Summary Paper: Relativistic Coordinate Systems for Clocks C₁ and C₂ in RF1 and RF2:

The paper provides a detailed exploration of coordinate systems governing the behavior of clocks C₁ and C₂ within reference frames RF1 and RF2. The paper's key points and mathematical presentation are summarized below.

Key Points:

Coordinate Systems: The paper introduces a 4-dimensional coordinate system (x, y, z, t) that combines spatial and temporal coordinates. This system is used to describe the positions of events 'p' associated with clocks C₁ and C₂.

Spatial Coordinates: The spatial positions of events 'p' of clock C₁ are represented as (x₁, y₁, z₁), while those of clock C₂ are represented as (x₂, y₂, z₂). These coordinates define the spatial location of the clocks within the chosen coordinate system.

Temporal Coordinates: The temporal coordinates, denoted as t₁ and t₂, are measured relative to the cosmic time origin 't₀.' 't₀' serves as a reference point for measuring time intervals and is associated with the cosmic dimension of time.

Spatial Origin on Earth: Clock C₁ is located at spatial origin 'o₁,' which serves as the reference point for spatial measurements. It is defined at mean sea level on Earth and represented as (0, 0, 0, t₁) within the coordinate system.

Introduction of Relative Elevated System: In scenarios involving elevation, clock C₂ is located at spatial origin 'o₂,' which is elevated to a height 'h' meters from 'o₁.' This introduces a relative gravitational potential difference (Ug) between C₁ and C₂.

Spatial Origin at Relative Height: Spatial coordinates for clock C₂ in the elevated system are represented as (x₂, y₂, z₂), with z₂ indicating the height 'h' above 'o₁.' This height difference results in the gravitational potential difference (Ug) between the two clocks.

Introduction of Relative Motion in System: In scenarios involving relative motion, clock C₂ is set in motion at a velocity 'v' meters/second from 'o₁.' This introduces a relative velocity (v) between C₁ and C₂.

Spatial Origin at Relative Motion: Spatial coordinates for clock C₂ in the moving system are represented as (x₂, y₂, z₂), with 'd' indicating the spatial displacement from 'o₁' due to motion. This motion results in a relative velocity (v) between C₁ and C₂.

Interplay of Spatial and Temporal Dimensions: The paper emphasizes the interplay between spatial and temporal dimensions, highlighting that spatial measurements are made relative to spatial origins ('o₁' or 'o₂'), while temporal measurements are made relative to the cosmic time origin ('t₀').

Mathematical Presentation:

The mathematical equations and calculations presented in the paper provide a rigorous foundation for understanding how the coordinates of clocks C₁ and C₂ are determined in different scenarios involving elevation and motion.

Conclusion:

In conclusion, the paper offers a unified framework for comprehending coordinate systems and their interplay, particularly in the context of relativistic effects on clocks C₁ and C₂. It underscores the importance of distinguishing between spatial and temporal coordinates and how they relate to cosmic time 't₀.' The study of gravitational potential difference and relative motion exemplifies the practical applications of these coordinate systems in modern physics and astronomy.
The paper's comprehensive approach enhances our understanding of how spatial and temporal dimensions interact within relativistic systems, providing valuable insights into the behavior of objects and clocks in different scenarios.

[References included in the original paper]

List of Entities:

