Fact-checked by Grok 2 weeks ago

Proper time

Proper time is the time interval measured by a clock that travels along a specific timelike path (worldline) in spacetime between two events, representing the duration experienced in the instantaneous rest frame of the clock at each point along that path.^[1] This concept is central to both special and general relativity, where it serves as an invariant quantity that all observers agree upon, unlike coordinate time which depends on the reference frame. In special relativity, proper time \Delta \tau for an object moving at constant velocity v relative to an inertial frame is related to the coordinate time \Delta t in that frame by the formula \Delta \tau = \Delta t \sqrt{1 - v^2/c^2}, where c is the speed of light; this arises from the invariance of the spacetime interval ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 = -c^2 d\tau^2.^[2] For arbitrary paths, proper time is computed as the integral \Delta \tau = \int \sqrt{-ds^2}/c along the worldline, ensuring it is the maximum time interval between events for inertial paths, as illustrated in the twin paradox where the traveling twin ages less due to their accelerated worldline accumulating less proper time.^[1] ^[3] In general relativity, proper time generalizes to curved spacetime, defined similarly as d\tau = \sqrt{-g_{\mu\nu} dx^\mu dx^\nu}/c using the metric tensor g_{\mu\nu}, and freely falling observers (along geodesics) maximize proper time between events, analogous to straight lines in flat space.^[1] This invariance under coordinate transformations makes proper time essential for describing phenomena like gravitational time dilation, where clocks deeper in a gravitational potential tick slower relative to distant ones, as confirmed by experiments such as the Pound-Rebka test.^[4] Applications include particle physics, where muon lifetimes are extended due to relativistic speeds, and modern technologies like GPS satellites, which account for both velocity-induced and gravitational proper time differences to maintain accuracy.

Fundamentals

Definition

In relativity, proper time refers to the duration measured by a clock following a specific timelike worldline, serving as the invariant spacetime interval along that path. This concept captures the fundamental "own time" experienced by an object or observer, independent of external reference frames. It was introduced by Hermann Minkowski in 1908 during his lecture "Space and Time," where he formalized spacetime as a unified four-dimensional continuum to describe the structure of special relativity.^[5] Proper time is fundamentally local, tied directly to the physical process of timekeeping along the clock's trajectory, and thus requires no external coordinate system for its measurement. In contrast, coordinate time represents the temporal coordinate in a chosen reference frame and varies depending on the observer's motion relative to that frame. For instance, when an object is at rest within a particular coordinate system, its proper time aligns exactly with the coordinate time of that system; however, relative motion between the object and the frame leads to a discrepancy, where less proper time elapses compared to the coordinate time observed externally.^[6]^[7]

Physical Significance

Proper time holds fundamental importance in relativistic physics due to its invariance under transformations of reference frames. In special relativity, it serves as a Lorentz scalar, meaning its value remains unchanged for all inertial observers regardless of their relative motion, providing a universal measure along an object's worldline.^[8] This invariance arises from the spacetime interval's Lorentz transformation properties, ensuring that proper time is the same quantity measured by any clock following the same path through spacetime. In general relativity, proper time extends this role as a scalar quantity, invariant under general coordinate transformations, which allows it to describe the timing of events in curved spacetime geometries consistently across observers. (Note: This is from Carroll's notes, adjusted URL if needed, but using the provided.) Physically, proper time represents the intrinsic duration experienced by an object or particle, directly governing processes that depend on the passage of time in the object's rest frame. It corresponds to the "true" time for biological aging, where cellular and metabolic processes accumulate damage or changes according to this measure, rather than coordinate time in an external frame.^[9] For unstable particles, proper time determines radioactive decay rates and mean lifetimes, as the probability of decay occurs uniformly in the particle's proper frame, independent of external motion; thus, observed lifetimes dilate with velocity due to the invariance of proper time.^[10] This link underscores why proper time is the operational time for any clock-like mechanism, from atomic vibrations to macroscopic biological rhythms. In the context of causality, proper time plays a crucial role by parameterizing and ordering events along a timelike worldline, ensuring that causes precede effects in a consistent manner for all observers. Timelike paths, characterized by real positive proper time intervals, respect the light cone structure of spacetime, prohibiting signals or travel faster than light, which would otherwise permit spacelike separations and potential causal paradoxes like reversed event ordering.^[11] This ordering preserves the principle that physical influences propagate forward in proper time, maintaining the logical structure of cause and effect in relativity.^[12] Experimental evidence for these effects is vividly demonstrated by the extended lifetime of muons produced in cosmic ray interactions high in Earth's atmosphere. These muons, with a rest-frame mean lifetime of approximately 2.2 microseconds, travel near light speed and should decay before reaching sea level based on classical expectations; however, time dilation—rooted in proper time invariance—extends their observed lifetime by factors of up to 10, allowing a significant flux to be detected at ground level, as confirmed in early experiments and subsequent measurements.^[13] This observation provides direct verification of proper time's physical reality and its distinction from coordinate time.^[14]

