Equivalence Definition and 747 Threads

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation.
Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other, if and only if they belong to the same equivalence class.

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  1. L

    Statistical physics - does total energy matter, or only differences?

    In statistical physics, in a two level system, I'll give an example to show what I am talking about: situation 1) energy 0 and energy E situation 2) energy E/2 and energy -E/2 Are these two situations equivalent? Computing the partition function for both of them it seems they are different...
  2. Lotto

    B What is a general definition of a limit?

    I suppose that it is because we are in extended real numbers. But the definition of a limit when ##x_0 = \infty## and let's say ##L=\infty## is different. Why are these definitions equivalent? Isn't the key that ##U_{\varepsilon}(\infty)=\left(\frac {1}{\varepsilon},\infty\right)##?
  3. P

    I Unit sphere is compact in 1-norm

    In Introduction to Topology by Gamelin and Greene, I'm working an exercise to show the equivalence of norms in ##\mathbb R^n##. This exercise succeeds another exercise where various equivalent formulations of "equivalent norms" have been given, e.g. that two norms ##\|\cdot\|_a,\|\cdot\|_b## are...
  4. Ben2

    How Do Right-Side Homotopies Affect Homotopy Equivalence?

    "[A] map f: X-->Y is called a \mathbf{homotopy~equivalence} if there is a map g: Y-->X such that fg\cong\mathbb{1} and gf\cong\mathbb{1}," where "cong" means "is homotopic." "The spaces X and Y are said to be \mathbf{homotopy~equivalent}..." Additional definitions are in Hatcher, "Algebraic...
  5. C

    I Equivalence Question between when-then statement and if then statement

    Dear Everybody, I am working on my homework. I am trying to prove a problem that was written by my professor in an odd way: Prove that when p is true, then q is true. Which proposition statement should I assume? I personally thought that I should assume the first one. But reading my...
  6. PhysicsRock

    Binary operation on equivalence classes

    So, my approach and solution are as follows: $$ [x * y] = \{ z \in M : z \sim (x * y) \} $$ Since we know that ##a * b \sim a^{\prime} * b^{\prime}## we can rewrite ##z## as ## x^{\prime} * y^{\prime} ##. Plugging this in yields: $$ [x * y] = \{ x^{\prime}, y^{\prime} \in M : x^{\prime} *...
  7. agm

    Finding the Equivalence Ratio of Combustion (Methane + Air)

    I'd like to do some experiments with flames at different Equivalence Ratios - but I'm confused as to how I can find the Equivalence Ratio for different conditions. Wiki-article So the stoichiometric balance for Methane + Air is CH4 + 2(O2 + 3.76N2) -> CO2 + 2H2O + 7.52N2 Referencing equations...
  8. Filip Larsen

    B Weak Equivalence Principle Confirmed by MICROSCOPE Satellite to 10^-15

    https://arstechnica.com/science/2022/09/einstein-wins-again-space-satellite-confirms-weak-equivalence-principle/ See also http://dx.doi.org/10.1103/PhysRevLett.129.121102 (limited access)
  9. mopit_011

    B Equivalence of Frictional and Applied Force

    The following passage is from Halliday Resnick Krane in Chapter 3 which is about dynamics in one dimension. "We can measure frictional forces. By placing the body on a horizontal surface where it experiences a frictional force, we could attach a spring and pull the body with just the right...
  10. L

    MHB Equivalence Relations: Solving & Proving Reflexivity, Symmetry & Transitivity

    Dear All, I am trying to solve the attached two questions. In both I need to determine if the relation is an equivalence relation, to prove it if so, and to find the equivalence classes. In both cases it is an equivalence relation, and I managed to prove both relations are reflexive. Now I...
  11. qnt200

    I Weak principle of equivalence (Galileo, Newton)

    Many tutorials that explain the weak principle of equivalence (Galileo, Newton) do not clearly state whether the body is affected by the force of inertia during free fall as a result of the gravitational acceleration of the body. In other words, the question is whether, during the free fall of a...
  12. karmion

    A Cross section-temperature equivalence

    It's assumed that interaction rate between a species of particule m and l is expressed as: Γm=<nlσv>, where nl is the density of the species l, σ the cross-section of species m (=probability of interaction) and v the relative velocity between the two particles. It's also assumed that...
  13. ergospherical

    I Wilbert Zweistein's Objection to Equivalence Principle

    I'd like to share the following competition problem: My first reaction was that the thermometer will be able to measure heating due to the Unruh effect whilst his lab is accelerating.
  14. shivajikobardan

    Comp Sci Are Equivalence Classes Confusing?

