Researcher ORCiD: 0000-0003-1871-7803: Tuesday, September 05, 2023

05 September 2023

The Significance of Origins in Spacetime (v2). Integrating Local Time with Cosmic Time:

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.

GPS and Relativity, Relativistic Error Effect on the GPS Time...,

explain in detail the formulas for the relativistic corrections to be implemented in high-speed aircraft, or when using other

Analysis of Relativistic Error Effect on the GPS Time and the Receiver Position Accuracy

The Significance of Origins in Spacetime (v1). Integrating Local Time with Cosmic Time:

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' (t), 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:

3. Separate Origins: A Prerequisite:

4. A complete representation thus entails differentiating these origins:

Event 'p' is positioned at coordinates (x1, y1, z1, t) in the spacetime coordinate system. Spatial coordinates (x1, y1, z1) are measured relative to the origin 'o' for spatial dimensions, while 't' 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. Local Time vs. Cosmic Time:

An intriguing facet of spacetime is the duality between 'local time' (t) and 'cosmic time' (t). 'Local time' refers to the temporal dimension that originates at specific moments or events. It progresses from these local origins and marks the duration of events within specific frames of reference.

On the other hand, 'cosmic time' represents the broader temporal fabric governing the entire universe. It is akin to a cosmic clock that ticks uniformly, independent of local variations. 'Cosmic time' permeates all of spacetime, providing a universal reference for temporal measurement.

6. Relationship between 'Local Time' and 'Cosmic Time':

The relationship between 'local time' and 'cosmic time' is a crucial aspect of spacetime. 'Local time' (t) is inherently tied to the observer's.

Conversely, 'cosmic time' (t) remains constant across the universe. It serves as the ultimate arbiter of temporal progression, unifying disparate regions of spacetime under a single temporal framework.

In summary, 'local time' and 'cosmic time' are intimately connected yet distinct in their origins and implications. 'Local time' emerges from specific events or reference points, while 'cosmic time' transcends local variations, providing a universal timekeeping system for the cosmos.

7. Mathematical Presentation:

In the context of spacetime, the choice of origin for the time dimension ’t’ is an important consideration, and it's not necessarily the same as the origin 'o' for the spatial coordinates (x, y, z).

Origin of Time ’t’:

The origin or reference point for the time dimension ’t’ is typically not explicitly mentioned. In physics, time is often measured relative to a different reference point, often referred to as the "origin of time." This reference point could be the moment of an event, such as the start of an experiment or the origin of a particular coordinate system.

Separate Origins:

In practice, the origin for time ’t’ is distinct from the origin 'o' for spatial coordinates (x, y, z). These origins serve different purposes. The origin for spatial coordinates defines the reference point for measuring distances in space, while the origin for time defines the reference point for measuring time intervals.

So, in the context of a complete spacetime description, one would typically have separate origins for space and time.

For example:

Origin for spatial coordinates (x, y, z) is 'o' with coordinates (0, 0, 0).

Origin for time ’t’ may be a specific moment (e.g., the start of an experiment) or another reference time.

In the statement, it's important to clarify that 'o' represents the origin for spatial coordinates (x, y, z), and if 't' is included as the fourth coordinate, it should be measured relative to its own distinct origin for time.

So, the complete representation might be something like:

Event 'p' is located at coordinates (x1, y1, z1, t) in the spacetime coordinate system. The spatial coordinates (x1, y1, z1) are measured relative to the origin 'o' for spatial coordinates, and 't' is measured relative to its own origin for time, which could be the start of an experiment or another reference moment.

This way, it is clearly differentiated between the origins for spatial and temporal dimensions.

While in scientific representations:

In the context of scientific representations, it's important to clarify the reference frame and assumptions being made. If the reference to time ’t’ in a scenario where one wants to consider its origin as the starting point of an event or measurement, he can make that explicit in his representation.

Here's how one can represent time ’t’ with its axial position in the (x, y, z) coordinate system with an origin at its beginning:

Spatial Coordinates of 'p':

The spatial position of point 'p' is represented by its coordinates (x1, y1, z1) with respect to the origin 'o' in the (x, y, z) coordinate system. These coordinates describe where 'p' is located in space.

Time ’t’ with an Origin:

To represent time ’t’ with its axial position, one can specify that ’t’ starts at a particular moment or event, and it increases as time progresses. One should include this information as part of his representation:

The x-coordinate (x1) tells how far 'p' is from the origin 'o' along the x-axis.

The y-coordinate (y1) tells how far 'p' is from 'o' along the y-axis.

The z-coordinate (z1) tells how far 'p' is from 'o' along the z-axis.

The time ’t’ is measured from a specified origin or starting point and indicates when the event associated with 'p' occurs.

The representation might look like this:

Event 'p' is located at coordinates (x1, y1, z1) in the (x, y, z) coordinate system, with time ’t’ starting at a specific origin and increasing from that point.

By specifying that ’t’ starts at a particular origin; one emphasizes the reference point for time measurement within his chosen coordinate system. This approach aligns with the concept of time as a progression from a defined starting moment.

8. Conclusion:

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. . 'Local time' and 'cosmic time' are intimately connected yet distinct in their origins and implications

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.