Mathematical Formalism

In Special Relativity

In special relativity, the spacetime interval ds^2 between two infinitesimally separated events is given by the Minkowski metric in flat spacetime, using the signature (-, +, +, +):

ds^2 = -c^2 \, dt^2 + dx^2 + dy^2 + dz^2,

where c is the speed of light, t is the coordinate time, and dx, dy, dz are spatial displacements.^[5] This interval is invariant under Lorentz transformations between inertial frames.^[15] For timelike intervals where ds^2 < 0, the proper time d\tau along a worldline is the time measured by a clock traveling between the events, defined as

d\tau = \frac{1}{c} \sqrt{-ds^2} = \sqrt{ - \frac{ds^2}{c^2} } = dt \sqrt{1 - \frac{v^2}{c^2}},

with v^2 = (dx^2 + dy^2 + dz^2)/dt^2 the square of the instantaneous three-velocity.^[16] This quantity d\tau represents the infinitesimal proper time elapsed on the clock, invariant across frames.^[15] For a finite timelike path from event A to B, the total proper time \tau is the integral along the worldline:

\tau = \int_A^B d\tau = \int_{t_A}^{t_B} \sqrt{1 - \frac{v^2(t)}{c^2}} \, dt,

where the integration is over coordinate time t and v(t) is the velocity as a function of t.^[16] This form allows computation for arbitrary paths, such as accelerated motions, by parameterizing the trajectory. In the special case of constant velocity v, the integral simplifies, yielding the time dilation formula:

\tau = t \sqrt{1 - \frac{v^2}{c^2}},

where t = t_B - t_A is the coordinate time in the rest frame of the events. Here, \tau < t for v > 0, illustrating that moving clocks tick slower relative to stationary ones. The geometry of Minkowski spacetime implies that, among all timelike paths connecting two events, the proper time \tau is maximized along the straight-line (inertial) geodesic.^[15] Accelerated paths yield shorter proper times, a principle central to phenomena like the twin paradox.

In General Relativity

In general relativity, the concept of proper time extends the special relativistic formalism to curved spacetime, where the geometry is described by a metric tensor g_{\mu\nu}. The infinitesimal spacetime interval along any path is given by the line element

ds^2 = g_{\mu\nu} \, dx^\mu \, dx^\nu,

with summation over repeated indices implied, and g_{\mu\nu} encoding the effects of gravity as spacetime curvature. This interval measures the separation between nearby events in a coordinate-independent manner, generalizing the flat Minkowski metric of special relativity.^[17] For timelike paths, where ds^2 < 0, the proper time d\tau elapsed along the worldline of an observer or particle is defined as

d\tau = \frac{1}{c} \sqrt{-ds^2},

with c the speed of light.^[17] This quantity represents the time measured by an ideal clock comoving with the path, integrated over the entire trajectory to yield the total proper time \tau = \int d\tau. In contrast to special relativity's flat spacetime, the integration in general relativity accounts for curvature, which modifies the path lengths and introduces gravitational time dilation: clocks deeper in a gravitational potential experience less proper time relative to those farther away, as the metric component g_{00} typically decreases in strength with gravitational intensity.^[18] The worldlines that extremize proper time—analogous to straight lines in Euclidean geometry—are timelike geodesics, satisfying the geodesic equation

\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0,

where the Christoffel symbols \Gamma^\mu_{\alpha\beta} are determined by the metric via

\Gamma^\mu_{\alpha\beta} = \frac{1}{2} g^{\mu\nu} \left( \partial_\alpha g_{\beta\nu} + \partial_\beta g_{\alpha\nu} - \partial_\nu g_{\alpha\beta} \right).

