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.
"The dual space is the space of all linear maps from the original vector space to the real numbers." Spacetime and Geometry by Carroll.
Dual space can be anything that maps a vector space (including matrix and all other vector spaces) to real numbers.
So why do we picked only a vector as a...
If I'm given a set of four vectors, such as A={(0,1,4,2),(1,0,0,1)...} and am given another set B, whose vectors are given as a form such as (x, y, z, x+y-z) all in ℝ, what steps are needed to show A is a basis of B?
I have calculated another basis of B, and found I can use linear combinations...
Let V = C[x] be the vector space of all polynomials in x with complex coefficients and let ##W = \{p(x) ∈ V: p (1) = p (−1) = 0\}##.
Determine a basis for V/W
The solution of this problem that i found did the following:
Why do they choose the basis to be {1+W, x + W} at the end? I mean since...
Let ## \mathcal{S} ## be a family of probability distributions ## \mathcal{P} ## of random variable ## \beta ## which is smoothly parametrized by a finite number of real parameters, i.e.,
## \mathcal{S}=\left\{\mathcal{P}_{\theta}=w(\beta;\theta);\theta \in \mathbb{R}^{n}...
I draw the graph like this:
For (b), I divided each force vector to e from p1 and p2 as x and y parts.
I computed them and got
Fx=-4.608*10^(-15)N
Fy=-2.52*10^(-15)N
However, I am not sure whether I did it correctly or not...
I appreciate every help from all of you!
Thank you!
Lets go through the example problem until we get to the part I don't understand. Figure 25-17 can be used as a reference to all questions. From part (a) to part (b) we eventually find the charge q on one plate (and by default the charge -q on the other). No problem there. The battery is then...
Sorry if the question is not rigorously stated.Statement: Let ##(q,p)## be a set of local coordinates in 2-dimensional symplectic space. Let ##\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})## be a set of local coordinates of certain open set of a differentiable manifold ##\mathcal{M}.## For...
I would like to show that fixing the orientation of k-manifold smooth connected ##S## in ##\mathbb {R} ^ n ## is equivalent to fixing a frame for one of its tangent spaces.
What I know is that different orientations correspond to orienting atlases containing maps that cannot be consistent with...
Rookie question; for a vector space ##V##, with basis ##v_1, v_2, \dots, v_n##, the dual space ##V^*## is the set of linear functionals ##\varphi: V \rightarrow \mathbb{R}##. Dual basis will satisfy ##\varphi^i(v_j) = \delta_{ij}##. Is the action of any dual vector on any vector always an inner...
hi
i was recently introduced to the Dirac notation and i guess i am following it really well , but can't get my head around the idea that the bra vector
said to live in the dual space of the ket vectors , i know about linear transformation and the structure of the vector spaces , and i realize...
In Dynamical Systems Theory, a point in phase space is interpreted as the state of some system and the system does not exist in two states simultaneously. Can some phase spaces be given an additional interpretation as describing a field of values at different locations that exist...
I was recently working on the two body problem and what I can say about solutions without solving the differential equation. There I came across a problem:
Lets consider the Kepler problem (the two body problem with potential ~1/r^2). If I use lagrangian mechanics, I get two differential...
If Evangelista Torricelli truly created a vacuum, then there would be nothing in it, yet you can see through it which means light is obviously still in there (and who knows what else), right?
If there was truly nothing in it, and glass is a highly viscous fluid, and fluids conform to fill empty...
Is it possible to create following two shapes from glass using currently known glass mass production techniques?
Shape #1: bottle with a prolonged neck that continues into the inner space, like this (cross section):
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/| |\
/ | | \
/ \
/ \
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|...
At the core of the earth, or sun, there is no net force of gravity...every direction is up...does this mean that space is not curved, or more generally, becomes less curved from the surface of an object towards its center?
I have heard of phase space path integrals, but couldn't find anything in Wikipedia about it, so I am wondering, what does it compute ? In particular, are the endpoints points of definite position and momentum? If so, how does one convert them to quantum states ? Also, how is it related to...
Hello everybody, my question may sound stupid, especially speaking of such a mind-blowing and important theory... but here I am!
I'm 17 and I'm reading a fabulous book by Stephen Hawking, "A Brief History of Time", and it introduced me to relativity theories... I literally started looking the...
Does the concept of the angle between two vectors make sense in Minkowski space?
Does the concept of orthogonal basis for Minkowski space make sense? If it does, how is it defined?
When we start with the usual (time, distance) basis for 2-D Minkowski space, the axes as drawn make a right...
Did energy begin to exist at the Big Bang? Can energy exist without space and time?
Or don't we know?
When I've tried to research this I get a mix of different answers. I have virtually no understanding of science or physics in general FYI.
Where do I start. I want to write the matrix form of a single or two qubit gate in the tensor product vector space of a many qubit system. Ill outline a simple example:
Both qubits, ##q_0## and ##q_1## start in the ground state, ##|0 \rangle =\begin{pmatrix}1 \\ 0 \end{pmatrix}##. Then we...
[I urge the viewer to read the full post before trying to reply]
I'm watching Schuller's lectures on gravitation on youtube. It's mentioned that spacetime is modeled as a topological manifold (with a bunch of additional structure that's not relevant to this question).
A topological manifold is...
Hello
As you know, the geometric definition of the dot product of two vectors is the product of their norms, and the cosine of the angle between them.
(The algebraic one makes it the sum of the product of the components in Cartesian coordinates.)
I have often read that this holds for Euclidean...
Hi
I believe I understand the concept of a vector space V and its dual V*. I also understand that for V finite dimensional, there is a natural isomorphism between V and V**.
What I am struggling to understand is - Does this natural isomorphism mean that V** is always IDENTICAL to V (identical...
