rips? Have we observed Virtual particles in an area and counted them? Does the expansion of space time affect the amount of Virtual particles in a particular area? Is it theoretically possible to expand space time to a...

Homework Statement The following task causes me problems: The science fiction writer R.A. Heinlein describes in the novel "Friday" a satellite ("space elevator"), which consists of a long rope, placed directly above the equator. The rope is aligned along the Earth's radius. It moves so that...

It's said that expectations drive the economy. So, how do you maintain interest in something that has no immediate bearing/benefit/utility of a population? I would think the answer to that question is to manifest a 'goal', in this case being an interplanetary species or even just building a...

This may belong to the computing subforum, let me know if this is more true than having it here in the math forum :) My questions are 1) Suppose there is a plane in 3D space and I have 3 points to define it: p1 = {x1, y1, z1} p2 = {x2, y2, z2} p3 = {x3, y3, z3} and I want to put a particular...

Homework Statement A space station (moment of inertia 5,000,000 kg•m2) is rotating with angular speed 0.30 rad/s when the 10 inhabitants are in the outer ring (radius 100 m). One half of the inhabitants move to the inner ring (radius 50 m). The average mass of each inhabitant is 70 kg. a...

It makes me wonder... wikipedia says about a basis: [FONT=Times New Roman]In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this...

Before I get to a question that's been on my mind for a while, I would like to make it clear that I am a high school student with no understanding of physics beyond my school's curriculum and some books that I've read, so excuse me if this sounds rather absurd. I know that the expansion of...

Good Morning I am having some trouble categorizing a few concepts (I made the one that is critical to this post to be BOLD) Remote parallelism: the ability to move coordinate systems and frames around in space. Euclidean Space Coordinate systems: Cartesian vs. cylindrical I am aware that if...

Does space expand only at the outer edges of the universe? What is the mechanism that creates more space particles? Is the huge volume of outer space pulling our universe apart? Please enlighten or comment. Kurt Ludwig

Greg Bernhardt submitted a new PF Insights post Mathematical Quantum Field Theory - Phase Space Continue reading the Original PF Insights Post.

https://www.sciencedaily.com/releases/2017/11/171122113013.htm A University of Geneva researcher has recently shown that the accelerating expansion of the universe and the movement of the stars in the galaxies can be explained without drawing on the concepts of dark matter and dark energy…...

Hi, I'm trying to understand Einstein's field equations conceptually, does it describe space density in a region of space by any chance? Like there is more space in this region compared to this other region. Thanks.

Hello, I would like your help understanding how to map a region of the space \mathbb{C}^2 spanned by two complex conjugate variables to the real plane \mathbb{R}^2 . Specifically, let us think that we have two complex conugate variables z and \bar{ z} and we define a triangle in the...

Hello Folks, I have two points on Earth at each end of a great circle path, for which I know the length in km and direction in degrees True. Also I have the RA and DEC of an object in space as seen from one of the previous points. The RA and DEC come from the setting circles of a telescope...

I have a matrix, [ a, ib; -1 1] where a and b are constants. I have to represent and analyse this matrix in a Hilbert space: I take the space C^2 of this matrix is Hilbert space. Is it sufficient to generate the inner product: <x,y> = a*ib -1 and obtain the norm by: \begin{equation}...

Hi! I'm doing a research project titled "Is it possible to colonise Mars?" I'm currently researching how to raise the temperature and pressure - and I've discovered to do this that re-directing asteroids towards Mars containing Ammonia would be vital. However, where do these asteroids that...

Assume a person is at space, and assume there's no friction or gravitational force in that space. The person has a bow and arrow. He stretches the arrow on the bow, and then shots the arrow out in the space. Since there are no friction and gravitational force in that space, the arrow will have...

Homework Statement Find ##f:R \to X##, f-continuous, where X is the discrete space. Homework EquationsThe Attempt at a Solution f is continuous at p if for any ##\epsilon > 0## there is ##\delta >0## such that ##d(f(x),f(p))<\epsilon## for all x such that ##d(x,p)<\delta##. Let ##\epsilon =...

Hello, I have derived the matrix form of one ODE, and found a complex matrix, whose phase portrait is a spiral source. The matrix indicates further that the ODE has diffeomorphic flow and requires stringent initial conditions. I have thought about including limits for the matrix, however the...

Suppose ##q: E-->X## is a covering map (not necessarily normal). Let ##E' = E/ Aut_{q}(E)## be the orbit space, and let ##\pi: E-->E'## be the quotient map. Then there is a covering map ##q': E' --->X## such that ##q' * \pi = q## where ##*## is composition of functions. I am confused why ##E'##...

I am trying to understand the derivation of the Casimir energy from https://en.wikipedia.org/wiki/Casimir_effect#Derivation_of_Casimir_effect_assuming_zeta-regularization. At one point, the derivation writes the following: The vacuum energy is then the sum over all possible excitation modes...

I was watching a movie called Everything and Nothing. It got me thinking. For the sake of argument pretend that you could make the sun pop in and out of existence. If you were to measure the distance between two points on the other side of the solar system with no sun, then if the sun were to...

Near space travels with balloons. http://edition.cnn.com/2015/03/05/tech/balloons-fly-edge-of-space/index.html look at above link. balloons goes just 36 km above Earth surface. but Earth orbit is 100 km away. so my thinking is if we pump oxygen outside near helium balloon artificially when u...

Hello. As many theories and hypotheses that there are about a secure, reliable and effective future space exploration and colonisation that there are, I would like to dedicate this thread to the discussion of this topic. Some of the suggestions are: Von Neumann probes are an effective strategy...

Homework Statement Let ##A## be a 2x3 matrix. If Nul(##A##) is a line through the origin in ℝ3, then Col(##A##) = ℝ2. Explain why. Hint: Think about the number of pivots in ##A##. Homework EquationsThe Attempt at a Solution So, Nul(##A##) is the set of all solutions to the equation ##Ax=0##...

Hello, I Have a particle with wavefunction Psi(x) = e^ix and would like to find its Hilbert space representation for a period of 0-2pi. Which steps should I follow? Thanks!

Homework Statement [/B] A rope with uniform mass distribution is attached to a spaceship. At the end of this rope, there is a mass, m. The length of the rope - l. Point A - where the rope is attached to a spaceship Point B - in the middle of the rope. The whole system moves with the same...

Hi I can't see the error in this can someone please explain where I went wrong? A man is in a spaceship traveling at a constant velocity He makes 2 identical tubes of length L with a mirror at one end, tube a and tube b He has a single light bulb. Next to the bulb is a detector. He carefully...

The term 'phase space' is often used in the study of nonlinear dynamics.What is it.