Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803
05 September 2023
1. Abstract:
In the realm of spacetime, the concept of origin takes on a critical role, especially when dealing with the dimensions of space and time. This article explores the importance of distinguishing between the origins of spatial coordinates (x, y, z) and the temporal dimension 't' within the context of a complete spacetime description. It emphasizes the need to recognize that 't' often has its own distinct origin, separate from the spatial coordinates.
Additionally, this article sheds light on the intriguing relationship between 'local time' (t) and 'cosmic time' (t₀), and how they are measured relative to different reference points. It delves into the idea that 't' can begin at specific moments or events, known as 'local time,' and elucidates how this local time relates to the broader concept of 'cosmic time' that governs the universe.
2. Introduction:
Spacetime, a fundamental concept in the realm of physics, combines the dimensions of space and time into a seamless continuum. In this intricate interplay, the choice of origin for the temporal dimension 't' becomes a pivotal consideration. Often, 't' is measured relative to what we might term the "origin of time" or "observer's frame." This origin could be defined by a significant event, the initiation of an experiment, or the establishment of a specific coordinate system.
It's crucial to distinguish the origin for time 't' from the origin for spatial coordinates (x, y, z), often represented as 'o.' These origins serve distinct purposes. The spatial origin 'o' sets the reference point for measuring distances in the spatial dimensions, while the temporal origin 't' sets the reference point for measuring time intervals.
3. Separate Origins: A Prerequisite:
In the context of a comprehensive spacetime description, separate origins for space and time are practically indispensable. Consider the following scenario: the origin for spatial coordinates (x, y, z) is 'o,' defined with coordinates (0, 0, 0). In contrast, the origin for time 't' may commence at a specific moment, such as the inception of an experiment or another reference time.
4. A complete representation thus entails differentiating these origins:
Event 'p' is positioned at coordinates (x1, y1, z1, t₁) in the spacetime coordinate system. Spatial coordinates (x1, y1, z1) are measured relative to the origin 'o' for spatial dimensions, while 't₁' is measured from its own distinct origin for time. This temporal origin could be the initiation of an experiment or any other significant reference moment.
This separation of origins allows for a precise understanding of where an event occurs in space and when it transpires in time.
5. Time ’t₁’ with Cosmic Origin t₀:
Event 'p' is located at coordinates (x₁, y₁, z₁, t₁) in the (x, y, z) system, originating from 'o' in spatial dimensions, while the time coordinate 't₁' originates from 't₀' in the cosmic dimension.
In this presentation, we have an event denoted as 'p' situated in the three-dimensional spatial coordinate system (x, y, z) with an origin labeled as 'o.' This origin 'o' serves as the reference point for measuring distances and positions within the spatial dimensions.
However, the temporal dimension, represented by the time coordinate 't₁,' operates with a distinct reference point. The origin for 't₁' is specified as 't₀,' which is a reference associated with the cosmic dimension of time. In essence, while spatial measurements are made relative to 'o,' temporal measurements are made relative to 't₀,' highlighting the separation between spatial and temporal origins.
This presentation distinguishes between the spatial origin 'o' and the cosmic time origin 't₀,' underlining the concept that time is not measured from the same reference point as spatial dimensions.
6. Mathematical Presentation:
Spatial Coordinates:
Spatial position of event 'p' in the (x, y, z) coordinate system:
x₁ represents the displacement along the x-axis.
y₁ represents the displacement along the y-axis.
z₁ represents the displacement along the z-axis.
Temporal Coordinate:
The temporal dimension, represented by 't₁,' is measured relative to its own origin, 't₀':
t₁ denotes the time coordinate of event 'p' and is measured from 't₀.'
In mathematical notation:
Spatial Coordinates:
(x1, y1, z1) represents the spatial position of event 'p' relative to the spatial origin 'o' in the (x, y, z) coordinate system.
Temporal Coordinate:
t₁ represents the time coordinate of event 'p' relative to the cosmic time origin 't₀.'
7. Conclusion:
The choice of origin in spacetime is a fundamental consideration, distinguishing the reference points for spatial coordinates from the temporal dimension 't.' A comprehensive spacetime description necessitates separate origins for space and time, ensuring precision in locating events in both dimensions.
This mathematical representation captures the separation of origins between spatial and temporal dimensions, as described in the presentation.
The relationship between 'local time' and 'cosmic time' underscores the nuanced nature of temporal measurement. 'Local time' serves as a dynamic, observer-dependent component, while 'cosmic time' remains an unchanging, universal entity. Recognizing the significance of origins in spacetime elucidates the intricate interplay between space and time and deepens our understanding of the fundamental fabric of the universe.
8. References:
[1] Einstein, A. (1915). General Theory of Relativity. Annalen der Physik, 354(7), 769-822.
[2] Hawking, S. W. (1988). A Brief History of Time: From the Big Bang to Black Holes. Bantam Books.
[3] Minkowski, H. (1908). Space and Time: An Introduction to the Special Theory of Relativity. Princeton University Press.
[4] Penrose, R. (1965). Gravitational Collapse and Space-Time Singularities. Physical Review Letters, 14(3), 57-59.
[5] Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. W. H. Freeman and Company.
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