  15. J

    B A difficulty with the equivalence of all inertial reference frames

    Although I am not a physicist, I am interested in physics, and recently I've been reading about special relativity. I have a doubt about it, a difficulty I see in the equivalence of all inertial reference frames which I haven't found solved anywhere, and I've thought perhaps you in this Forum...
  16. gregthenovelist

    I Equivalence relation and different sample spaces

    It is a theorem that: two propositions implying each other, in the sense that the set of outcomes making one true is the same as the one making the other true) have the same probability. this comes from the fact that if p --> q, the P(p&q) = P(p), we have that if p <-> q, then P(p&q) = P(p)=...
  17. M

    I Equivalence of Universe Accelerating Expansion?

    Without suggesting that there actually is a massive spherical shell enclosing the universe, I am trying to explore whether the observed expansion of the universe would be the same if there was a spherical shell with a diameter greater than the universe, with some arbitrary mass producing a...
  18. e2m2a

    A Why is the Principle of Equivalence Necessary for GTR?

    I am studying the general theory of relativity(GTR). Covariance and the principle of equivalence are foundational pillars for the theory. I can understand the need for covariance but I don't see why the need for the principle of equivalence (POE). What I have seen so far is that the...
  19. e2m2a

    I Analyzing Dynamics in Constant Acceleration w/Rindler & Equivalence

    Not sure when to use Rindler coordinates to analyze dynamics in a constant accelerating reference system. Rindler coordinates seem messy because they are always changing. Wouldn't it be easier to invoke the principle of equivalence and treat the environment of an accelerating system as a...
  20. only1god

    I What if Einstein equivalence principle is proven wrong one day?

    What would be the consequences of such thing? How it will affect physics theories and the world?
  21. P

    Find R.H. Dicke's Paper: "Gravitation without a Principle of Equivalence

    Summary:: Searching for Paper by R H Dicke Good day all. Does anyone know where I might obtain a copy of the Paper R. H. Dicke. Gravitation without a principle of equivalence. Review of Modern Physics, 29(3):363–376, 1957 I am prepared to pay for it. Thanks Paul
  22. P

    A Proving the Equivalence of Local and Global Maxima for Concave Functions

    Consider the following theorem: Theorem: Let ##f## be a concave differentiable function and let ##g## be a concave function. Then: ##y \in argmax_{x} {f(x)+g(x)}## if and only if ##y \in argmax_{x} {f(y)+f'(y)(x-y)+g(x)}.## The intuition is that local maxima and global maxima coincide for...
  23. P

    MHB Proving Equivalence of f(x) and g(x)

    Consider the two functions f(x)=(x²+3x+10)+(2x²+2x-17) and g(x)=(4x²+4x+4)-(x²+x+11). If they are equivalent, prove they are; if they are not equivalent, prove they aren't.
  24. P

    MHB Proving Equivalence of f(x) and g(x)

    Consider the two functions f(x)=(x-1)(x4+x³+x²+x+1) and g(x)=x5-1. If they are equivalent, prove they are; if they are not equivalent, prove they aren't.
  25. S

    B Equivalence Principle and Rindler Horizons

    I just thought of this. I'm not an expert, so cut me some slack if I get something wrong. Given: 1) Acceleration is supposed to produce a Rindler horizon. 2) If I understand it correctly, under Relativity and/or the Equivalence Principle, standing on (for example) the Earth is the same as...
  26. H

    A Can holography provide a complete description of a space-time?

    I read that the parameters of each these two theories can be obtained from parameters of the other theory and if one event may appear in ads5 with a given probability there is another event in the CFT side which will get the same probability. But is there a one to one map between the two space...
  27. M

    I Equivalence principle question

    The equivalence principle states that a person stood on Earth would experience “gravity” the same as if he was in an elevator in space traveling at 1g. I get this. but when Einstein was first exploring this, I read he came to the realisation that a person free falling on Earth (if in a vacuum)...
  28. WMDhamnekar

    MHB When to use equivalence relations? How to write it in octave?

