Flat Definition and 469 Threads

Using York decomposition as a foundation, can we reformulate General Relativity to treat space as flat (##R_{ij} = 0##), redistributing all curvature into a vector field for time (##\vec{t}_i##)? The York decomposition: $$\tilde{h}_{ij} = \phi^{-4} h_{ij}$$ $$\tilde{K}_{ij} = \phi^2 K_{ij}$$...

Please excuse my ignorance I hear some people say that space is flat. What exactly do they mean by this?

Years ago, I filled some latex party balloons with Freon 12. I haven't repeated the experiment with the Freon 134 that is available nowadays. It was fun to throw them around because Freon is so much heavier than air--you can throw them across the room and they land on the floor and stay there...

The earth is a sphere. I can stand anywhere on its surface and look up directly above my head at the night sky and see lots of stars many light years away. I see this view wherever I stand on the earth's surface although the stars will be different ones, depending on where I'm standing...

Let ##M## be a flat module over a commutative ring ##A##. Suppose ##X_1## and ##X_2## are submodules of an ##A##-module ##X##. Prove that ##(X_1 \cap X_2) \otimes_A M = (X_1 \otimes_A M) \cap (X_2 \otimes_A M)## as submodules of ##X\otimes_A M##.

I started by expanding ##dx## and ##dt## using chain rule: $$dt = \frac{dt}{dX}dX+\frac{dt}{dT}dT$$ $$dx = \frac{dx}{dX}dX+\frac{dx}{dT}dT$$ and then expressing ##ds^2## as such: $$ds^2 =...

Question: A ball is hit with a certain force and it starts with an initial velocity of V m/sec decelerating due to friction along a flat path and then up an inclined ramp coming to stop at the top of the ramp where the ramp becomes flat again (as per diagram A). The total distance traveled along...

Can anyone tell me how to solve this problem? I have the stem of a tree that is X feet tall. It's just a cylinder as the top and all branches have been cut off. I want to cut off the top portion such that when it falls, it will do precisely a 3/4 rotation and land perfectly flat. What fraction...

Current flow through a wire is considered (or assumed) to be flown uniform across the cross section of the wire (or is it more near skin? or at core?). However I wonder if that's the case with a sheet or plate or circular disc. e.g. If I apply voltage at two opposite corners of the square or...

I understand, given dark energy, given the current expansion of the universe, given its ~13.8 billion year old age, we calculate the universe is like 93 billion light years across currently. Something like that. But I was listening to something the other day, and when they test the curvature...

Are they equivalent to each other?Thank you.

Hi everyone I got 36 cm as the answer for the following problem, but it's supposed to be 32cm. These are the cuts I have 4 x 5cm = 20cm 2 x 3cm = 6 cm 1 x 10cm = 10cm which adds up to 36cm. I can get 32cm with 4x3=12 2x5=10 1x10=10 But I don't think that would be the correct net. Is...

The implication seems to be that from the beginning of the post expansion era, there was everywhere an average velocity of a large volume of matter which was (very near) zero everywhere with respect to a common fixed coordinate system (with a spacially uniform time expansion of distances)...

Hi, Also, I read this article, What Do You Mean, The Universe Is Flat? (Part I), on Scientific American; URL: https://blogs.scientificamerican.com/degrees-of-freedom/httpblogsscientificamericancomdegrees-of-freedom20110725what-do-you-mean-the-universe-is-flat-part-i/ I have few questions about...

5/18/22 I am an MS in physics. I need to find out if the following CONFORMAL METRIC produces zero or nonzero curvature? I suspect the curvature is zero, but others have said it's probably not? MAXIMA sometimes says it is, and other times produces a Ricci scalar that looks like the FRW scalar...

Global Newtonian mechanics seems to be compatible with 1) Hubble's law, and 2) the cosmological principle: take a vector space, set the velocity of a galaxy at x to be v=x, where x is the position vector. Then from any galaxy, the other galaxies seem to go away with velocity v2-v1 = x2-x1. That...

References (1) https://www.physicsforums.com/threads/what-is-the-probability-that-the-universe-is-absolutely-flat.971984/ (2) https://www.physicsforums.com/threads/calculating-the-probability-that-the-universe-is-finite.1011826/ Suppose the Friedmann equation is used to analyze two models. (1)...

