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.
So there's two spaceships in deep space. at rest with respect to each other. Then one of them shoots off at some huge speed and everyone feels it. Then they shut the engines off suddenly. No one is going to suddenly lurch forward, they will at the instant the engines are shut off effectively be...
Is it possible to build a skyscraper that's so tall that it enters the ozone layer? I was imagining such a tower where customers could go to the top in elevators and view the Milky Way in bed with a glass roof in their luxury apartments :)
Obviously not just a tower without supports. Maybe with...
If one shows that ##U\cap V=\{\textbf{0}\}##, which is easily shown, would that also imply ##\mathbf{R}^3=U \bigoplus V##? Or does one need to show that ##\mathbf{R}^3=U+V##? If yes, how? By defining say ##x_1'=x_1+t,x_2'=x_2+t,x_3'=x_3+2t## and hence any ##\textbf{x}=(x_1',x_2',x_3') \in...
I read in my textbook Calculus on Manifolds by Spivak that a set ##A\subset \mathbb{R}^n## is bounded iff there is a closed n-rectangle ##D## such that ##A\subset D##. It should be plain that if I wanted to define unboundedness, I should just say something along the lines of: "A set ##A\subset...
I'm reading a book about analytical mechanics and in particular, in a chapter on hamiltonian Mechanics it says:
"In the state space (...) the complete solutionbof the canonical equations is pictured as an infinite manifold of curves which fill (2n+1)-dimensional space. These curves never cross...
Most sources of pop culture & even engineering ones have failed to provide the key problems of space traveling. For example, dv ain't that hard as aerodynamics of Space. For 1st glance, it looks ridiculous == Space is Vacuum. Mostly, yes.. but Space has a lot of dust & gaseous clouds. Just one...
I've got the solution to the question but I just need more detail. I can't work out the first step of the solution to the second step.
That should read, I don't know what they multiplied ih-bar by to make it (i/h-bar)^2?
Given that the Set of 1-Forms is a Vector Space distinct from, but complimentary to, the Linear Vector Space of Vectors. And given that there is an Isomorphism between the linear space of vectors and the dual vector space of 1-forms, does it make mathematical sense to combine a vector space and...
If we suppose the following 8-dimensional manifold given by
$$a_1=cos(x)cos(y)cos(z)$$
$$a_2=cos(x)cos(y)sin(z)$$
$$a_3=cos(x)sin(y)cos(z)$$
$$a_4=cos(x)sin(y)sin(z)$$
$$a_5=sin(x)cos(y)cos(z)$$
$$a_6=sin(x)cos(y)sin(z)$$
$$a_7=sin(x)sin(y)cos(z)$$
$$a_8=sin(x)sin(y)sin(z)$$
Then obviously...
--##ker(T^2)=ker(T)## if ##T(V)=T^2(V)##--
Suppose that ##T^2(V)=T(V)##. So ##T:T(V)\mapsto T^2(V)=T(V)##. Hence, ##T## is one-to-one and so ##ker(T)=\{0\}##. Suppose that ##T^2(w)=0## for some ##w\in ker(T^2)##. Then ##T^2(w)=T(T(w))=0## which implies that ##T(w)\in ker(T)## and so ##T(w)=0##...
Sorry if the question has been already answered, but I didn't manage to find it. Let's go back to ligo detection of gravitational waves, my question is the following: if space time changes its texture due to a gravitational effect, all the rulers (and clocks) in that spot will be affected, so...
When the C drive gets limited in space, I take some space from the D drive and the audio files in D drive corrupted and can not open. How do I open these files?
Interesting article ...NEW SOURCE OF SPACE RADIATION: Astronauts are surrounded by danger: hard vacuum, solar flares, cosmic rays. Researchers from UCLA have just added a new item to the list. Earth itself.“A natural particle accelerator only 40,000 miles above Earth’s surface is producing...
Consider an ##N_2## molecule. Chemists say that the triple bond is due to one ##p_x - p_x## overlap, one ##p_y - p_y## overlap and one ##p_z - p_z## overlap. The x-axis (the label is not important; I’m sure you know what I mean) is clear because it’s the longitudinal axis of the molecule. But...
