7.1 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Force | 9/11 | https://en.wikipedia.org/wiki/Force | reference | science, encyclopedia | 2026-05-05T09:33:04.607426+00:00 | kb-cron |
remains valid because it is a mathematical definition. But for momentum to be conserved at relativistic relative velocity,
v
{\displaystyle v}
, momentum must be redefined as:
p
=
m
0
v
1
−
v
2
/
c
2
,
{\displaystyle \mathbf {p} ={\frac {m_{0}\mathbf {v} }{\sqrt {1-v^{2}/c^{2}}}},}
where
m
0
{\displaystyle m_{0}}
is the rest mass and
c
{\displaystyle c}
the speed of light. The expression relating force and acceleration for a particle with constant non-zero rest mass
m
{\displaystyle m}
moving in the
x
{\displaystyle x}
direction at velocity
v
{\displaystyle v}
is:
F
=
(
γ
3
m
a
x
,
γ
m
a
y
,
γ
m
a
z
)
,
{\displaystyle \mathbf {F} =\left(\gamma ^{3}ma_{x},\gamma ma_{y},\gamma ma_{z}\right),}
where
γ
=
1
1
−
v
2
/
c
2
.
{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.}
is called the Lorentz factor. The Lorentz factor increases steeply as the relative velocity approaches the speed of light. Consequently, the greater and greater force must be applied to produce the same acceleration at extreme velocity. The relative velocity cannot reach
c
{\displaystyle c}
. If
v
{\displaystyle v}
is very small compared to
c
{\displaystyle c}
, then
γ
{\displaystyle \gamma }
is very close to 1 and
F
=
m
a
{\displaystyle \mathbf {F} =m\mathbf {a} }
is a close approximation. Even for use in relativity, one can restore the form of
F
μ
=
m
A
μ
{\displaystyle F^{\mu }=mA^{\mu }}
through the use of four-vectors. This relation is correct in relativity when
F
μ
{\displaystyle F^{\mu }}
is the four-force,
m
{\displaystyle m}
is the invariant mass, and
A
μ
{\displaystyle A^{\mu }}
is the four-acceleration. The general theory of relativity incorporates a more radical departure from the Newtonian way of thinking about force, specifically gravitational force. This reimagining of the nature of gravity is described more fully below.
=== Quantum mechanics ===
Quantum mechanics is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is different from that of classical physics. Instead of thinking about quantities like position, momentum, and energy as properties that an object has, one considers what result might appear when a measurement of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result. The expectation value for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence. In quantum mechanics, interactions are typically described in terms of energy rather than force. The Ehrenfest theorem provides a connection between quantum expectation values and the classical concept of force, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, the Born rule is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law, with a force defined as the negative derivative of the potential energy. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance. Quantum mechanics also introduces two new constraints that interact with forces at the submicroscopic scale and which are especially important for atoms. Despite the strong attraction of the nucleus, the uncertainty principle limits the minimum extent of an electron probability distribution and the Pauli exclusion principle prevents electrons from sharing the same probability distribution. This gives rise to an emergent pressure known as degeneracy pressure. The dynamic equilibrium between the degeneracy pressure and the attractive electromagnetic force give atoms, molecules, liquids, and solids stability.
=== Quantum field theory ===