One recent example of a thread discussing flat or not is: https://www.physicsforums.com/threads/could-the-universe-be-infinite.1011228/ . I found an interesting 2021 article regarding the Hubble constant tension...

I would like to attached a picture here...

In a textbook, which is not in Englisch language unfortunately, I found a passage saying that intrinsic curvature of spacetime is just a specific definition. The alternative definition is that spacetime is flat, whereas clocks and rods have variable lengths - which is just Feynman’s bug...

I am a knitter and am interested using physics to explain the negative ease we see in knitting especially in rib [ (k1, p1) rib for example). I think some could be explained by the stretch-ability of the materials, e.g. wool, acrylic, but is there a movement of the yarn in between the "live"...

Given a certain manifold in ##R^3## I've been told that at every location ##p## it is possible to encounter a reference frame from which the metric is the euclidean at zero order from that point and its first correction is of second order. This, nevertheless does not match with the following...

It is mathematically proven that any intrinsically curved manifold can be derived from embedding in a flat space of sufficient number of dimensions. I heard somewhere, but lost that reference, that for our universe 10 flat dimensions are needed in the most general case.. May I ask - Is the...

It is recently claimed that between 1800 and 800 million years ago, Earth was flat. This is supposedly indicated by lack of minerals that should have formed in mountains´ roots. However, from that period, there are rocks formed in orogeny - mountain forming processes. For example Grenvillean...

Off the back of a recently closed thread where there was some discussion about the gravitational field of an infinite flat slab, I decided to have a play at investigating that. I've found a few interesting things. It's fairly straightforward to solve for this situation. You use Cartesian-esque...

General relativity. Curvature of spacetime: ok. time dilation: ok. What about space? Curvature is intrinsic and given by complex equations. But could we definitely say is there more space between 2 points along curved space through the star than would be through flat space (no star there) or...

I would appreciate if someone check my work: I tried to simplify the answer a lot: I imagined that, if we have this ds between two points different than the distance that should be if the space was flat, so it would be enough to generalize and say that space is not flat. So, using this...

When parallel transporting a vector along a straight line on flat space, does the connection (when calculating the covariant derivative) always equal zero? Do things change at all when using an arbitrary connection, rather than Christoffel symbols?

I would like if my procedure is correct ... Due to the symmetry of the problem, I only worry about the vertical coordinate of the field, so I will work with the magnitude of the field, and I will treat the problem in polar coordinates. ##E= \int_{R_1} ^ {R_2} \int_{0} ^ {\pi} \frac {\sigma...

I was just reading about de Sitter space and the following question occurred to me: de Sitter spacetime is curved despite containing no mass-energy, because it has a positive cosmological constant. Does it have a foliation into spatially flat, constant-time hypersurfaces though? Maybe it's...

[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...

Vehicle Mass : 3 tonnes Manufacturer's instruction is to test vehicle's parking brake effectiveness is to engage parking brake on 30.96 degrees slope. If it holds, it is effective. I do not have 30.96 degrees slope & I want to replicate the same test on flat ground. I was thinking of...

The RW metric reads $$ds^2 = -dt^2 + a^2(t) \Big( \frac{dr^2}{1-kr^2} + r^2 d\theta^2 + r^2 sin(\theta)^2 d\phi^2 \Big)$$ The value of k determines the model is flat/open/closed. But say if we have a model on a completely different form, with no explicit k-dependence. How would I determine if...

When we say the universe is "flat," do we mean: 1) If you could hypothetically "zoom" out to the edge of the universe it would appear roughly as a flat like a sheet of paper with local fluctuations / bumps to indicate the presence of matter. 2) If we draw 2 parallel lines from 1 location they...

"Mad Mike" Hughes wanted to see for himself if the Earth was a sphere or flat. Went up in a rocket, when it failed, he ejected without his parachute and died in the fall. https://www.polygraph.info/a/flat-earth-believer-mad-mike-hughes-dies-in-diy-rocket-crash/30454704.html Just think how...

