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Simultaneity

Simultaneity refers to the property of two or more events occurring at the same time, a concept that appears straightforward in everyday experience but reveals profound complexities in physics and philosophy.^[1] In classical mechanics, simultaneity is considered absolute, meaning that if two spatially separated events happen simultaneously in one inertial frame, they do so in all frames, independent of relative motion between observers.^[2] However, Albert Einstein's special theory of relativity, introduced in 1905, revolutionized this understanding by establishing the relativity of simultaneity, where the perceived simultaneity of distant events depends on the observer's state of motion.^[3] This relativity arises from two foundational postulates of special relativity: the laws of physics are the same in all inertial frames, and the speed of light in vacuum is constant for all observers, regardless of the motion of the source or observer.^[1] To determine simultaneity, observers typically use light signals: two events are deemed simultaneous in a given frame if light emitted from each event reaches a midpoint observer at the same time, assuming equal distances.^[2] For example, consider two light flashes emitted from the ends of a rail car at rest relative to a platform observer at the midpoint; both flashes arrive simultaneously, indicating simultaneity in that frame.^[1] Yet, for an observer moving with the rail car, the midpoint shifts during light travel, causing the flashes to arrive at different times, thus judging the events non-simultaneous.^[2] Philosophically, the conventionality of simultaneity underscores that defining simultaneity for distant events involves a choice, as Einstein noted that such relations lack an absolute answer without convention.^[3] Hans Reichenbach later formalized this with the ε-synchronization parameter, where the standard Einstein synchrony (ε = 1/2) assumes isotropic light speed, but other values yield equivalent but nonstandard frames, all consistent with relativity's causal structure.^[3] These implications extend to spacetime geometry, where simultaneity defines "planes of simultaneity" slicing the Minkowski spacetime, affecting notions of presentism versus eternalism in the philosophy of time.^[3] Neither observer's judgment is privileged; both are valid within their frames, challenging intuitive absolute time and highlighting relativity's core insight that time is not universal.^[2]

Etymology and Basic Concepts

Etymology

The term "simultaneity" derives from the Latin adverb simul, meaning "at the same time" or "together," combined with the suffix "-taneity," a variant of "-ity" that indicates a state, condition, or quality.^[4]^[5] This formation parallels the adjective "simultaneous," which entered English in the 1650s from Medieval Latin simultaneus.^[6] The suffix integration reflects a common pattern in English for abstract nouns describing temporal or relational properties, emphasizing the notion of concurrence.^[7] The word's first recorded use in English appears in the mid-17th century, prior to 1651, in the philosophical and theological writings of Nathaniel Culverwell, an English thinker known for his work An Elegant and Learned Discourse of the Light of Nature (published posthumously in 1652).^[5] This early adoption occurred amid growing interest in temporal concepts during the Scientific Revolution, where English scholars began incorporating Latin-derived terms to articulate ideas about time and events.^[5] Culverwell's usage likely drew from broader Continental influences, marking the term's transition into philosophical lexicon as a descriptor for events occurring without temporal succession.^[5] By the late 17th century, "simultaneity" gained traction in English philosophical texts, influenced by Gottfried Wilhelm Leibniz's explorations of time, coexistence, and relational order in works such as his correspondence with Samuel Clarke (1715–1716).^[8] Leibniz's relational view of time, where simultaneity arises from the non-successive alignment of states across substances, helped embed the term in debates on absolute versus relative temporality, though his writings were primarily in Latin and French.^[8] English translations and commentaries on Leibniz's ideas, including those by philosophers like George Berkeley, further popularized the word in intellectual circles.^[8] In the late 19th and early 20th centuries, the term evolved within scientific literature, particularly through discussions of time measurement and synchronization by physicists such as Henri Poincaré, whose analyses in works like Science and Hypothesis (1902) refined its application to precise temporal relations.^[9] Poincaré's emphasis on the conventional aspects of time determination integrated "simultaneity" into emerging frameworks for understanding event coordination, bridging philosophy and empirical inquiry.^[9] This period saw the word's usage expand beyond metaphysics into technical contexts, setting the stage for its 20th-century prominence while retaining its core etymological sense of co-temporality.^[5]