• C₁ (Clock C₁): A specific clock used as a reference point, contributing to the study's spatial and temporal coordinates.
• C₂ (Clock C₂): Another clock used for comparison, experiencing scenarios such as elevation or motion, leading to changes in its coordinates.
• Coordinate System Used: A 4-dimensional system (x, y, z, t) integrating spatial (x, y, z) and temporal (t) coordinates for describing event positions.
• Coordinates of C₁: Spatial and temporal position coordinates within the chosen system, often represented as (x₁, y₁, z₁, t₁).
• Coordinates of C₂: Similar coordinates to C₁ but varying in scenarios involving elevation or motion, denoted as (x₂, y₂, z₂, t₂).
• Cosmic Origin ('t₀'): The reference point for temporal coordinates, associated with the cosmic time dimension.
• h (Height): Represents the vertical distance between spatial origins 'o₁' and 'o₂' in elevation scenarios, affecting gravitational potential differences.
• o₁ (Spatial Origin 'o₁'): The spatial reference point for spatial coordinates, typically linked to the starting position of clock C₁.
• o₂ (Spatial Origin 'o₂'): The spatial reference point in scenarios involving Clock C₂, potentially different from 'o₁.'
• p (Event 'p'): A specific spacetime event associated with either C₁ or C₂, with coordinates of interest in understanding clock positions.
• RF1 (Reference Frame 1): One of the reference frames used in the study, providing context for analyzing clock positions and movements.
• RF2 (Reference Frame 2): The second reference frame used in the study, offering a framework for analyzing clock behavior in various scenarios.
• Spatial Origin: A spatial reference point ('o₁' or 'o₂') defining the starting point for distance and position measurements.
• t₀ (Cosmic Time Origin): The cosmic time origin serving as the reference point for temporal coordinates.
• t₁ (Temporal Coordinate of Event 'p' of C₁): Temporal coordinates associated with event 'p' of Clock C₁, measured from 't₀.'
• t₂ (Temporal Coordinate of Event 'p' of C₂): Temporal coordinates for event 'p' of Clock C₂, measured from 't₀.'
• Ug (Gravitational Potential Difference): Represents the difference in gravitational potential between C₁ and C₂, arising from elevation or gravity.
• v (Velocity): The speed at which Clock C₂ is set in motion relative to 'o₁,' influencing spatial coordinates.

Relativistic Coordinate Systems for Clocks C₁ and C₂ in RF1 and RF2:

Date 10-09-2023. Soumendra Nath Thakur. ORCiD: 0000-0003-1871-7803

This paper elucidates the intricacies of coordinate systems governing the behavior of clocks C₁ and C₂ within reference frames RF1 and RF2. Offering a comprehensive exploration, it delves into the realms of spatial and temporal coordinates, the origins of spatial reference on Earth, and the incorporation of relative elevated systems, relative height, and relative motion. Each facet's profound influence on the coordinate systems and their intrinsic connection with cosmic time 't₀' is thoroughly examined.
Through a series of illustrative examples, this study illuminates the procedure for computing the coordinates of o₂ and C₂ in diverse scenarios, encompassing instances where C₂ is elevated to varying heights or set into motion. Notably, it underscores the pivotal significance of disentangling the common cosmic time in favor of an unwavering focus on spatial and temporal dimensions.
This paper serves as an effective conduit for conveying the profound interplay between spatial and temporal dimensions, fostering a unified framework that seamlessly encapsulates the descriptions of object positions and the intricate choreography of their movements within the fabric of spacetime.

Mathematical Presentation:

The list of entities and coordinate systems used in this paper is described below under the heading "List of entities".

Spatial Coordinates:

The spatial position of event 'p' of clock 'C₁' in the (x, y, z) coordinate system is represented as follows with the understanding that mass-to-energy conversion through nuclear reactions or radioactive decay is not considered:

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' by an atomic clock located at mean sea level on Earth:

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 'p' of clock 'C₁' relative to the spatial origin 'o' in the (x, y, z) coordinate system.

Temporal Coordinate:

t₁ represents the time coordinate of event 'p' of clock 'C₁' relative to the cosmic time origin 't₀.'

1.0. Spatial Origin on Earth:

Event 'p' of 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, t₁) = (x₁, y₁, z₁, t₁) in the (x, y, z, t) system with 'o₁.'

2.1. Introduction of Relative Elevated System:

Event 'p' of 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,' and now there is a relative gravitational potential difference (Ug) between the clocks 'C₁' and 'C₂.'