In short, I was trying to look into feasibility of deflecting an asteroid with a collision of ~32MJ of direct energy. I wanted to know how many collisions are necessary to deflect at a given time out (distance away.)
I found this link, where the collision is perpendicular to the motion of the...
In another forum, some people argue that time and space are discrete, due to Planck time and Planck length.
However, I disagree with this idea. I think, the Planck time and Planck length are just some scales that we can measure, but they do not forbid continuous time and space shorter than...
Aristotle's absolute space and time can be represented as ordered pairs (s, t) but not as fibers π(s) = t of time as is the case of Galileo and Newton's space time. That is to say that the space of Galileo and Newton is the projection π(s) = t on the time axis. The time space of Galileo and...
If you viewed my most recent thread before this one, then you know that I have been studying curves in spacetime (timelike/spacelike/lightlike), and I have especially been looking into the CTCs (closed timelike curves) that the Godel metric is famous for. During my studies I found that I had to...
I am reading N. L. Carothers' book: "Real Analysis". ... ...
I am focused on Chapter 3: Metrics and Norms ... ...
I need help with a remark by Carothers concerning convergent sequences in \mathbb{R}^n ...Now ... on page 47 Carothers writes the following:
In the above text from Carothers we...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand Example 3.10 (b) on page 95 ... ... Example 3.10 (b) reads as follows:
My question is as...
My question is about space and the multiverse. I was reading work by Max Tegmark and he sees the multiverse as Level 1-4. A Level 1 multiverse seems like it's self evident and I was wondering about the evidence against it. It's simple:
Space expands faster than we can observe it so we exist in...
I am reading N. L. Carothers' book: "Real Analysis". ... ...
I am focused on Chapter 3: Metrics and Norms ... ...
I need help Exercise 32 on page 46 ... ... Exercise 32 reads as follows:
I have not been able to make much progress ...
We have ...B_r(x) = \{ y \in M \ : \ d(x, y) \lt r \}...
Summary: Expansion of matter in the visible universe, total volume of space outside the visible universe, black hole mechanics, and general questions from an uneducated but extremely interested evolved monkey.
Hello everyone! I do not know the rules of this forum, or any forum for that matter...
There's no thermodynamics forum, so I'll post this here.
Things "freezing" in space has always bothered me ever since Tim Robbins removed his helmet while in orbit around Mars...
I mean the first question has derivative form and the second is linear form so what the difference here in steps of converting both to transfer function... please need some ellaboration to make sure i am solving correctly or not... is it correct to apply the same rule on both:
Transfer function=...
My questions are now... Do the steps of converting this space to transfer function include any laplace ? or just we do get [SI-A]-1 and then transfer function is = C* [SI-A]-1 * B As [1 0] * [s-1/det -0.5/det ; 0.5/det s-0.5/det] * [0; 1] = -0.5/s^2+s+0.5 I mean do we need any laplace after that...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Lemma 3.44 on page 105 ... ... Lemma 3.44 and its proof read as follows:
In the above...
The standard definition of the basis for a vector space is that all the vectors can be defined as finite linear combinations of basis elements. Consider the vector space consisting of all sequences of field elements. Basis vectors could be defined as vectors which are zero except for one term in...
Suppose I have a system of two (possibly interacting) spins of 1/2. Then the state of each separate spin can be written as a ##\mathbb{C}^2## vector, and the spin operators are made from Pauli matrices, for instance the matrices
##\sigma_z \otimes \hat{1}## and ##\hat{1} \otimes \sigma_z##...
I've been struggling with a somewhat-recent paper by Charles Francis, "A construction of full QED using finite dimensional Hilbert space," available here: https://arxiv.org/pdf/gr-qc/0605127.pdf
But also published in...
If i get this right: In space when electrons leave its atoms, do they attach to something else? Or is the electrons bouncing freely in space when they get detached? Is this because of dark energy? And if so, where do they go? Do they move between atoms or are the free to go anywhere? And where...
A free photon can have any wavelength and energy; no discreteness there. Just because something is quantized, or fundamentally quantum in nature, doesn’t mean everything about it must be discrete.