Core Definition

Simultaneity refers to the coincidence in time of two or more events, where they occur at the same perceived instant. This concept captures the exact temporal overlap of occurrences, independent of any spatial separation between them in everyday human intuition.^[10]^[11] Unlike synchrony, which implies a deliberate coordination or alignment of timing—such as in synchronized movements or clock adjustments—simultaneity denotes mere temporal concurrence without requiring any orchestrated relation.^[12] Coincidence, by contrast, often encompasses chance occurrences that may involve both time and space, whereas simultaneity focuses strictly on shared timing.^[13] Common examples illustrate this intuitively: two people clapping hands precisely at the same moment during a performance demonstrates simultaneity in action. Likewise, multiple traffic lights in a city shifting from green to red at the identical instant helps regulate flow without implying prior coordination between each light. In classical physics, such simultaneity aligns with the assumption of absolute time, where temporal coincidence is universal across observers.^[14]

Simultaneity in Classical Physics

Absolute Simultaneity in Newtonian Mechanics

In Newtonian mechanics, simultaneity is conceived as an absolute and objective feature of the universe, grounded in Isaac Newton's framework of absolute space and absolute time. As outlined in the Scholium to the Definitions in his Philosophiæ Naturalis Principia Mathematica (1687), absolute time flows uniformly without dependence on external factors, such as the motion of bodies or observable phenomena, ensuring that durations and temporal coincidences are invariant across all reference frames.^[14] This uniformity implies that the "moment of duration" is identical at any location, rendering simultaneity a universal property independent of observers or their states of motion.^[15] Absolute space complements this by providing an immutable, homogeneous backdrop that remains similar and immovable, unaffected by the changes within it.^[14] Events occurring at the same absolute time are thus deemed simultaneous regardless of spatial separation or relative velocities, as time coordinates every point in the absolute framework without variation. Newton emphasized this distinction from relative measures, such as apparent solar days, which fluctuate and serve only as sensible approximations of the underlying absolute duration.^[16] For instance, the rotation of the Earth might suggest varying day lengths, but absolute time proceeds equably, guaranteeing that simultaneity holds objectively.^[15] The implications for classical mechanics are profound: this absolute simultaneity underpins the predictability and universality of physical laws, allowing for the consistent application of forces and motions across the cosmos. Events simultaneous in one inertial frame—such as the synchronized striking of distant clocks—are necessarily simultaneous in all others, facilitating the ideal of universal clock synchronization without need for frame-specific adjustments.^[17] In this view, mechanics operates within a single, all-encompassing temporal structure, where observer independence ensures that causal relations and dynamical equations remain unaltered by perspective.^[14]

Measurement and Synchronization in Classical Contexts

In classical physics, the synchronization of mechanical clocks relied on the assumption of absolute time, where clocks at rest relative to one another could be aligned without accounting for propagation delays in signals. Mechanical clocks, such as pendulum or balance-wheel designs, were typically synchronized by transporting them slowly between locations, ensuring they remained in phase due to the uniformity of time flow in Newtonian mechanics. For more precise alignment over short distances, a common method involved starting the clocks simultaneously using a visual or auditory cue, like a flash or sound from a central point equidistant from both, under the approximation that signal speeds were either infinite or sufficiently fast to neglect delays. This approach was foundational in laboratory settings and early engineering applications, where distances were limited to human scales.^[18] Light signals emerged as a practical tool for clock synchronization in classical contexts, particularly when assuming the speed of light was either instantaneous or its finite value resulted in negligible travel times for the distances involved. In pre-relativistic physics, light from a source placed midway between two clocks could trigger both devices via photocells or visual observation, with the equal-distance assumption ensuring the signals arrived simultaneously. This method, akin to an early form of the synchronization convention later formalized by Einstein, involved sending light signals over equal distances to align clocks, treating the propagation as effectively instantaneous for synchronization purposes. Such techniques were employed in 19th-century astronomical observations and precision measurements, where light's known speed (measured by Roemer in 1676) was considered but often ignored for practical short-range applications due to minimal delays.^[18]^[19] A prominent historical application of synchronization techniques occurred in 19th-century railroad networks, where telegraph systems transmitted time signals to coordinate schedules across vast distances. Starting in the 1840s, British railways adopted "Railway Time" based on Greenwich Mean Time, with electric telegraphs invented by Cooke and Wheatstone in 1837 enabling station masters to adjust mechanical clocks using precise signals from the Royal Observatory. By 1852, daily time signals were sent nationwide via telegraph, standardizing operations and preventing collisions; this system assumed near-instantaneous signal propagation over wires, analogous to the negligible-delay approximation for light signals. In the United States, on November 18, 1883, railroads synchronized clocks across four time zones using telegraph signals from observatories, marking a continental shift to standard time and influencing global practices formalized at the 1884 International Meridian Conference. These efforts underscored the engineering necessity of synchronized clocks in classical frameworks, bridging Newtonian absolute time with practical signal-based methods.^[20]^[21]^[22]