2.2. Spatial Origin at Relative Height:

Event 'p' of 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, t₂) = (x₂, y₂, z₂, t₂) in the (x, y, z, t) 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, and so there is a gravitational potential difference (Ug) between the clocks 'C₁' and 'C₂.'

3.1. Introduction of Relative Motion in System:

Event 'p' of another clock 'C₂' is located at coordinates (x₁, y₁, z₁, t₂) in an (x, y, z) system with the present origin 'o₂,' which initially originated in the (x, y, z) system with origin 'o₁' until set in motion at 'v' meters/second from 'o₁,' and now there is a relative velocity (v) between the clocks 'C₁' and 'C₂.'

3.2. Spatial Origin at Relative Motion:

Event 'p' of clock 'C₂' is located at coordinates (x₂, y₂, z₂, t₂) in the (x, y, z) system, originating at 'o₂' in spatial dimensions, which is set in motion at 'v' meters/second from 'o₁,' defined with coordinates (0, 0, d, t₂) = (x₂, y₂, z₂, t₂) in the (x, y, z, t) system with 'o₂.' Initially, origin 'o₂' or the clock 'C₂' earlier originated and merged with origin 'o₁,' at an actual distance of (o₂ - o₁) = d meters, and so there is a motion of 'v' meters/second between the clocks 'C₁' and 'C₂.'

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,' and measured by an atomic clock located at mean sea level on Earth, 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, and measured by an atomic clock located at mean sea level on Earth. 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.

Relativistic Coordinates & Time: Classical Approach:

The Example questions and Solutions provided in the paper are as per the application of the Classical approach. As such, Time Distortion is not measured in the solutions of the examples. This approach is rooted in this chapter, which meticulously explores spatial and temporal dimensions and how coordinate systems govern the behavior of clocks C₁ and C₂ within reference frames RF1 and RF2. This classical approach emphasizes the profound influence of spatial and temporal coordinates on cosmic time 't₀' while avoiding the complexities of time distortion. Through illustrative examples, it demonstrates how to compute coordinates in scenarios involving elevation and motion, shedding light on the significance of separating spatial and temporal dimensions for precise analyses in the realm of relativistic physics.

Example question (1) (gravitational potential difference):

Clock C₁ is located at o₁ at (t₁=10:30 Hrs) on 09-09-2023;

Where: (t₁ - t₀) = 13.8 billion years,

at (t₂=11:21 Hrs) on 09-09-2023;

C₂ elevated to a height (h) = 403 km;

Decide coordinates of o₂ and C₂;

Solution:

Clock C₁ is located at o₁ at t₁ = 10:30 Hrs on 09-09-2023 (Spatial time).

Clock C₂ is elevated to a height (h) = 403 km at t₂ = 11:21 Hrs on 09-09-2023 (Spatial time).

Cosmic time 13.8 billion years is common to both equations, so ignored.

Now, let's proceed with the calculations without mentioning the common 13.8 billion years:

We want to find the coordinates for o₂ and C₂:

Let (x₁, y₁, z₁, t₁) be the coordinates of C₁ at o₁, where (x₁, y₁, z₁) represents the spatial position, and t₁ is the time coordinate relative to t₀.

Let (x₂, y₂, z₂, t₂) be the coordinates of C₂ at o₂, where (x₂, y₂, z₂) represents the spatial position, and t₂ is the time coordinate relative to t₀.

Given that C₂ is elevated by 403 km, we can calculate the coordinates of C₂ at o₂ as follows:

x₂ = x₁ (no change in horizontal position)

y₂ = y₁ (no change in horizontal position)

z₂ = z₁ + 403 km (accounting for the elevation)

Now, we can calculate the time coordinate t₂ for C₂ at o₂:

t₂ = t₁ + (t₂ - t₁) = t₁ + 00:51 Hrs.