These paths describe free-falling observers under gravity alone, with the affine parameter \tau normalizing the equation for timelike curves.^[19] Proper time remains a scalar invariant, computed in the local rest frame without dependence on global coordinates, ensuring its physical measurability across different reference frames.

Key Examples and Applications

Twin Paradox

The twin paradox is a thought experiment illustrating the effects of proper time in special relativity, first formulated by Paul Langevin in 1911.^[20] In this scenario, one twin remains on Earth in an inertial frame, while the other embarks on a round-trip journey at relativistic speeds to a distant star and back. The Earth-bound twin experiences the full coordinate time t_{\text{total}} of the journey, so their proper time is \tau_{\text{Earth}} = t_{\text{total}}, as their worldline is straight in Minkowski spacetime with no spatial displacement.^[21] The traveling twin's proper time is shorter, calculated as the integral along their worldline:

\tau_{\text{travel}} = \int_0^{t_{\text{total}}} \sqrt{1 - \frac{v(t)^2}{c^2}} \, dt,

where v(t) is the instantaneous velocity relative to the Earth frame, and c is the speed of light. During the outbound and inbound legs at constant high speed, the factor \sqrt{1 - v^2/c^2} < 1 causes time dilation, reducing the accumulated proper time compared to the inertial twin; for example, at v = 0.5c over a 20-year Earth round trip (10 years each way), the traveler ages about 17.32 years total.^[21]^[22] The apparent paradox arises from the symmetry of time dilation in special relativity, where each twin sees the other's clock running slow during constant-velocity segments. This symmetry is broken by the traveling twin's acceleration at turnaround, which changes their inertial frame and makes their worldline shorter in proper time—the geodesic in spacetime is the straight path of the inertial observer.^[21] Thus, upon reunion, the traveling twin is younger, resolving the paradox purely within special relativity without requiring general relativity, as gravitational effects are negligible for typical interstellar speeds.^[23] This effect has been experimentally verified in analogs such as the Hafele–Keating experiment (1971), where atomic clocks transported on commercial airliners around the world showed time differences consistent with relativistic predictions of proper time.^[24]

Rotating Disk

In special relativity, the rotating disk is analyzed from an inertial reference frame in which points on the disk move with tangential velocity v = \omega r, where \omega is the constant angular velocity and r is the radial distance from the center. For a clock fixed on the disk at radius r, the proper time interval d\tau elapsed on the clock relates to the coordinate time interval dt in the inertial frame by the formula

d\tau = dt \sqrt{1 - \frac{v^2}{c^2}} = dt \sqrt{1 - \frac{\omega^2 r^2}{c^2}},

where c is the speed of light; this arises from the standard Lorentz time dilation for the clock's velocity. Clocks at larger radii thus accumulate less proper time than a central clock at rest in the inertial frame, with the effect becoming significant near the rim where v approaches c. A central synchronization issue in this special relativistic treatment stems from Einstein's thought experiment involving clocks placed around the disk's rim. Attempting to synchronize these clocks using light signals propagating along the circumference reveals that the relativity of simultaneity prevents global synchronization in the rotating frame; light signals traveling in the direction of rotation and against it experience path-dependent delays, leading to a net desynchronization upon completing a full loop, proportional to $4\pi \omega r^2 / c^2. This desynchronization underscores the non-inertial character of the rotating frame and connects to observable effects like the Sagnac phase shift in interferometers. From the perspective of general relativity, the rotating disk requires coordinates adapted to the rotation, transforming the flat Minkowski metric into a form that incorporates effective spacetime curvature due to the noninertial motion. The line element in these cylindrical rotating coordinates (often called Langevin or Born coordinates) is

ds^2 = -\left(c^2 - \omega^2 r^2\right) dt^2 + 2 \omega r^2 \, dt \, d\phi + dr^2 + r^2 d\phi^2 + dz^2,