Simultaneity in Special Relativity

Relativity of Simultaneity Principle

In special relativity, the relativity of simultaneity principle asserts that the notion of two events occurring at the same time is not absolute but depends on the observer's inertial frame of reference. This principle arises directly from the postulate that the speed of light in vacuum is constant for all observers, regardless of their relative motion. In his seminal 1905 paper, Albert Einstein defined time and simultaneity operationally using light signals between synchronized clocks, showing that what constitutes "simultaneous" events varies between frames moving at constant velocity relative to each other.^[23] Specifically, Einstein argued that "we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system."^[23] The key insight is that events deemed simultaneous in one inertial frame—meaning light signals from those events reach a central observer at the same moment—will generally not appear simultaneous to an observer in a frame moving relative to the first. This relativity stems from the invariance of the speed of light, which precludes a universal "now" across space, challenging the classical assumption of absolute time. For instance, if two distant events are synchronized in a stationary frame, an observer moving parallel to the line connecting them will perceive one event as occurring before the other due to the differing paths light must travel to reach them.^[23] Einstein illustrated this principle through a thought experiment involving a train and lightning strikes, detailed in his 1916 popular exposition of relativity. Imagine a train moving at high speed past a platform, with an observer M stationary at the midpoint of the platform and another observer M' at the midpoint inside the train. Lightning strikes the track ends A and B simultaneously as seen by M, with light from both reaching M at the same time since M is equidistant. However, M' on the moving train sees the light from the forward strike B arrive first because the train's motion carries M' toward that light while away from the light from A, thus perceiving the strike at B as occurring before the one at A.^[24] This demonstrates how simultaneity is observer-dependent, with no preferred frame to resolve the discrepancy.

Mathematical Formulation via Lorentz Transformations

In special relativity, the Lorentz transformations provide the mathematical framework for relating spatial and temporal coordinates between two inertial reference frames moving at constant relative velocity v along the x-axis, with the origins coinciding at t = t' = 0. These transformations, derived from the constancy of the speed of light and the relativity principle, are given by

x' = \gamma (x - vt), \quad y' = y, \quad z' = z, \quad t' = \gamma \left( t - \frac{vx}{c^2} \right),

where \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} is the Lorentz factor, c is the speed of light, and the primed frame moves with velocity v relative to the unprimed frame.^[23] To quantify the relativity of simultaneity, consider two spatially separated events that are simultaneous in the unprimed frame, meaning they occur at the same time t_1 = t_2 = t but at different positions x_1 and x_2, so \Delta x = x_2 - x_1 and \Delta t = 0. In the primed frame, the time difference between these events is \Delta t' = t_2' - t_1' = \gamma \left( t_2 - \frac{v x_2}{c^2} \right) - \gamma \left( t_1 - \frac{v x_1}{c^2} \right) = -\gamma \frac{v \Delta x}{c^2}. This shows that \Delta t' \neq 0 unless v = 0 or \Delta x = 0, demonstrating that simultaneity is frame-dependent.^[23] Conversely, for events simultaneous in the primed frame (\Delta t' = 0), the Lorentz transformation implies a non-zero \Delta t in the unprimed frame. Setting \Delta t' = 0 yields \Delta t = \gamma \frac{v \Delta x'}{c^2} (where \Delta x' is the spatial separation in the primed frame), which is zero only if v = 0 or \Delta x' = 0. Thus, the condition for simultaneity—equal times for spatially separated events—holds only within the same frame or for coincident events, underscoring the loss of absolute simultaneity.^[25]

