Continuum Definition and 131 Threads

I cannot understand a corollary from this paper on surreal numbers: Does it say that there are no countable surreal numbers greater than the germs of Hardy fields?

Can the holographic principle be applied to spacetimes and metrics that are (fundamentally) continuous/smooth? Or only to discrete ones?

Question: When thinking of continuums the most notable seems to be space-time but they also mark a simplification to reality like in continuum mechanics, often taught when learning the tensor calculus needed for general relativity. The question is that for general relativity when a geodesic...

Hi, unfortunately, I am not getting anywhere with the following task and I try solve it

I’ve been reading “Interpreting astronomical spectra” and am still confused about the micro processes contributing to the continuum. In particular,the sun’s photosphere. I’m not interested in the black body thermodynamics approach because some processes should cause this. Bound-bound...

Is there more references for how accurate is the continuum approximation to discrete sums? Perhaps more mathematical. What I've found: https://lonitch.github.io/Sum-to-Int/ https://arxiv.org/pdf/2102.10941.pdf Some examples are: Sum to integral $$\sum_{\mathbf{k}} \to 2 \left ( \frac{L}{2...

How much does the equivalent width of a line change by the introduction of 5% scattered light? We know the equivalent width is defined as We know the equivalent width is defined as $$W = \int_{-\infty}^{\infty} \bigg(\frac{1-F_{\nu}}{F_c}\bigg) \, d\nu$$ where ##F_{\nu}## represents the flux in...

OK I've been stuck for a while in how to derive ##(1)##, so I better solve a simplified problem first: We work with Where $$\mathscr{L} = \mathscr{L}(\phi_a (\vec x, t), \partial_{\mu} \phi_a (\vec x, t)) \tag{3}$$ And ##(3)## implies that ##\mathscr{L}(\vec x, t)## We know that...

What are the types of thermodynamics processes that happen in solids or continuum medium

Can anybody suggest which books are good for Lagrangian/Hamiltonian formulations for continuum beyond The Classical Mechanics by Goldstein ( it seems a bit too complicated for my understanding.)?

How would one plot the space-time continuum graphically(if it were possible,obviously)?

Summary: The Continuum Hypothesis and Number e Now, I must ask a very stupid question: When taking: $$\lim_{_{n \to \infty} } (1+\frac{1}{n})^n=e\\$$ the ##n## we use take its values from the set: ## \left\{ 1,2,3 ... \right\} ## which has cardinality ## \aleph_0 ##, which is equivalent...

I've recently been reading about the 2-dimensional Ising model and its continuum limit from several sources, including https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture1.pdf https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture2.pdf As far as I understood it, the state...

I have a question regarding a paragraph in "Radiation detection and measurement" by Knoll. In the chapter about the discrete Gaussian it states that "Because the mean value of the distribution ##\bar{x}## is large , values of ##P(x)## for adjacent values of x are not greatly different from each...

I am a Phd student working in a technological center. My work is related to CFD simulations by using OpenFoam coupled to discrete element method. I am very interested in mathematical and physical background of continuum mechanics both solid and fluids. I am loking forward to solve my interests...

Maxwell stress tensor ##\bar{\bar{\mathbf{T}}}## in the static case can be used to determine the total force ##\mathbf{f}## acting on a system of charges contanined in the volume bounded by ##S## $$ \int_{S} \bar{\bar{\mathbf{T}}} \cdot \mathbf n \,\,d S=\mathbf{f}= \frac{d}{dt} \mathbf...

I came across this article about the near absence of continuum mechanics in university-level physics education: http://www.troian.caltech.edu/papers/Gollub_PhysToday_Dec03.pdf I have wondered this issue myself: why is continuum mechanics mainly studied by engineers rather than physicists, even...

Those treatments of Entropy in continuum mechanics that I've viewed on the web introduce Entropy abruptly, as if it is a fundamental property of matter. For example the current Wikepedia article on continuum mechanics ( https://en.wikipedia.org/wiki/Continuum_mechanics ) says: Are other...

For example, I am following the below proof: Although the above derivation involves a projection on the position basis, it appears one can generalize this result by using any complete basis. So despite it not being explicitly mentioned here, are all wave functions with any continuum basis...

Hi, people of PF I'm trying to decide between concentrating on continuum mechanics or design/manufacturing for my master's degree. My goal is to ultimately work in the industry, so design/manufacturing seems to make a lot of sense. However at the same time, continuum mechanics (and physics in...

This negation of CH is based on Woodin's work: https://en.wikipedia.org/wiki/%CE%A9-logic Of course, you can only believe his result because you need to believe his axioms first. But for me it is really convincing for multiple reasons: 1. While it is, of course, a negation of CH, it does not...

john baez submitted a new PF Insights post Struggles with the Continuum - Conclusion Continue reading the Original PF Insights Post.

john baez submitted a new PF Insights post Struggles with the Continuum - Part 7 Continue reading the Original PF Insights Post.

Wikipedia: "The hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers" ; i.e., let cardinality of integers = ℵ0 and cardinality of reals = ℵℝ; then there is no ℵ such that ℵ0 < ℵ < ℵℝ . But what about...

john baez submitted a new PF Insights post Struggles with the Continuum - Part 6 Continue reading the Original PF Insights Post.

john baez submitted a new PF Insights post Struggles with the Continuum - Part 5 Continue reading the Original PF Insights Post.

john baez submitted a new PF Insights post Struggles with the Continuum - Part 4 Continue reading the Original PF Insights Post.