This is section 16.3 of QFT for the Gifted Amateur. I understand the concept of the spacetime propagator ##G^+(x, t, x', t')##, but the following propagator is introduced without any explanation I can see:
$$G^+(x, y, E) = \sum_n \frac{i\phi_n(x)\phi_n^*(y)}{E - E_n}$$
It would be good to have...
I'm studying 'A Most Incomprehensible Thing - Notes towards a very gentle introduction to the mathematics of relativity' by Collier, specifically the section 'More detail - contravariant vectors'.
To give some background, I'm aware that basis vectors in tangent space are given by...
Thank you all for clearing my doubt before. There is a question I want to ask on space and time and this time it is not about the absolute time as I have understood fairly that space and time are always relative.
So Here it is according to GR It is very beautifully explained that Very massive...
Suppose I'm traveling inside a spaceship at speed comparable to light between two points A and B. According to me the distance between the two points will be shortened due to length contraction. But actually my spaceship passes through every point between A and B so the distance measured by...
Good Morning
Recently, I asked why there must be two possible solutions to a second order differential equation. I was very happy with the discussion and learned a lot -- thank you.
In it, someone wrote:
" It is a theorem in mathematics that the set of all functions that are solutions of a...
I'm studying 'Core Principles of Special and General Relativity' by Luscombe - the chapter on tensors.
Quoting:
The book goes on to talk about a switch to the spherical coordinate system, in which ##\mathbf{r}## is specified as:
$$\mathbf{r}=r\sin\theta\cos\phi\ \mathbf{\hat...
Summary:: Inner Product Spaces, Orthogonality.
Hi there,
This my first thread on this forum :)
I encountered the above problem in Schaum’s Outlines of Linear Algebra 6th Ed (2017, McGraw-Hill) Chapter 7 - Inner Product Spaces, Orthogonality.
Using some particular values for u and v, I...
This is an experiment I would have liked to do from the ISS, but an approximation could be done in a vacuum chamber on Earth. How big a soap bubble, polymer bubble, or glass bubble could you blow in the vacuum of space? How to calculate the evaporation rate in vacuum?
A liquid exposed to vacuum...
Imagine this question in 2 dimensions, time (t) and distance (x), that is (t,x). Alice (A) is at the origin, x=0. Bob (B) begins at x=c. Thus we have A(0,0) and B(0,c). Both Alice and Bob send a light signal towards the other but let's say the signal changes colour every second by the colours of...
Presume we look at a two-dimensional view of space time, with no local masses, and we draw a grid of equidistance spaced lines. The intent is to look at space but not time.
As we begin, we look in all directions and the grid lines are evenly spaced.
Begin adding mass to the center of the grid...
Since they started this I have been somewhat amazed by the ability to not only launch but also peacefully get back a rocket intact and landing on it's vertical axis.
I'll admit I haven't read a ton of material with regards to this so pardon if this has been asked already.
To me it seems that...
In Henley and Garcia's Subatomic Physics, they introduce phase space in chapter 10 by considering all the possible locations a particle can occupy in a plot of ##p_x \ vs. x##, ##p_x## being the momentum of the particle in the x direction. They next consider an area pL on this plot, and state...
In "The Geometry of Minkowski Space in Terms of Hyperbolic Angles" by Chung, L'yi, & Chung in the Journal of the Korean Physical Society, Vol. 55, No. 6, December 2009, pp. 2323-2327 , the authors define an angle ϑ between the respective inertial planes of two observers in Minkowski space with...
I'm often involved in projects where Dropbox is used to share documents, and members are invited in order to gain access to a folder established by one of the members (not me). I recently got a message that my Dropbox quota (2 GB) is us up. I don't sync anything locally, I only use the documents...
I'm reading about the geometry of spacetime in special relativity (ref. Core Principles of Special and General Relativity by Luscombe). Here's the relevant section:
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Minkowski space is a four-dimensional vector space (with points in one-to-one correspondence with those of ##\mathbb{R}^4##)...
Hello All :)
I am a student of mathematics, but I have only one semester of physics in college. I can't solve one with homework. Will there be anyone wise who can solve this?
The task is as follows:
Calculate: at what height the artificial satellite must move (orbit height): geostationary...
One proposal that I have read (but cannot re-find the source, sorry) was to identify a truth value for a proposition (event) with the collection of closed subspaces in which the event had a probability of 1. But as I understand it, a Hilbert space is a framework which, unless trivial, keeps...
I am surrounded by Space and so I am curious to know what the name of this Space is. Is this Space given any name by a Mathematician? I suspect that this Space may be called Euclidean Space because Euclidean Space knows how to present point in three-dimensions. Euclidean Space talks about...
Recently I asked a question about the curvature of the universe.
https://www.physicsforums.com/threads/constant-curvature-and-about-its-meaning.977841/
In that context I want to ask something else.
Is this curvature (##\kappa##) different than the Gaussian Curvature ? Like it seems that we...
So, I'm a little confused and I thought I might get some help here.
I have just started learning about manifolds and its super confusing because I've always worked with Euclidean spaces, too much that I didn't even realize it's euclidean and that it has different properties from others.
So my...
Metaphorical depictions of the universe in the shape of a 2-sphere are very common.
Now, let's consider mapping the three dimensions of space onto the "surface" of a 3-sphere. Like Non-Euclidean de Sitter geometry.
The surface of a sphere is positionally symmetrical. However, the positional...
Hello everybody, new here. Sorry in advance if I didn't follow a specific guideline to ask this.
Anyways, I've got as a homework assignment two cannonical transformations (q,p)-->(Q,P). I have to obtain the hamiltonian of a harmonic oscillator, and then the new coordinates and the hamiltonian...