Simultaneity in General Relativity

Effects of Curved Spacetime

In general relativity, formulated by Albert Einstein in 1915, simultaneity is conceptualized through spacelike hypersurfaces that foliate the curved spacetime manifold, extending the relativity of simultaneity from special relativity to account for gravitational effects. These hypersurfaces represent slices of "now" for observers, but their geometry is distorted by the curvature induced by mass-energy, making global simultaneity observer-dependent and frame-specific.^[26] In the local limit, where curvature is negligible, this reduces to the flat spacetime description of special relativity.^[26] For observers following geodesic congruences—bundles of timelike geodesics representing worldlines in free fall—simultaneity is established along a foliation of these hypersurfaces orthogonal to the four-velocity field.^[27] However, the proper time measured along these geodesics varies with gravitational potential, as the norm of the Killing vector (in stationary spacetimes) decreases deeper in the potential well, leading to differential aging and altered simultaneity relations between events.^[27] This variation arises because the connection form defining simultaneity incorporates gravitational influences, such as acceleration and the Riemann curvature tensor, ensuring that events deemed simultaneous on one foliation may not align on another due to the inhomogeneous curvature.^[26] A concrete illustration occurs with atomic clocks placed at different altitudes in Earth's gravitational field, where the higher clock accumulates proper time faster due to weaker curvature.^[28] In a 2010 NIST experiment, two aluminum-ion clocks separated by 33 cm showed the upper clock ticking faster by about 4 parts in 10^17 per second, resulting in a desynchronization of roughly 90 billionths of a second over 79 years.^[28] A more recent 2022 JILA experiment using strontium atomic clocks measured the effect at an even smaller scale of 1 mm separation, detecting a fractional frequency shift of approximately 2 × 10^{-18}, 75 times smaller than the 2010 result and confirming general relativity's predictions with unprecedented precision.^[29] These demonstrations show how spacetime curvature tilts the "now" slices, rendering simultaneity relative even for stationary observers at varying heights.^[28]

Synchronization in Gravitational Fields

In general relativity, the Einstein synchronization procedure, originally developed for flat spacetime, is adapted to gravitational fields by employing light signals propagated along null geodesics to coordinate clocks. This method, known as radar synchronization, assigns a time T to an event as the average of the emission and reception times of a light signal: T = \frac{1}{2}(t_e + t_r), where t_e is the coordinate time of emission from a reference clock and t_r is the reception time, with the spatial separation defined similarly as R = \frac{1}{2} c (t_r - t_e). In the Schwarzschild metric describing the spacetime around a spherically symmetric mass, such as a star or black hole exterior, null geodesics allow for local synchronization within regions where light rays connect events uniquely, typically outside the photon sphere at r = 3M (in units where G = c = 1, with M the mass). However, gravitational lensing and the focal properties of the metric limit global applicability, as events in shadow regions beyond a focal radius r_f \approx \frac{r_*^2}{4M} (for observer at radial coordinate r_*) cannot be reached by bidirectional light signals.^[30] A practical implementation of synchronization accounting for gravitational effects is seen in the Global Positioning System (GPS), where atomic clocks on satellites must be adjusted for time dilation due to Earth's gravitational field. Satellite clocks, orbiting at altitudes of approximately 20,200 km, experience a weaker gravitational potential, causing them to run faster by about 45.7 microseconds per day relative to ground clocks purely from general relativity, though special relativistic velocity effects partially offset this by slowing them by 7.2 microseconds per day, yielding a net gain of 38.5 microseconds per day. To maintain simultaneity for precise positioning, satellite clocks are preset at launch with a frequency offset of -4.4647 \times 10^{-10} relative to their nominal 10.23 MHz rate, ensuring that proper time on the satellites aligns with coordinate time on Earth; receivers then apply additional corrections for orbital eccentricity and other perturbations, achieving synchronization accuracy within nanoseconds. These adjustments demonstrate how gravitational redshift is compensated to define a shared time frame across varying potentials.^[31] In strong gravitational fields near black holes, establishing global simultaneity becomes problematic, as event horizons prevent the return of light signals necessary for two-way synchronization. In the Schwarzschild geometry, while local radar synchronization is possible outside the horizon using ingoing and outgoing null geodesics, crossing the event horizon renders bidirectional communication impossible: signals can be sent inward but not received back, breaking the standard Einstein procedure and prohibiting a consistent global hypersurface of simultaneity that foliates the entire spacetime without causal violations. This failure arises because the horizon acts as a one-way null surface, where time-like hypersurfaces cannot extend globally without intersecting or becoming space-like inside the horizon, complicating the definition of "simultaneous" events across the boundary.^[32]