The idea that space (or space and time, since they’re inextricably linked by Einstein’s theories...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.6 on page 94 ... ... Theorem 3.6 and its proof read as follows:
In the above...
A theorem from Axler's Linear Algebra Done Right says that if 𝑇 is a linear operator on a complex finite dimensional vector space 𝑉, then there exists a basis 𝐵 for 𝑉 such that the matrix of 𝑇 with respect to the basis 𝐵 is upper triangular.
In the proof, he defines U=range(T-𝜆I) (as we have...
If I'm trying to solve the problem of a particle in free space (H = P2/2m ).
the eigenfunctions of the Hamiltonian cannot be normalized.
now assume I have a legitimate wave function expressed in terms of the eigenfunction of H and I want to measure its energy.
what will happen to the...
Due to my lab work
I want to try map the phase space that created with critical external magnetic field H_c
and the critical current J_c through the superconductor of type II. the critical transition happen from the Abrokosov phase to the non-superconductor phase, occurred by the fact that in...
Centrifuges have long been viewed as the means to provide artificial gravity in the zero-g of deep space. Space colony structures like the O'Neill cylinder and the Stanford torus.
Centrifuges could also work their magic on the surfaces of low-gravity planets. Conical or paraboloidal shaped...
I thought about this pretty hard, can windows frost over in space? I'm pretty sure they don't or can't after giving it a bit of thought. This is just a curiosity of mine.
So first I believe the correct question that we're trying to answer is:
Can a window get colder than interior air faster...
If two events are occurring in a space like hypersurface...is it that the two events will appear to be simultaneous from a frame moving with velocity ##v\lt c## and will appear to be occurring at the same spatially position from a frame with velocity ##v\gt c##??
Spin in Physical Space, Internal Space, and Hilbert Space:
Spin is already a great unifying principle in theoretical physics, but its potential is far from exhausted. Here I discuss – very concisely – the role of spin in particle physics, in space-time physics, and in ideas for unified field...
I am reading a text which talks about the WIMP speed distribution in the galactic halo in the frame of the Sun and Earth. The point where I am stuck it is trying to explain the concept of Gravitational Focusing of WIMPs at the location of the Earth due to the gravitational well of the Sun...
Hi! I am trying to change the hydrogen ground state wave funcion from position to momentum space, so i solved the integral
Ψ(p)=(2πħ)^(-3/2) (πa^3)^(-1/2)∫∫∫e^(prcosθ/ħ) e^(-r/a) senθ r^2 dΦdθdr
and got 4πħ(2πħ)^(-3/2) p^(-1) (πa^3)^(-1/2) I am [(ip/ħ-1/a)^(-2)], which according to the...
Trying to understand the effects of a deep space nuclear explosion.
Starfish prime was detonated at 400km, ie still within Earth's atmosphere, and the explosion effects are well described.
https://en.wikipedia.org/wiki/High-altitude_nuclear_explosion
A significant outcome is the high MeV...
Determine if the set of vectors
$\begin{bmatrix}
x\\y\\5
\end{bmatrix}\in \Bbb{R}^3$
form a vector space
ok if I follow the book example I think this is what is done
$\begin{bmatrix} x_1\\y_2\\5 \end{bmatrix}
+\begin{bmatrix} x_2\\y_2\\5 \end{bmatrix}
+\begin{bmatrix} x_2\\y_2\\5...
Hello,
consider a full-cone (let me say a cone including bottom half, upper half and the vertex) embedded in ##E^3##. We can endow it with the topology induced by ##E^3## defining its open sets as the intersections between ##E^3## open sets (euclidean topology) and the full-cone thought itself...
So Juno is on her downhill run towards Jupiter. Only doing about 8k right now, but by the time she rounds Jupiter she'll be doing 100k or more.
We are going to get so much science from Juno.
Personally I am eagerly awaiting the results of what Jupiter looks like under all those clouds.
Is there any theory in physics that can be modeled in any type of space (Hilbert space, Euclidean, Non-Euclidean...etc)? And if yes, could that theory also contain/be compatible with all types of (physical) symmetries?