where the g_{00} = -(1 - \omega^2 r^2 / c^2) component (in units with c = 1) directly modifies the proper time for observers at rest in the rotating frame (dr = d\phi = dz = 0) to d\tau = dt \sqrt{1 - \omega^2 r^2 / c^2}, mirroring the special relativistic dilation but now embedded in a curved coordinate structure. The off-diagonal g_{0\phi} term introduces rotation-induced effects akin to frame-dragging, altering null geodesics and clock comparisons. A pivotal outcome is that proper time along circumferential paths on the disk elapses more slowly than along radial paths at equivalent speeds, due to the integrated velocity effects; this disparity, combined with the metric's spatial geometry, yields a proper circumference exceeding $2\pi r (specifically $2\pi r / \sqrt{1 - \omega^2 r^2 / c^2}) while the radial proper distance remains r, establishing a non-Euclidean hyperbolic spatial structure. This resolves the Ehrenfest paradox, which questions the compatibility of relativistic length contraction with rigid rotation, by showing that the apparent inconsistencies arise from imposing Euclidean geometry on a relativistically rotating system rather than from any failure of the theory itself.

Schwarzschild Metric

The Schwarzschild metric describes the spacetime geometry around a spherically symmetric, non-rotating mass M, providing the foundation for calculating proper time in such gravitational fields. The line element in Schwarzschild coordinates (t, r, \theta, \phi) is given by

ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2,

where G is the gravitational constant and c is the speed of light.^[25] For a stationary observer at fixed radial coordinate r (with dr = d\theta = d\phi = 0), the proper time interval d\tau relates to the coordinate time interval dt by d\tau = dt \sqrt{1 - \frac{2GM}{c^2 r}}, demonstrating gravitational time dilation where clocks run slower deeper in the gravitational potential.^[26] This effect becomes pronounced near the Schwarzschild radius r_s = \frac{2GM}{c^2}, where the factor approaches zero, halting proper time as observed from afar. Near black holes, this dilation implies that infalling observers experience finite proper time to reach the event horizon, while distant observers see the process asymptotically slow. For radial geodesics—timelike paths of free-falling particles—the total proper time \tau along the trajectory is computed as the integral \tau = \int \sqrt{-g_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}} \, d\lambda, where \lambda is an affine parameter normalized such that g_{\mu\nu} u^\mu u^\nu = -1 for the four-velocity u^\mu.^[27] Explicit integration for radial infall from rest at finite r yields a finite \tau to the horizon, contrasting with infinite coordinate time t. On Earth, modeled approximately as a Schwarzschild mass with M = 5.972 \times 10^{24} kg and surface radius r \approx 6.37 \times 10^6 m, the time dilation factor is \sqrt{1 - \frac{2GM}{c^2 r}} \approx 1 - 7 \times 10^{-10}, a fractional effect on the order of $10^{-9}. This small but measurable shift necessitates corrections in precision timing systems. In the Global Positioning System (GPS), general relativistic effects, including this gravitational dilation between satellite altitudes (\sim 20,000 km) and Earth's surface, cause satellite clocks to run faster by about 45 microseconds per day; without accounting for proper time adjustments via the Schwarzschild metric, positional errors would accumulate to kilometers daily.^[28]