Philosophical and Conceptual Implications

Epistemological Challenges

In special relativity, the relativity of simultaneity establishes that whether two spatially separated events occur at the same time depends on the observer's inertial frame, posing a fundamental epistemological challenge to verifying an absolute "now." This frame-relativity implies that empirical confirmation of simultaneity is inherently observer-dependent, as no universal standard exists to adjudicate between conflicting assessments from different frames, undermining claims to objective knowledge of temporal order for distant events.^[33] For instance, light-signal synchronization methods, while practical within a given frame, yield varying results across frames, highlighting the limits of observational access to spacetime structure.^[34] Einstein's epistemological perspective on time perception was significantly shaped by the influences of Henri Poincaré and Ernst Mach, who emphasized the conventional and empirical underpinnings of temporal concepts. Poincaré's analysis of scientific conventions, particularly in geometry and measurement, led Einstein to view time coordination as reliant on definitional choices rather than absolute truths, influencing his rejection of Newtonian absolute time in favor of relational frameworks.^[34] Similarly, Mach's critique of mechanistic absolutes and advocacy for concepts grounded in sensory experience prompted Einstein to prioritize empirical verifiability in redefining simultaneity, though he diverged from Mach's full sensationalism by integrating constructive theoretical elements.^[34] These influences fostered Einstein's holistic epistemology, where knowledge of time emerges from coordinated theoretical and observational practices rather than isolated perceptions.^[34] The conventionality of simultaneity debate further complicates epistemological claims, questioning whether simultaneity constitutes a discoverable physical fact or merely a coordinate choice within a frame. Hans Reichenbach argued that synchronization conventions, such as his ε-parameter for light signals (where ε=1/2 yields Einstein's standard but other values are empirically equivalent), introduce an element of arbitrariness, as no observation can uniquely determine the one-way speed of light or distant simultaneity without presupposing a convention. This view implies that our knowledge of temporal relations is partially conventional, limited by the underdetermination of empirical data, and challenges realist interpretations of spacetime.^[35] Counterarguments, like David Malament's causal structure analysis, contend that Minkowski spacetime's conformal properties fix a unique simultaneity relation, rendering alternatives non-physical and restoring objectivity to verification processes. Despite such critiques, the debate persists, underscoring the tension between empirical accessibility and theoretical interpretation in understanding simultaneity.^[33]

Metaphysical Interpretations

The relativity of simultaneity in special relativity has profound metaphysical implications, leading to interpretations of time as a static, four-dimensional block where all events—past, present, and future—coexist eternally without a privileged "now." This block universe theory posits that spacetime forms a fixed structure, akin to a loaf of bread where every slice represents a possible moment, but the entire loaf exists timelessly; observers merely experience different "slicings" of this block depending on their frame of reference.^[36] In this view, simultaneity is subjective, a perceptual artifact rather than an objective feature of reality, challenging the intuitive notion of time as flowing from past to future.^[36] These implications extend to the debate between presentism and eternalism, two competing metaphysical theories of time's ontology. Presentism asserts that only the present exists, with the past no longer real and the future not yet real, relying on an absolute, universal "now" defined by global simultaneity.^[37] Eternalism, conversely, holds that all temporal points are equally real, treating time like space where past, present, and future events persist unchangingly in the spacetime manifold.^[37] Special relativity favors eternalism because the relativity of simultaneity undermines any objective present; events simultaneous for one observer are not for another, implying no frame-independent "now" to privilege the present over other times.^[37] This alignment is formalized in the Rietdijk–Putnam argument, which demonstrates that if an event is real in one frame, it must be equally real across all frames, entailing the co-reality of all spacetime events and thus eternalism over presentism.^[38]^[39] Further metaphysical challenges arise from general relativity's allowance of closed timelike curves (CTCs), paths in spacetime that loop back on themselves, permitting an observer to return to their own past.^[40] Such curves enable causal loops, where an effect precedes its cause in a self-consistent cycle without external origin—for instance, an object or information passed backward in time becomes the source of its own existence, as in the bootstrap paradox.^[41] These loops disrupt intuitive simultaneity by blurring temporal order: events along a CTC are causally connected yet cyclically simultaneous from the traveler's perspective, violating linear causality and suggesting a reality where cause and effect are not strictly sequenced.^[40]^[41] The Novikov self-consistency principle resolves potential paradoxes by positing that only consistent histories occur in CTCs, preserving the block universe's timeless coherence but at the cost of intuitive notions of temporal precedence.^[41]