    Sometimes to help describe one expression, another expression is shown that produces identical results. The exact equivalence of expressions is indicated with ‘ ≡’. For example: rot90 ([1, 2; 3, 4], -1) ≡ rot90 ([1, 2; 3, 4], 3) ≡ rot90 ([1, 2; 3, 4], 7) What is the meaning of 'rot90;? What...
  29. Ebi Rogha

    B Exploring Energy-Mass Equivalence: Converting Energy to Matter in Everyday Life?

    a) Can we convert energy to mass (matter) in every day life? b) When we charge a phone battery, its mass (weight) increases according to E=mc2 . Does it mean we convert energy to matter? If not, how its mass increases?
  30. L

    B Simple reasoning that the equivalence principle suggests curvature

    I am a high-school teacher and a PhD. student. I am looking for ways to introduce my students to GR. In my treatment, I speak about the equivalence principle and later about curvature in general and consequently that of spacetime. I am missing a connection of these two parts that would be...
  31. AdvaitDhingra

    B Question regarding the equivalence principle

    Hello, So I read that a person in a rocket accelerating at 9.8m/s^2 would feel the same pull downwards as a person standing on the Earth's surface. However, I can think of a few instances where you could tell the difference: - If you measure g (9.8N/kg) on Earth, you will notice that the...
  32. E

    A Gravitational Potential Energy & the Equivalence Principle

    First, in section 20.4, after listing all the things gravitational potential energy does not do, they say the equivalence principle forbids it being localized. I thought I understood the equivalence principle, but maybe I don’t. Any comments explaining that would be appreciated. Second, they...
  33. J

    B Special Relativity & Einstein's Equivalence Principle

    I was recommended a paper: <https://digital.library.adelaide.edu.au/dspace/bitstream/2440/101285/3/hdl_101285.pdf>. And in the opening sentence read: "The Einstein equivalence principle (EEP) is at the heart of special relativity." To me this didn't make sense. Firstly because Einstein...
  34. S

    MHB Proving $(A\cap B)\cup(B\cap X')=0$ Equivalence

    Prove that: $(A\cap B)\cup(B\cap X')=0$ is equιvalent with $B\subseteq A'$ and $B\subseteq X\subseteq A'$ 0 is for the emty set and $X'$ is for the complement of $X$
  35. L

    I Equivalence of these quantum circuits

    In the attached image, there are two quantum circuits that are equivalent. I am trying to understand how. Let's call the top qubit ##q_1## and the bottom one ##q_2##, and the outputs ##q_1'## and ##q_2'##. From what I understand, the C-NOT gate doesn't affect the control qubit. Because Hadamard...
  36. S

    MHB Proving Equivalence of 0A=0, (-A)B=-AB, (-A)(-B)=AB

    Given the following axioms: For all A,B,C...we have: 1) A=A 2) A=B <=> B=A 3) A=B & B=C => A=C 4) A=B => A+C= B+C 5) A=B=> AC =BC ( NOTE :Instead of writing A.C or B.C e.t.c we write AB ,BC e.t.c) 6) A+B= B+A..........AB=BA 7) A+(B+C) = (A+B)+C............A(BC)=(AB)C 10)...
  37. ohwilleke

    I Evidence of Strong Equivalence Principle Violations?

    This paper appears to be a major break though in observational evidence of a strong equivalence principle violation, something predicted in MOND and contrary to general relativity. The analysis is model dependent, but it seems to rule out the most plausible conventional GR based alternatives...
  38. il postino

    Chemistry Calculate the potential at the equivalence point

    Data ##E^{0}_{I_2/I-}= 536mV## ##E^{0}_{Cr_2O_7^{2-}/Cr^{3+}}= 1330mV## ##I_{2} + 2e- \Leftrightarrow 2I^{-}## ##Cr_2O_7^{2-} + 14H^+ + 6e^- \Leftrightarrow 2Cr^{3+} + 7H_2O## Total Reaction ##6I^{-} + Cr_2O_7^{2-} + 14H^+ \Leftrightarrow 3I_{2} + 2Cr^{3+} + 7H_2O##...
  39. R

