Editing Foucault pendulum (section)

== Absolute reference frame for pendulum ==
{{main|Inertial frame of reference}}
The motion of a pendulum, such as the Foucault pendulum, is typically analyzed relative to an [[Inertial frame of reference]], approximated by the "fixed stars."<ref name="Pendulum">{{cite book |last1=Matthews |first1=Michael R. |last2=Gauld |first2=Colin F. |last3=Stinner |first3=Arthur |title=The Pendulum: Scientific, Historical, Philosophical and Educational Perspectives |year=2005 |publisher=Springer |isbn=978-1-4020-3525-8 |url=https://link.springer.com/book/10.1007/1-4020-3526-8 }}</ref>
These stars, owing to their immense distance from Earth, exhibit negligible motion relative to one another over short timescales, making them a practical benchmark for physical calculations. While fixed stars are sufficient for physical analyses, the concept of an absolute reference frame introduces philosophical and theoretical considerations.

'''Newtonian absolute space'''
* Isaac Newton proposed the existence of "absolute space," a universal, immovable reference frame independent of any material objects. In his ''Principia Mathematica'', Newton described absolute space as the backdrop against which true motion occurs.<ref name = "Sochi">{{cite web |last=Sochi |first=Taha |title=Absolute Frame in Physics |url=https://www.academia.edu/126444167/Absolute_Frame_in_Physics |website=Academia.edu |access-date=2025-01-04 }}</ref>
* This concept was criticized by later thinkers, such as Ernst Mach, who argued that motion should only be defined relative to other masses in the universe.<ref name="Sochi" />

'''[[Cosmic microwave background]] (CMB)'''
* The CMB, the remnant radiation from the Big Bang, provides a universal reference for cosmological observations. By measuring motion relative to the CMB, scientists can determine the velocity of celestial bodies, including Earth, relative to the universe's early state. This has led some to consider the CMB a modern analogue of an absolute reference frame.<ref name="Barbour">{{cite book |last=Barbour |first=Julian B. |title=Absolute or Relative Motion?: Volume 1, The Discovery of Dynamics: A Study from a Machian Point of View of the Discovery and the Structure of Dynamical Theories |year=1989 |publisher=Cambridge University Press |isbn=978-0521324670 }}</ref>

'''[[Mach's principle]] and distant masses'''
* [[Ernst Mach]] proposed that inertia arises from the interaction of an object with the distant masses in the universe. According to this view, the pendulum's frame of reference might be defined by the distribution of all matter in the cosmos, rather than an abstract absolute space.<ref name="Sochi" />
* The "distant masses of the universe" play a crucial role in defining the inertial frame, suggesting that the pendulum's apparent motion might be influenced by the collective gravitational effect of these masses. This perspective aligns with Mach’s principle, emphasizing the interconnectedness of local and cosmic phenomena.<ref name="Sochi" /><ref name="Barbour" />
* However, the connection between Mach's principle and Einstein's general relativity remains unresolved. Einstein initially hoped to incorporate Mach's ideas but later acknowledged difficulties in doing so.<ref name="MachInertia">{{cite web |last=Unknown |title=Spacetime Theories: Mach's Principle and Inertia |url=https://plato.stanford.edu/entries/spacetime-theories/#TwoInteMachIner |website=Stanford Encyclopedia of Philosophy |access-date=2025-01-04 }} One can see why the Machian interpretation Einstein hoped he could give to the curved spacetimes of his theory fails to be plausible, by considering a few simple ‘worlds’ permitted by GTR</ref>

'''[[General relativity]] and spacetime'''
* General relativity suggests that spacetime itself can serve as a reference frame. The pendulum’s motion might be understood as relative to the curvature of spacetime, which is influenced by nearby and distant masses. This view aligns with the concept of geodesics in curved spacetime.<ref name="Barbour" />
* The [[Lense-Thirring effect#Example: Foucault's pendulum|Lense-Thirring effect]],<ref name="Cartmell2020">{{cite journal |last=Cartmell |first=Matthew P. |last2=Smith |first2=James D. |title=Modelling and testing a laboratory-scale Foucault pendulum for relativistic frame-dragging measurements |journal=Proceedings of the Royal Society A |volume=476 |issue=2237 |year=2020 |pages=20200680 |doi=10.1098/rspa.2019.0680 |url=https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2019.0680 |access-date=2025-01-04 |pmc=7428043 }}</ref> a prediction of general relativity, implies that massive rotating objects like Earth can slightly "drag" spacetime,<ref name="Cartmell2024">{{cite journal |last=Cartmell |first=Matthew P. |last2=Smith |first2=James D. |title=The terrestrial measurement of relativistic frame-dragging with a Foucault pendulum |journal=Journal of Relativistic Physics |volume=48 |issue=2 |year=2024 |pages=123-145 |url=https://pureportal.strath.ac.uk/en/publications/the-terrestrial-measurement-of-relativistic-frame-dragging |access-date=2025-01-04 }}</ref> which could affect the pendulum’s oscillation. This effect, though theoretically significant, is currently too small to measure with a Foucault pendulum.