Applications in Other Disciplines

In Computing and Information Theory

In parallel computing, simultaneity is achieved through mechanisms like simultaneous multithreading (SMT), which enables multiple threads to execute concurrently on multi-core processors by issuing instructions from independent threads to the processor's functional units in the same cycle. This approach maximizes on-chip parallelism, potentially delivering up to four times the throughput of traditional superscalar processors and twice that of fine-grain multithreading systems, by improving resource utilization amid varying thread workloads.^[42] Distributed systems face significant challenges in establishing simultaneity due to network delays and the lack of a shared global clock, leading to difficulties in determining whether events occur concurrently across nodes. Leslie Lamport's logical clocks address this by imposing a partial ordering on events via the "happened before" relation, where events are timestamped to reflect causal dependencies rather than physical time, allowing for consistent total ordering without assuming synchronized physical clocks.^[43] The Network Time Protocol (NTP) further mitigates these issues by synchronizing clocks across internet-scale networks to within milliseconds, using hierarchical time servers and delay-minimizing algorithms to filter out variable propagation delays.^[44] These protocols handle network asymmetries in ways that mimic the relativity of simultaneity in physics, where observers disagree on event timing due to relative motion; recent work formalizes this analogy in relativistic distributed computing models, showing how light-speed limits and simultaneity relativity impact algorithm correctness in interplanetary or high-velocity networks.^[45] In quantum computing, superposition enables qubits to exist in multiple states simultaneously until measurement, allowing quantum algorithms to explore vast solution spaces in parallel, a capability unattainable by classical bits confined to single states. Richard Feynman's seminal proposal for quantum simulators highlighted how this superposition permits efficient computation of quantum mechanical evolutions by processing superposed configurations concurrently, addressing the exponential complexity that hampers classical simulations.^[46] David Deutsch extended this by defining a universal quantum computer that operates on superposed inputs to perform inherently quantum computations, such as evaluating functions across all possible inputs at once, thereby establishing the foundational architecture for scalable quantum parallelism.

In Logic and Mathematics

In logic and mathematics, simultaneity is formalized as a relation capturing events or states that occur at the same temporal instant within structured models of time. In temporal logic, particularly Linear Temporal Logic (LTL), simultaneity is represented through propositional conjunction, where multiple propositions hold true concurrently at a given time point in a linear sequence of instants. LTL, introduced by Amir Pnueli, extends classical propositional logic with temporal operators such as "globally" (G, denoting "always") and "eventually" (F, denoting "sometimes"), which quantify over future instants, while conjunction (∧) ensures simultaneity within each instant.^[47] For example, a formula like p \land q asserts that propositions p and q are true simultaneously at the current time step, enabling the specification of concurrent behaviors in reactive systems without interleaving assumptions. This framework supports model checking to verify properties over infinite execution paths, where simultaneity aids in defining atomic snapshots of system states.^[47] In set-theoretic models of concurrency, simultaneous events are elements within a partial order augmented by a simultaneity relation, distinguishing them from causally ordered or independent occurrences. Leslie Lamport's foundational work on event ordering defines a partial order via the "happens-before" relation (→), where events a and b are concurrent (and thus potentially simultaneous) if neither a → b nor b → a holds, forming equivalence classes without temporal precedence.^[48] Extending this, Peter H. Mills proposes prossets—partial orders with an explicit simultaneity relation (=)—to model concurrent systems, where unrelated events in the partial order can be grouped via = to represent true simultaneity, as in linear sequences of simultaneous event multisets.^[49] Timestamps in such structures, often assigned via logical clocks, label events with equal values to denote simultaneity, preserving the partial order while resolving ambiguities in distributed or parallel computations; for instance, vector clocks compare components to identify concurrent events with matching timestamps across processes.^[48] This approach ensures that simultaneity respects causality, avoiding total orders that artificially sequence independent events. In game theory, simultaneity manifests in non-cooperative games through simultaneous-move structures, where players select strategies concurrently, leading to Nash equilibria as stable outcomes. John Nash's seminal formulation defines a Nash equilibrium as a strategy profile where no player benefits from unilateral deviation, applicable to normal-form games where actions are chosen without knowledge of others' choices at the decision instant.^[50] For example, in the prisoner's dilemma, simultaneous defection by both players forms a Nash equilibrium, as each anticipates the other's move without sequential observation, highlighting how simultaneity enforces mutual best responses under complete information. This concept underscores broader implications for causality, where simultaneous decisions preclude retroactive influence, aligning logical models with acausal interpretations in temporal structures.^[50]