    B Understanding Strong Equivalence Principle Errors & Misunderstandings

    Please help me understand all the errors and misunderstandings, contained in the following paragraphs: 1. When a gravitational wave passes through earth, it 'squashes' the form of earth, hence gravity geometry is changed, hence gravity at any given point on Earth change at that moment, be it a...
  40. L

    Equivalence of various forms of flux linkage

    Hello all, Apologies if this has already been asked before, but I tried researching this question for a while with no results. I was reading Grainger's Power System Analysis' derivation of the inductance of a single wire and got confused by his definition of magnetic flux linkage. He seems to...
  41. U

    I Equivalence principle and the Uniqueness theorem

    We work with Maxwell's equations in the frequency domain. Let's consider a bounded open domain ## V ## with boundary ## \partial V ##. 1. The equivalence theorem tells me that if the field sources in ## V ## are assigned and if the fields in the points of ## \partial V ## are assigned, then I...
  42. C

    I Equivalence Principle & Geometry: Ben Crowell's General Relativity

    In the first sentence of Chapter 2 in Ben Crowell's "General Relativity" he states: "The geometrical treatment of space, time, and gravity only requires as its basis the equivalence of inertial and gravitational mass". This is stated as if it's an obvious fact, but I don't understand why. Why...
  43. G

    I Equivalence of interpretations in more elaborate setups

    Hi. One of the first things I was told about quantum interpretations was that they are all equivalent, i.e. make the same experimentally testable predictions. This seems reasonable for simple experiments whose mathematical description is straightforward, like preparation – unitary evolution –...
  44. H

    Prove Q is the equivalence relation on A

    I can’t understand it.
  45. Stephen Bulking

    I The Tritium Puzzle: Unravelling Mass-Energy Equivalence

    I was finding the energy required to separate tritium into it's component parts, the binding energy when it hit me that how could 1amu= 931.2 MeV and 1.66e-27 kg at the same time?
  46. J

    I Equivalence principle: an electric charge and a coil

    The near-range magnetic field ##\vec{B}## of a point charge ##q## at distance ##\vec{r}##, moving at a non-relativistic velocity ##\vec{v}##, is given by $$\vec{B}=\frac{q}{4\pi\epsilon_0c^2}\frac{\vec{v}\times\hat{r}}{r^2}.$$ Faraday's law of induction for the induced EMF ##V_c## in a coil...
  47. M

    MHB For which n is the term an integer & Calculate the equivalence

    Hey! 😊 Question 1: We consider $\frac{2n-1}{n+7}$. For which $n$ is this term an integer? I have done the following: We set $n+7=m \Rightarrow n=m-7$. Then we get $$\frac{2n-1}{n+7}=\frac{2(m-7)-1}{(m-7)+7}=\frac{2m-15}{m}$$ So $m$ has to be a divisor of $15$, i.e. $m\in \{1,3,5,15\}$...
  48. J

    I Falling EM system contradicts the equivalence principle?

    The following is an improved version of my previous post https://www.physicsforums.com/threads/falling-electric-dipole-contradicts-the-equivalence-principle.964594/ Consider the following system comprising a particle on the left with charge ##+q## that is a large distance ##d## away from two...
  49. nomadreid

    I Equivalence of two different definitions of quasicrystals

    https://en.wikipedia.org/wiki/Riemann_hypothesis#Quasicrystals a quasicrystal as "a distribution with discrete support whose Fourier transform also has discrete support." https://en.wikipedia.org/wiki/Quasicrystal#Mathematicsdefines a quasicrystal as "a structure that is ordered but not...
  50. Yassin98

    The Equivalence Principle -- Is this a way to distinguish between a gravitational field and an accelerated rocket?

    If we are in a cabine in a gravitational field and inside, we have a racket and a ball. We put strings in each side of the racket and we connect the racket to the ceiling of the cabine. This strings only allows us to keep the weight of the racket. Then, we drop a ball to the racket. We do this...
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