So, the coordinates for o₂ and C₂, without mentioning the common 13.8 billion years, are:

Coordinates of o₂: (x₂, y₂, z₂, t₁ + 00:51 Hrs)

Coordinates of C₂: (x₂, y₂, z₂, t₁ + 00:51 Hrs)

Consequences under gravitational potential difference:

After elevating C₂ to a height of 403 km at t₂ relative to t₀, both C₂ and o₂ share the same spatial position in the (x, y, z) coordinate system at a specific spatial time (09-09-2023). Their temporal coordinates are also the same, with a difference of 00:51 Hrs from t₁, measured relative to the cosmic time origin t₀. This highlights the interrelationship between spatial and temporal dimensions, allowing us to describe the positions and movements of objects in a unified framework.

This paper reflects that both o₂ and C₂ share the same spatial coordinates, which is in line with the scenario where C₂ is elevated vertically without any change in horizontal position. Additionally, their temporal coordinates remain the same, with a time difference of 51 minutes from t₁, consistent with the previous analysis.

Example question (2) (Clock is in motion):

Clock C₁ is located at o₁ at (t₁=10:30 Hrs) on 09-09-2023;

Where: (t₁ - t₀) = 13.8 billion years,

at (t₂=11:21 Hrs) on 09-09-2023;

C₂ is set in motion at a distance (d) = 403 km from o₁;

Solution:

Clock C₁ is located at o₁ at t₁ = 10:30 Hrs on 09-09-2023, where (t₁ - t₀) = 13.8 billion years.

Clock C₂ is set in motion at a spatial distance (d) = 403 km from o₁ at t₂ = 11:21 Hrs on 09-09-2023.

In this scenario, we will focus on the spatial and temporal coordinates, ignoring the common 13.8 billion years, to determine the coordinates of o₂ and C₂.

First, let's calculate the spatial coordinates of o₂ and C₂:

Since C₂ is set in motion at a distance (d) of 403 km from o₁, the spatial coordinates of C₂ will be determined based on the original position of o₁ (x₁, y₁, z₁) with an additional 403 km in the direction of motion:

Coordinates of o₂: (x₁, y₁, z₁)

Coordinates of C₂: (x₁ + 403 km, y₁, z₁)

Next, we need to calculate the temporal coordinates. Since we have (t₁ - t₀) = 13.8 billion years, and we want to find the temporal coordinates for o₂ and C₂ at t₂, we can calculate t₂ as follows:

t₂ = t₁ + (t₂ - t₁)

To calculate the time difference (t₂ - t₁) between t₂ and t₁, we can subtract the hours and minutes:

t₂ - t₁ = (11:21 Hrs) - (10:30 Hrs)

Now, calculate the difference in hours and minutes:

t₂ - t₁ = 0 Hrs 51 mins

Now, add this time difference to t₁ to get the temporal coordinates:

Temporal coordinate of o₂: t₁ + 0 Hrs 51 mins

Temporal coordinate of C₂: t₁ + 0 Hrs 51 mins

Consequences of clock is in motion:

In this scenario where Clock C₂ is set in motion at a distance of 403 km from its original position at o₁, we find the following consequences:

Spatial Coordinates: The spatial coordinates of o₂ remain the same as those of o₁ (x₁, y₁, z₁), indicating that o₂ and o₁ share the same spatial position.

Spatial Displacement of C₂: The spatial coordinates of C₂ (x₂, y₂, z₂) are calculated based on the original position of o₁ and the additional distance traveled (403 km) in the direction of motion. Therefore, the spatial coordinates of C₂ are (x₁ + 403 km, y₁, z₁), indicating that C₂ has moved a distance of 403 km in the x-axis direction relative to o₁. 

Temporal Coordinates: Both o₂ and C₂ share the same temporal coordinates, with a time difference of 51 minutes from t₁. This demonstrates that the temporal dimension remains synchronized between o₂ and C₂.