Hi! Let ##T^{ik}## be the stress-energy-tensor, and ##v_k## some future-pointing, time-like four vector. How can I see that the object ##T^{ik}v_k## is future-pointing and not space-like? Thank you for your help!

Homework Statement Determine the thermodynamic restrictions for a rigid heat conductor defined by the constitutive equations: \DeclareMathOperator{\grad}{grad}\psi = \hat{\psi}\left(\theta,\grad \theta, \grad \grad \theta\right) \\ \eta = \hat{\eta}\left(\theta,\grad \theta, \grad \grad...

I have fond almost the same question here with some answers. Anyhow, I am not satisfied with the answers, or couldn't understand them well. Beside that, I would formulate the question a little bit different. Unfortunately the thread is already closed (from 2012). Provided there is only one...

john baez submitted a new PF Insights post Struggles With the Continuum - Part 3 Continue reading the Original PF Insights Post.

Hello Please forgive me if i am not posting in the correct forum. Also you may find my English a bit rusty since i am basically French Ok so i want to solve some exercises in continuum mechanics . The first exercise states : we have a stress tensor in a Cartesian coordinate system with the...

john baez submitted a new PF Insights post Struggles With The Continuum - Part 2 Continue reading the Original PF Insights Post.

This is a diagram of a pitot-static tube. My question is however not related to its applications but rather, what causes the liquid to rise up the static tube? The static tube is at right angles to the fluid flow. I understand that this is a very basic question but I can't seem to get my head...

john baez submitted a new PF Insights post Struggles With the Continuum - Part 1 Continue reading the Original PF Insights Post.

Hello! I started learning relativity recently and I'm an engineering student. I can't stop drawing similarities between the nature of gravity and behaviour of continuous media in the field of continuum mechanics. Is there some direct connection and if so, has something like this been explored...

Space in quantum mechanics seems to be modeled as a triplet of real numbers, i.e. a continuum. Same happens in special relativity. General relativity I do not know (nor field theories). And then we apply the Pythagorean theorem and triangle inequality and so forth... I have a few general...

Hi all, I've got two questions about the emissions spectrum from solids. Question #1: I feel like I have a reasonable understanding of line absorption and emission spectrum of low density gases based on transitions of electrons between discrete allowed energy levels in a gas. I'm trying to...

I have always wondered if spacetime is actually smooth, or if it is granular such as a quantized spacetime. I remember learning that if you draw the smallest possible square around a circle, you will, of course, get a perimeter, 8 times the radius of the circle. Then if you draw in the...

Hi all.. I am currently doing some works on the continuum mechanics. And trying to study the macroscopic behavior of solids ( for simplicity, taken homogeneous materials) upon the action of external force ( which is the stress; pressure). How is it possible to account for the changes that can be...

Homework Statement In Cartesian coordinates ##x##, ##y##, where ##x## is the horizontal and ##y## the vertical coordinate, the velocity in a small-amplitude standing surface wave on water of depth ##h## is given by; $$v_x = v_0 sin(\omega t) cos(kx) cosh[k(y + h)]$$ $$v_y = v_0 sin(\omega t)...

Hi guys, Here is a little puzzle I have been wondering about. I can't solve it, perhaps you can help. We know there are infinitely many real numbers on the number line. Indeed, already between 0 and 1 there are infinitely many real numbers. So if a real number is a point on the number line...

Hi there. I'm reading Gurtin's 'the mechanics and thermodynamics of continua', and working some exercises of his book. In the section 21: 'The first law: balance of energy', after the derivation of the balance equation, he uses an identity to rewrite the balance of energy. The balance of energy...

We know that in order to be integrated a function must be continuous. Does this imply that space and time must be a continuum? If they were considered discrete, say at the level of Planck's unit, would this affect the integrability of functions? It it would not, would it affect the precision...

Relativity and continuum I have read contrasting things about the issue. Can you explain briefly why the assumption of discrete space and time is absolutely incompatible with relativity? What is the main problem? Lorenz invariance or what? Thanks

Is there any book that does what Landau does in Fluid Mechanics and Theory of Elasticity, only using a Lagrangian/Action-principles the whole way through? I can really only find brief tiny descriptions like this one in books on other topics, is there nothing that does for fluids/elasticity...

Homework Statement I have a question that looks so stupid that I have never dared to ask. If I want to measure the time average from t=0s to t=1s of a given f(t), the solution is compute the following integral: TA = 1/T*∫F(t)dt However, I have some doubts about this calculus.Homework...

Putting the following three statements together: (a) Assuming that the continuum hypothesis is false, the power of the continuum 2\aleph0 is real-valued measurable. (b) The existence of a real-valued measurable and the existence of a measurable (= real-valued measurable & inaccessible)...

If we accept that there does indeed exist a set whose cardinality is between \aleph_0 and \aleph_1, what would such a set look like? I know that in ZM-C we can choose to either add the continuum hypotheses or not, but if we chose to negate it, that means that there definitely is a set greater...

If I am in my office and the space-time continuum is further bent, would I realize that time is passing slower or faster and why?PS: I am new to physics, so please provide a complete answer.