In relativity, a flat space is always regraded to be endowed with an invariant metric field ##g_{\mu\nu}(x)= \eta_{\mu\nu}##, So in a flat space the corresponding connection ##\Gamma_{\mu\nu}^\rho(x)=0## It means that if we parallel transport a vector ##v^\nu(x_0)## in the space. Then it...

Hi all, I have ran into some confusion about the equivalence principle; perhaps I should state what I understand and then proceed to ask questions. It is my understanding that the equivalence principle states that spacetimes are locally Minkowski, and so the rules of SR apply in that locality...

All the accounts which I have read (and which are accessible to my limited knowledge of General Relativity and its mathematics) on the holographic principle says vaguely that the AdS/CFT correspondence is very enlightening, but with the caveat that, well, we don't happen to live in an AdS space...

Let me begin by stating that I'm aware of the fact that this is a metric of de Sitter spacetime, aka I know the solution, my problem is getting there. My idea/approach so far: in the coordinates ##(u,v)## the metric is given by $$g_{\mu\nu}= \begin{pmatrix}1 & 0\\ 0 & -u^2\end{pmatrix}.$$ The...

Since torque= |force||radius|sin(theta) : î = |71E6||5000m|sin(30)= 177.5E9 N*m ĵ = |0||5000m|sin(30) = 0 k̂ = |-158E6||5000m|sin(30)= 395 N*m

I am trying to prove that for a single component flat universe $$\frac{dz}{dt_0} = H_0(1 + z) - H_0(1 + z)^{\frac{3 + 3w}{2}}$$ For a single component flat universe, ##q = \frac{2}{3 + 3w}## ##a(t) = (t/t_0)^q## ##t_0 = qH_0^{-1}## ##1 + z = (t_0/t_e)^q## Now here is my approach...

Summary: is it possible to heat a coiled spring If I had a spirally coiled spring like inserted pic, spiral coiled spring approx. 300mm dia and spring section +-12mm dia with +-1mm gauge wire, with a total length of spring approx. 4mtr-6mtr , is it possible to heat this with induction heating...

My questions are based on material from the following source. https://arxiv.org/pdf/1502.01589.pdf pg 26 Eqs 27 & 28 In the base CDM model, the Planck data constrain the Hubble constant H0 and matter density Ωm to high precision: H0=67.3±1.0 km s−1 Mpc−1 Ωm=0.315±0.013...

We need to prove that a3(t)= ρo/2Λ [cosh(sqrt(24πGΛ)*t) -1] by changing into a variable of u, where u=2Λa3/ρo From Friedmann's second equation we know that Λ= ρm/ 2 Also ρm= ρo/ a3 [First attempt] I begin from Friedmann's equation where (for here), ρtotal= ρm + Λ and k=0; a'2/a2 = 8πG(ρm +...

In a real case (not ideally rigid bodies), a (e. g.) hard metal sphere is on a flat (e. g.) hard metal surface (a table) and the sphere is "charged" vertically on the table by a vertical force directed downward. In this situation, an engineer told me that the maximum pressure on the table is not...

From the geodesic equation d2xμ/dΓ2+Γμ00(dt/dΓ)2=0,for non-relativistic case ,where Γ is the proper time and vi<<c implying dxi/dΓ<<dt/dΓ. Now if we assume that the metric tensor doesn't evolve with time (e,g gij≠f(t) ) then Γμ00=-1/2gμs∂g00/∂xs. If we here assume that the metric components of...

From friedmann equation And For a flat universe with k=0 and ρ=ρc ,da/dt becomes undefined and d2a/dt2 becomes 0 But for the present time we know that our universe is flat and expanding with a acceleration (q -ve) , Therefore is it here something I am missing??

Hi, I have become very interested in cosmology recently and I had a question about the possible shapes of the Universe. I understand the critical density plays a role in the shape. This is what I know (or think I know). At the critical density of energy and matter, the universe is flat. If it...

The Guardian - Adam Gabbatt @adamgabbatt Thu 10 Jan 2019 01.00 ESTLast modified on Thu 10 Jan 2019 11.02 EST A group of people who believe the Earth is flat have announced their “biggest, boldest, best adventure yet”: a Flat Earth cruise scheduled for 2020. Article continues...