Length of an object in the object's rest frame
Proper length
[1]
or
rest length
[2]
is the length of an object in the object's
rest frame
.
The measurement of lengths is more complicated in the
theory of relativity
than in
classical mechanics
. In classical mechanics, lengths are measured based on the assumption that the locations of all points involved are measured simultaneously. But in the theory of relativity, the notion of
simultaneity
is dependent on the observer.
A different term,
proper distance
, provides an invariant measure whose value is the same for all observers.
Proper distance
is analogous to
proper time
. The difference is that the proper distance is defined between two spacelike-separated events (or along a spacelike path), while the proper time is defined between two timelike-separated events (or along a timelike path).
Proper length or rest length
[
edit
]
The
proper length
[1]
or
rest length
[2]
of an object is the length of the object measured by an observer which is at rest relative to it, by applying standard measuring rods on the object. The measurement of the object's endpoints doesn't have to be simultaneous, since the endpoints are constantly at rest at the same positions in the object's rest frame, so it is independent of Δ
t
. This length is thus given by:
![{\displaystyle L_{0}=\Delta x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/574aac22e36bacd527b0885695b59f2e7432bfd4)
However, in relatively moving frames the object's endpoints have to be measured simultaneously, since they are constantly changing their position. The resulting length is shorter than the rest length, and is given by the formula for
length contraction
(with
γ
being the
Lorentz factor
):
![{\displaystyle L={\frac {L_{0}}{\gamma }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca69ed71aaf610930bc8e4ea1dbf0b3fa919740)
In comparison, the invariant proper distance between two arbitrary events happening at the endpoints of the same object is given by:
![{\displaystyle \Delta \sigma ={\sqrt {\Delta x^{2}-c^{2}\Delta t^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc33b3d5c379e31b0b84906682be174e38742da2)
So Δ
σ
depends on Δ
t
, whereas (as explained above) the object's rest length
L
0
can be measured independently of Δ
t
. It follows that Δ
σ
and
L
0
, measured at the endpoints of the same object, only agree with each other when the measurement events were simultaneous in the object's rest frame so that Δ
t
is zero. As explained by Fayngold:
[1]
- p. 407: "Note that the
proper distance
between two events is generally
not
the same as the
proper length
of an object whose end points happen to be respectively coincident with these events. Consider a solid rod of constant proper length
l
0
. If you are in the rest frame
K
0
of the rod, and you want to measure its length, you can do it by first marking its endpoints. And it is not necessary that you mark them simultaneously in
K
0
. You can mark one end now (at a moment
t
1
) and the other end later (at a moment
t
2
) in
K
0
, and then quietly measure the distance between the marks. We can even consider such measurement as a possible operational definition of proper length. From the viewpoint of the experimental physics, the requirement that the marks be made simultaneously is redundant for a stationary object with constant shape and size, and can in this case be dropped from such definition. Since the rod is stationary in
K
0
, the distance between the marks is the
proper length
of the rod regardless of the time lapse between the two markings. On the other hand, it is not the
proper distance
between the marking events if the marks are not made simultaneously in
K
0
."
Proper distance between two events in flat space
[
edit
]
In
special relativity
, the proper distance between two spacelike-separated events is the distance between the two events, as measured in an
inertial frame of reference
in which the events are simultaneous.
[3]
[4]
In such a specific frame, the distance is given by
![{\displaystyle \Delta \sigma ={\sqrt {\Delta x^{2}+\Delta y^{2}+\Delta z^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e017202d028f39be2f6845e4c73d787f1e67deeb)
where
The definition can be given equivalently with respect to any inertial frame of reference (without requiring the events to be simultaneous in that frame) by
![{\displaystyle \Delta \sigma ={\sqrt {\Delta x^{2}+\Delta y^{2}+\Delta z^{2}-c^{2}\Delta t^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fd213e66d8d64b76266a99d7d16a95bb1a21101)
where
The two formulae are equivalent because of the invariance of
spacetime intervals
, and since Δ
t
= 0 exactly when the events are simultaneous in the given frame.
Two events are spacelike-separated if and only if the above formula gives a real, non-zero value for Δ
σ
.
Proper distance along a path
[
edit
]
The above formula for the proper distance between two events assumes that the spacetime in which the two events occur is flat. Hence, the above formula cannot in general be used in
general relativity
, in which curved spacetimes are considered. It is, however, possible to define the proper distance along a
path
in any spacetime, curved or flat. In a flat spacetime, the proper distance between two events is the proper distance along a straight path between the two events. In a curved spacetime, there may be more than one straight path (
geodesic
) between two events, so the proper distance along a straight path between two events would not uniquely define the proper distance between the two events.
Along an arbitrary spacelike path
P
, the proper distance is given in
tensor
syntax by the
line integral
![{\displaystyle L=c\int _{P}{\sqrt {-g_{\mu \nu }dx^{\mu }dx^{\nu }}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f300fa3c14928ab14fdceece2e37562e976c1eb1)
where
In the equation above, the metric tensor is assumed to use the
+???
metric signature
, and is assumed to be normalized to return a
time
instead of a distance. The ? sign in the equation should be dropped with a metric tensor that instead uses the
?+++
metric signature. Also, the
should be dropped with a metric tensor that is normalized to use a distance, or that uses
geometrized units
.
See also
[
edit
]
References
[
edit
]
- ^
a
b
c
Moses Fayngold (2009).
Special Relativity and How it Works
. John Wiley & Sons.
ISBN
978-3527406074
.
{{
cite book
}}
: CS1 maint: location missing publisher (
link
)
- ^
a
b
Franklin, Jerrold (2010). "Lorentz contraction, Bell's spaceships, and rigid body motion in special relativity".
European Journal of Physics
.
31
(2): 291?298.
arXiv
:
0906.1919
.
Bibcode
:
2010EJPh...31..291F
.
doi
:
10.1088/0143-0807/31/2/006
.
S2CID
18059490
.
- ^
Poisson, Eric; Will, Clifford M. (2014).
Gravity: Newtonian, Post-Newtonian, Relativistic
(illustrated ed.). Cambridge University Press. p. 191.
ISBN
978-1-107-03286-6
.
Extract of page 191
- ^
Kopeikin, Sergei; Efroimsky, Michael; Kaplan, George (2011).
Relativistic Celestial Mechanics of the Solar System
. John Wiley & Sons. p. 136.
ISBN
978-3-527-63457-6
.
Extract of page 136