Difference between revisions of "Dimensions in Physics"

Revision as of 17:20, 7 February 2024

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Beyond the 3 space dimensions and 1 time dimension other dimensions have been proposed to deal with issues not addressed in these 4.

Context

In understanding physics, dimensions play a crucial role in describing physical quantities.

The common use of calculus in Mathematics assumes continuity, the real world can surprise us with its discrete and varied dimensions. Physics bridges this gap by ensuring that equations remain dimensionally consistent, allowing us to explore the universe with mathematical precision.^[1]

1. Dimensional Analysis

  - The dimension of any physical quantity expresses its dependence on the base quantities (such as length, mass, time, etc.). Each dimension is represented by a symbol (or a power of a symbol) corresponding to the base quantity.
  - For instance:
    - Length has dimension **L** or **L^1**.
    - Mass has dimension **M** or **M^1**.
    - Time has dimension **T** or **T^1**.
  - We can express the dimension of any physical quantity as:
    \[ \text{Dimension} = L^a M^b T^c I^d \Theta^e N^f J^g \]
    where \(a\), \(b\), \(c\), \(d\), \(e\), \(f\), and \(g\) are powers associated with each base quantity.
  - Quantities with dimensions that can be written with all seven powers equal to zero (i.e., \(L^0 M^0 T^0 I^0 \Theta^0 N^0 J^0\)) are called **dimensionless** or **pure numbers**.

1. Continuity and Dimensionality

  - In mathematics, we often assume continuity. However, in the real world, physical dimensions need not always be continuous.
  - For example, consider motion. We postulate that motion is continuous, but this assumption doesn't necessarily hold in reality.
  - Physicists use square brackets around the symbol for a physical quantity to represent its dimensions. For instance:
    - If \(r\) represents the radius of a cylinder and \(h\) represents its height, we write \([r] = L\) and \([h] = L\) to indicate that both dimensions are lengths.
    - Surface area (\(A\)) has dimensions \(L^2\), and volume (\(V\)) has dimensions \(L^3\).
    - Mass (\(m\)) has dimensions \(M\), and density (\(\rho\)) has dimensions \(M/L^3\).

3. Dimensional Consistency

  - Any mathematical equation involving physical quantities must be dimensionally consistent.
  - Rules for dimensional consistency:
    - Every term in an expression must have the same dimensions.
    - Adding or subtracting quantities of differing dimensions is nonsensical (like adding apples and oranges).
    - Equations must have the same dimensions on both sides.

(1) 1.4 Dimensional Analysis | University Physics Volume 1 - Lumen Learning. https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/1-4-dimensional-analysis/.

(2) 2.2: Units and dimensions - Physics LibreTexts. https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_%28Martin_Neary_Rinaldo_and_Woodman%29/02%3A_Comparing_Model_and_Experiment/2.02%3A_Units_and_dimensions.

(3) 6.1: An Analytic Definition of Continuity - Mathematics LibreTexts. https://bing.com/search?q=do+all+physical+dimensions+need+to+be+continuous.

(4) 1.7: Limits, Continuity, and Differentiability. https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_%28Boelkins_et_al.%29/01%3A_Understanding_the_Derivative/1.07%3A_Limits_Continuity_and_Differentiability.

Kaluza–Klein is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to string theory. In their setup, the vacuum has the usual 3 dimensions of space and one dimension of time but with another microscopic extra spatial dimension in the shape of a tiny circle. Gunnar Nordström had an earlier, similar idea. But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential, and writing the Maxwell equations in five dimensions.

Problems

1. For the Schrodinger equation to work, the space is typically defined as only piece-wise continuous, but a single photon may travel for a billion years or more, which seems to imply that all of 4-space must be continuously differentiable, not just in little patches.

Types of Information

• Laplace - the clockwork universe is set in motion and runs deterministically by itself (Laplace's daemon)
• Gaussian - error bounds prevent us from make any exact measurement
• Boltzmann statistics of distinguishable (known) particles
• Planck & Bose - statistics of indistinguishable particles
• Pauli & Dirac - statistics of particles subject to exclusion principle
• Fisher - population statistics
• Shannon - information content

Entropy

The entropy of the universe is constantly increasing. The second law of thermodynamics states that the state of entropy of the entire universe, as an isolated system, will always increase over time. Energy always flows downhill, and this causes an increase of entropy. Entropy is the spreading out of energy, and energy tends to spread out as much as possible. It flows spontaneously from a hot (i.e. highly energetic) region to a cold (less energetic) region. As a result, energy becomes evenly distributed across the two regions, and the temperature of the two regions becomes equal. The same thing happens on a much larger scale. The Sun and every other star are radiating energy into the universe. However, they can’t do it forever. Eventually the stars will cool down, and heat will have spread out so much that there won’t be warmer objects and cooler objects. Everything will be the same very cold temperature. Once everything is at the same temperature, there’s no reason for anything to change what it’s doing. The universe will have run down completely, and the entropy of the universe will be as high as it is ever going to get.^[2]

The change in entropy of an isolated system during an irreversible process is > 0; while for a reversible process, it is = 0.the change in entropy of an isolated system during an irreversible process is > 0; while for a reversible process, it is = 0.

The collapse of the wave function is generally considered to be irreversible. But there does not seem to be anyway to prove that this is true.

References

1. ↑ Khan Academy Differentiability implies continuity https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-4/a/proof-differentiability-implies-continuity.
2. ↑ Ernest Z. Why is entropy of the universe increasing? https://socratic.org/questions/why-is-entropy-of-universe-increasing

Other Material

• See wiki page on Information in Physics