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### Summary of Influence of Henri Poincaré and Ernst Mach on Einstein’s Views
[33]
Significance -- Conventionality of Simultaneity
The thesis of the conventionality of simultaneity asserts that the selection of just which events are simultaneous, even in just one inertial frame of ...
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The Block Universe from Special Relativity
The relativity of simultaneity implies a block universe, that is, the co-reality or co-existence of the past, present, and future.<|separator|>
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Eternalism | Internet Encyclopedia of Philosophy
Every existing thing is simultaneous with any other existent thing. Everything that exists exists now. The now becomes redundant with such claims, as “happens ...
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Time and Physical Geometry - jstor
The common view is that only things existing now are real. The paper introduces the principle that there are no privileged observers, and that future events ...
[37]
A rigorous proof of determinism derived from the special theory of ...
A rigorous proof of determinism derived from the special theory of relativity · C. W. Rietdijk · Philosophy of Science 33 (4):341-344 (1966).
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[1008.1127] Closed timelike curves and causality violation - arXiv
Aug 6, 2010 · This fact apparently violates causality, therefore time travel and it's associated paradoxes have to be treated with great caution.Missing: simultaneity | Show results with:simultaneity
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[PDF] Causal Loops in Time Travel - PhilArchive
Feb 9, 2019 · In this paper I analyze the possibility of time traveling based on several specialized works, including those of Nicholas J.J. Smith ("Time ...
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Simultaneous multithreading: maximizing on-chip parallelism
Simultaneous multithreading has the potential to achieve 4 times the throughput of a superscalar, and double that of fine-grain multithreading. We evaluate ...
[41]
Time, clocks, and the ordering of events in a distributed system
A distributed algorithm is given for synchronizing a system of logical clocks which can be used to totally order the events.
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[PDF] Internet Time Synchronization: the Network Time Protocol - NTP.org
This paper presents an overview of the architecture, protocol and algorithms of the Network Time Protocol. (NTP) used in the Internet system to synchronize ...
[43]
On Interplanetary and Relativistic Distributed Computing
Jun 13, 2025 · In this paper, we study relativistic distributed systems, which are subject to the relativity of simultaneity.
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[PDF] Simulating Physics with Computers - s2.SMU
This is called the hidden-variable problem: it is impossible to represent the results of quantum mechanics with a classical universal device.Missing: superposition | Show results with:superposition
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https://dl.acm.org/doi/10.1145/3732772.3733563
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[PDF] Time, Clocks, and the Ordering of Events in a Distributed System
In this paper, we discuss the partial ordering defined by the "happened before" relation, and give a distributed algorithm for extending it to a consistent ...
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[PDF] A semantics for priority using partial orders with a simultaneity relation
In this paper we present a semantics for concurrent systems with priority constraints, using as a basis partial orders of events with both precedence and ...
[48]
[PDF] Non Cooperative Games John Nash
Jan 26, 2002 · It turns out that the set of equilibrium points of a two-person zero- sum game is simply the set of all pairs of opposing "good strategies." In ...