In summary, when Clock C₂ is in motion at a distance of 403 km from o₁, o₂ and C₂ share the same temporal coordinates, and C₂ undergoes a spatial displacement of 403 km relative to o₁. This illustrates the interplay between spatial and temporal dimensions in the context of relative motion within the chosen coordinate system.

Conclusion:

In conclusion, the paper 'Relativistic Coordinate Systems for Clocks C₁ and C₂ in RF1 and RF2' presents a comprehensive framework for understanding the coordinate systems and their interplay in the context of clocks C₁ and C₂. The examples provided illustrate the significance of spatial and temporal dimensions while accounting for scenarios involving gravitational potential difference and relative motion.

In both examples, it becomes evident that while spatial coordinates are influenced by elevations and motion relative to a common origin, temporal coordinates remain synchronized, referencing the cosmic time origin 't₀.' This unified framework facilitates precise descriptions of object positions and movements, emphasizing the intrinsic connection between spatial and temporal dimensions.

The study of gravitational potential difference and clock motion showcases the versatility of the proposed coordinate systems, allowing for accurate measurements and predictions in various scenarios. Overall, this paper contributes to a deeper understanding of relativistic coordinate systems and their practical applications in modern physics and astronomy.

References: 

[1] Weber, Hans J.; Arfken, George B. (2003). Essential Mathematical Methods for Physicists, ISE. London: Academic Press.

[2] Thakur, S. N., Samal, P., & Bhattacharjee, D. (2023). Relativistic effects on phaseshift in frequencies invalidate time dilation II. TechRxiv Org. https://doi.org/10.36227/techrxiv.22492066.v2

[3] Lee, J. M. (2013). Introduction to smooth manifolds. Springer Science & Business Media.

[4] Goldstein, H. (1950). Classical Mechanics

[5] Szekeres, P. (2004). A course in modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry. Cambridge University Press.

[6] Bergmann, P. G. (1976). Introduction to the Theory of Relativity. Courier Corporation.

List of Entities:

• C₁ (Clock C₁): A specific clock used as a reference point, contributing to the study's spatial and temporal coordinates.

• C₂ (Clock C₂): Another clock used for comparison, experiencing scenarios such as elevation or motion, leading to changes in its coordinates.

• Coordinate System Used: A 4-dimensional system (x, y, z, t) integrating spatial (x, y, z) and temporal (t) coordinates for describing event positions.

• Coordinates of C₁: Spatial and temporal position coordinates within the chosen system, often represented as (x₁, y₁, z₁, t₁).

• Coordinates of C₂: Similar coordinates to C₁ but varying in scenarios involving elevation or motion, denoted as (x₂, y₂, z₂, t₂).

• Cosmic Origin ('t₀'): The reference point for temporal coordinates, associated with the cosmic time dimension.

• h (Height): Represents the vertical distance between spatial origins 'o₁' and 'o₂' in elevation scenarios, affecting gravitational potential differences.

• o₁ (Spatial Origin 'o₁'): The spatial reference point for spatial coordinates, typically linked to the starting position of clock C₁.

• o₂ (Spatial Origin 'o₂'): The spatial reference point in scenarios involving Clock C₂, potentially different from 'o₁.'

• p (Event 'p'): A specific spacetime event associated with either C₁ or C₂, with coordinates of interest in understanding clock positions.

• RF1 (Reference Frame 1): One of the reference frames used in the study, providing context for analyzing clock positions and movements.

• RF2 (Reference Frame 2): The second reference frame used in the study, offering a framework for analyzing clock behavior in various scenarios.

• Spatial Origin: A spatial reference point ('o₁' or 'o₂') defining the starting point for distance and position measurements.

• t₀ (Cosmic Time Origin): The cosmic time origin serving as the reference point for temporal coordinates.

• t₁ (Temporal Coordinate of Event 'p' of C₁): Temporal coordinates associated with event 'p' of Clock C₁, measured from 't₀.'

