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 have some questions about the curvature of space (NB not of spacetime) near a planet like Earth. Unambiguously defining space curvature requires choice of a coordinate system, so I choose the Swarzschild system. Here are my questions:
Would constant-time hypersurfaces under the Swarzschild...
Homework Statement [/B]
The International Space Station (ISS) has a mass of 400,000 kg and orbits 408 km
above the Earth’s surface. The ISS is 109 m across.
Homework Equations : [/B]
R=a(semimajoraxis) cubedroot(m2/3M1)The Attempt at a Solution : [/B]
ive tried multiple ways with multiple...
Homework Statement
Derive, using the canonical commutation relation of the position space representation of the fields φ(x) and π(y), the corresponding commutation relation in momentum space.Homework Equations
[φ(x), π(y)] = iδ3(x-y)
My Fourier transforms are defined by: $$ φ^*(\vec p)=\int...
Has the spectrum of light that is reflected off planet Earth ever been measured from outer space? (In the same sense that we measure spectra of the light emitted/reflected from other celestial bodies in astronomy). If so, would it be possible that there could be dips in the spectrum resulting...
Homework Statement
So I have these two Matrices:
M = \begin{pmatrix}
a & -a-b \\
0 & a \\
\end{pmatrix}
and
N =
\begin{pmatrix}
c & 0 \\
d & -c \\
\end{pmatrix}
Where a,b,c,d ∈ ℝ
Find a base for M, N, M +N and M ∩ N.
Homework Equations
I know the 8 axioms about the vector spaces.
The...
the problem:
Say we have the entire space uniformly charged. Then, the E field experienced by any point is zero, from symmetry.*
But, it means that for any Gaussian surface, the flux though it is zero even though the charge enclosed is clearly not. Gauss' law seems to disagree with symmetry, but...
I am trying to follow modern QFT by Tom Banks and I am having an issue with literally the first equation.
He claims that beginning from ## |p_1 , p_2, ... , p_k> \: = \: a^\dagger (p_1) a^\dagger (p_2) \cdots a^\dagger (p_k)|0> ## with the commutation relation ##[a (p),a^\dagger (q)]_\pm \: =...
Hello
In our Quantum Mechanics lecture we have been discussing a simplified model of the Stern-Gerlach experiment. Let ##|+>## and ##|->## denote an electron that is "spin up" and "spin down" (with respect to ##\hat{z}##), respectively. Our professor then asserted that ##|+>## and ##|->## acted...
Regarding curvature of spacetime/space: At some given point in a gravitational field, spacetime is curved at that point and this is a constant. (I'm assuming this is true).
Although we can talk about the curvature of spacetime, I never hear anyone talking about the curvature of space. Can...
Homework Statement
A 50 g, 420 mw laser pointer is floating in outer space (don’t ask how it got there) at rest with respect to an observer. The laser pointer is turned on and let go. If the battery runs continuously for 250 hours before dying, what is the final speed achieved by the laser...
Hey I'm new to Physics. I have a question. Is it possible to use the force generated when air entera a vacuum tube to launch something into space? Can anybody throw some light on this in simple words please. Thank You.
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:
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 set.[1]
So what is the...
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
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 =...