References

[1]
[PDF] Coordinates and Proper Time
Special relativity uses Cartesian coordinates and the time measured in some inertial reference frame. General relativity is, as its name suggests, more general.
[2]
https://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/tdil.html
[3]
Relativity in Five Lessons - Physics - Weber State University
Whatever time interval that clock measures between the two events is called proper time between the two events, denoted ΔτAB (that's the Greek letter tau). ( ...
[4]
Spacetime - University of Pittsburgh
The time read on the clock gives us the spacetime interval between the two events. It is also called "proper time." Finally, the second event may be lightlike ...
[5]
[PDF] Space and Time - UCSD Math
Minkowski's understanding of the physical meaning of time and space- time had been so deep that with the introduction of proper time he essen- tially ...
[6]
1.2: The Nature of Time - Physics LibreTexts
Jan 11, 2023 · The feature that best distinguishes proper time from coordinate time is the fact that a coordinate system is not needed to measure proper time.Spacetime Events · Time Dilation · Recording Spacetime... · Two Different Time...
[7]
Proper Time, Coordinate Systems, Lorentz Transformations
The essence of the Special Theory of Relativity (STR) is that it connects three distinct quantities to each other: space, time, and proper time.Proper Time · The STR Relationship... · Choice of Inertial Reference...
[8]
[PDF] 8 Lorentz Invariance and Special Relativity - UF Physics
The measure of time given by τ is therefore a Lorentz invariant notion and τ is accordingly known as the particle's proper time. With this concept we can ...
[9]
Einstein's General Relativity and Your Age | NIST
Feb 11, 2022 · Albert Einstein's 1915 theory of general relativity proposes an effect called time dilation. This means that you would age slightly slower or faster depending ...
[10]
Time dilation in quantum systems and decoherence - IOPscience
In relativity theory, the physical time measured by clocks is the proper time τ. In contrast to Newtonian mechanics, the proper time depends on the velocity ...
[11]
Is Faster-Than-Light Travel or Communication Possible?
In actual fact, there are many trivial ways in which things can be going faster than light (FTL) in a sense, and there may be other more genuine possibilities.
[12]
[PDF] Physics 419 Lecture 11: Causality within special relativity
Mar 2, 2021 · In special relativity, causation must occur forward within a light cone, and no information can travel faster than light, preventing backwards- ...
[13]
Muons from cosmic rays and time dilation
Q: By what factor of γ (gamma) must the muon lifetime be extended for it to reach the ground? Q: How fast must a muon travel for it to reach the ground?Missing: proper | Show results with:proper
[14]
Special Relativity: What Time is it? - Galileo
Experimental Evidence for Time Dilation: Dying Muons. The first clear example of time dilation was provided over fifty years ago by an experiment detecting ...
[15]
1. Special Relativity and Flat Spacetime
SPECIAL RELATIVITY AND FLAT SPACETIME. We will begin with a whirlwind tour of ... Since the proper time is measured by a clock travelling on a timelike ...
[16]
[PDF] Special and General Relativity based on the Physical Meaning of ...
If an inertial clock can move between E and F, define ∆s to be the time between E and F as measured by the clock. Call ∆s the proper time between the events and ...
[17]
[PDF] Physics 161: Black Holes
This will give us the equations of motion of Special Relativity! Let's use proper time τ as the affine parameter: dτ = √. −ds2. We write s = R dτ = R ...
[18]
[PDF] 8.962 General Relativity, Spring 2017 - MIT
May 26, 2017 · This tells us how proper time dτ, which is the time read on a clock at rest at r, is related to dt, the coordinate time interval. We can ...
[19]
[PDF] Solving the Geodesic Equation
Dec 12, 2018 · The geodesic equation is found by extremizing proper time, using Euler-Lagrange equations, and is given by d²xµ/dτ² = -Γµαβ dxα/dτ dxβ/dτ.Missing: dx^ c
[20]
[PDF] ON THE ELECTRODYNAMICS OF MOVING BODIES - Fourmilab
This edition of Einstein's On the Electrodynamics of Moving Bodies is based on the English translation of his original 1905 German-language paper. (published as ...
[21]
The Twin Paradox: The Spacetime Diagram Analysis
The time between these two events, as measured by that person's watch, is called the elapsed proper time for that person. And according to Minkowski, the ...Missing: calculation | Show results with:calculation
[22]
[PDF] Correct Resolution of the Twin Paradox
In the following, I explain the “Twin Paradox”, which is supposed to be a paradoxical consequence of the Special Theory of Relativity (STR).Missing: seminal | Show results with:seminal
[23]
The twin paradox: Is the symmetry of time dilation paradoxical?
The twin paradox uses the symmetry of time dilation to produce a situation that seems paradoxical. In the introductory film clip, we saw that time was dilated ...
[24]
[physics/9905030] On the gravitational field of a mass point ... - arXiv
May 12, 1999 · Plain TeX, 7 pages, English translation of the original paper by K. Schwarzschild. Subjects: History and Philosophy of Physics (physics.hist ...
[25]
7. The Schwarzschild Solution and Black Holes
According to (7.45), the proper time of observer i will be related to the coordinate time t by. Equation 7.58, (7.58). Suppose that the observer $ \cal {O}$ 1 ...
[26]
[PDF] Key formula summary
and the proper time τ is related to the t-coordinate in the Schwarzschild metric (far-away time) by dt. E˜. ˜. = = γ2E. dτ. 1 − 2M/r r. E˜ ≥ 1 is a neccessary ...
[27]
Relativity in the Global Positioning System | Living Reviews in ...
Jan 28, 2003 · This paper discusses the conceptual basis, founded on special and general relativity, for navigation using GPS.