Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803
07 September 2023
1. Abstract:
In the realm of spacetime, the concept of origin plays a pivotal role, particularly when dealing with the dimensions of space and time. This comprehensive study delves into the critical importance of differentiating between the origins of spatial coordinates (x, y, z) and the temporal dimension 't' within the framework of spacetime.
Furthermore, it illuminates the intriguing relationship between 'local time' (t) and 'cosmic time' (t₀) and their measurements relative to distinct reference points. The research explores how 't' can have its own unique origin, separate from spatial coordinates, and how this 'local time' connects with the overarching concept of 'cosmic time' governing the universe.
This multidimensional analysis enhances our understanding of the profound interplay between space and time, highlighting the fundamental fabric of the universe.
2. Introduction:
Spacetime, a foundational concept in the realm of physics, seamlessly intertwines the dimensions of space and time, forming the fabric of our universe. Within this intricate tapestry, the selection of an origin for the temporal dimension 't' takes on profound significance. 't' is often measured relative to what we might term the "origin of time" or "observer's frame." This origin can be defined by a pivotal event, the commencement of an experiment, or the establishment of a specific coordinate system.
It is imperative to distinguish the origin for time 't' from the origin for spatial coordinates (x, y, z), which is typically represented as 'o.' These origins serve disparate functions. The spatial origin 'o' serves as the foundational reference point for measuring distances within the spatial dimensions, while the temporal origin 't' serves as the reference point for measuring intervals of time.
3. Separate Origins: A Prerequisite:
Within the framework of a comprehensive description of spacetime, the existence of distinct origins for space and time becomes indispensable. Consider a scenario where the origin for spatial coordinates (x, y, z) is 'o,' defined precisely at coordinates (0, 0, 0). Conversely, the origin for time 't' might commence at a specific moment, such as the inception of an experiment or another precisely defined reference time.
4. A complete representation thus entails differentiating these origins:
Imagine an event 'p' positioned at coordinates (x1, y1, z1, t₁) within the spacetime coordinate system. Spatial coordinates (x1, y1, z1) are measured in relation to the origin 'o' in the spatial dimensions, while 't₁' is measured from its own distinct origin in the temporal dimension. This temporal origin could correspond to the initiation of an experiment or any other momentous reference point.
This separation of origins is fundamental for achieving precision in understanding both 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) within the (x, y, z) system, originating from 'o' in the spatial dimensions. Simultaneously, the time coordinate 't₁' originates from 't₀' within the cosmic dimension.
In this representation, we find an event labeled as 'p' situated within the three-dimensional spatial coordinate system (x, y, z), with 'o' as its foundational reference point for measuring spatial distances and positions.
However, the temporal dimension, as denoted by the time coordinate 't₁,' operates with its own unique reference point. This reference point is identified as 't₀,' which is a reference deeply entwined with the cosmic dimension of time. Effectively, while spatial measurements are anchored in reference to 'o,' temporal measurements find their basis in 't₀,' underlining the fundamental distinction between the origins of space and time.
This presentation serves to underscore the crucial differentiation between the spatial origin 'o' and the cosmic time origin 't₀,' emphasizing the principle that time is not measured from the same reference point as spatial dimensions.
6. Mathematical Presentation:
Spatial Coordinates:
The spatial position of event 'p' in the (x, y, z) coordinate system is represented as follows:
- 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.'
Spatial Origin on Earth:
Clock 'c₁' is located at coordinates (x1,y1,z1,t1) in the (x, y, z) system, originating from 'o₁' in spatial dimensions, which is located at mean sea level on Earth, defined with coordinates (0,0,0) = (x1,y1,z1) in the (x, y, z) system with 'o₁.'
Introduction of Elevated System:
Another clock 'c₂' is located at coordinates (x1,y1,z1,t2) in an elevated (x, y, z) system with the present origin 'o₂,' which initially originated in the (x, y, z) system with origin 'o₁' until elevated to a height 'h' meters from 'o₁.'
Spatial Origin at a Height:
Clock 'c₂' is located at coordinates (x2,y2,z2,t2) in the (x, y, z) system, originating at 'o₂' in spatial dimensions, which is located at a height 'h' meters from 'o₁,' defined with coordinates (0,0,h) = (x2,y2,z2) in the (x, y, z) system with 'o₂.' Initially, origin 'o₂' or the clock 'c₂' earlier originated and merged with origin 'o₁,' at an actual distance of (o₂ - o₁) = h meters.
Both temporal origins 'o₁' and 'o₂' of these coordinate systems for the respective clocks 'c₁' and 'c₂' are in a common scale of cosmic time relative to 't₀,' while origins 'o₁' and 'o₂' serve as the reference points for measuring distances and positions within the spatial dimensions.
However, the temporal dimension, represented by the time coordinates 't₁' and 't₂,' operates with a common and distinct reference point. The origin for 't₁' and 't₂' is specified as 't₀,' which is a reference associated with the cosmic dimension of time. In essence, while spatial measurements are made relative to 'o₁' and 'o₂,' temporal measurements are made relative to 't₀,' highlighting the separation between spatial and temporal origins.
7. In Conclusion:
The exploration of spatial origins on Earth and the introduction of elevated coordinate systems underscore the critical role of distinguishing between spatial and temporal dimensions within the context of spacetime.
The study begins by establishing 'o₁' as the spatial origin at mean sea level on Earth, serving as the reference point for measuring distances in the (x, y, z) system. 'c₁' is located at coordinates (x1,y1,z1,t1) relative to this spatial origin.
The introduction of the elevated system, represented by 'c₂,' introduces the concept of an elevated spatial origin 'o₂.' Initially, 'o₂' originates within the same (x, y, z) system as 'o₁' and is later elevated to a height 'h' meters above 'o₁.' Consequently, 'c₂' is situated at coordinates (x2,y2,z2,t2) in this elevated system, defined relative to 'o₂' and located 'h' meters above 'o₁.'
The critical distinction lies in the temporal dimension, represented by 't₁' and 't₂.' Both 't₁' and 't₂' operate within a common scale of cosmic time relative to 't₀,' emphasizing their shared temporal framework. However, the reference points for measuring distances and positions within the spatial dimensions are 'o₁' and 'o₂,' highlighting the separation between spatial and temporal origins.
This research accentuates the fundamental concept that while spatial measurements are made relative to spatial origins, temporal measurements are made relative to a distinct temporal origin, 't₀,' associated with the cosmic dimension of time. This distinction is paramount in understanding the intricate interplay between space and time within the framework of spacetime.
In essence, the significance of spatial and temporal origins elucidates the complexity of spacetime, enriching our comprehension of the fundamental fabric of our 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.