• t₂ (Temporal Coordinate of Event 'p' of C₂): Temporal coordinates for event 'p' of Clock C₂, measured from 't₀.'

• Ug (Gravitational Potential Difference): Represents the difference in gravitational potential between C₁ and C₂, arising from elevation or gravity.

• v (Velocity): The speed at which Clock C₂ is set in motion relative to 'o₁,' influencing spatial coordinates.

Coordinate Systems for Clocks C₁ and C₂ in RF1 and RF2:

Date 09-09-2023. Soumendra Nath Thakur. ORCiD: 0000-0003-1871-7803

This paper described the coordinate systems for clocks C₁ and C₂ in reference frames RF1 and RF2. It provides explanations of spatial and temporal coordinates, spatial origins on Earth, and the introduction of relative elevated systems, relative height, relative motion, and their respective effects on the coordinate systems and the relationship with cosmic time 't₀.'

The examples provided in the paper also demonstrate how to calculate the coordinates of o₂ and C₂ in different scenarios, whether C₂ is elevated to a height or in motion, while highlighting the importance of ignoring the common cosmic time to focus on spatial and temporal dimensions. The text effectively conveys the interrelationship between spatial and temporal dimensions, allowing for a unified framework to describe the positions and movements of objects. 

Mathematical Presentation:

Spatial Coordinates:

The spatial position of event 'p' of clock 'C₁' in the (x, y, z) coordinate system is represented as follows with the understanding that mass-to-energy conversion through nuclear reactions or radioactive decay is not considered:

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' by an atomic clock located at mean sea level on Earth:

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 'p' of clock 'C₁' relative to the spatial origin 'o' in the (x, y, z) coordinate system.

Temporal Coordinate:

t1 represents the time coordinate of event 'p' of clock 'C₁' relative to the cosmic time origin 't₀.'

1.0. Spatial Origin on Earth:

Event 'p' of 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.'

2.1 Introduction of Relative Elevated System:

Event 'p' of 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' and, now there is a relative gravitational potential difference (Ug) between the clocks 'C₁' and 'C₂.'

2.2. Spatial Origin at Relative Height:

Event 'p' of 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, and so there is a gravitational potential difference (Ug) between the clocks 'C₁' and 'C₂.'

3.1. Introduction of Relative Motion in System:

Event 'p' of another clock 'C₂' is located at coordinates (x1,y1,z1,t2) in an (x, y, z) system with the present origin 'o₂,' which initially originated in the (x, y, z) system with origin 'o₁' until set in motion of 'v' meters/second from 'o₁' and, now there is a relative velocity (v) between the clocks 'C₁' and 'C₂.'

3.2. Spatial Origin at Relative Motion:

Event 'p' of clock 'C₂' is located at coordinates (x2,y2,z2,t2) in the (x, y, z) system, originating at 'o₂' in spatial dimensions, which is set in motion at 'v' meters/second from 'o₁,' defined with coordinates (0,0,d) = (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) = d meters, and so there is a motion of 'v' meters/second between the clocks 'C₁' and 'C₂.'

Both the 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,' and measured by an atomic clock located at mean sea level on Earth, 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, and measured by an atomic clock located at mean sea level on Earth. 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.

Example question (1) (gravitational potential difference):

Clock C₁ is located at o₁ at (t1=10:30 Hrs) on 09-09-2023; 

Where: (t₁ - t₀) = 13.8 billion years, 

at (t2=11:21 Hrs) on 09-09-2023;

C₂ elevated to a height (h) = 403 km;

Decide coordinates of o₂ and C₂; 

Solution:

Clock C₁ is located at o₁ at t₁ = 10:30 Hrs on 09-09-2023 (Spatial time).

Clock C₂ is elevated to a height (h) = 403 km at t₂ = 11:21 Hrs on 09-09-2023 (Spatial time). 

Cosmic time 13.8 billion years is common to both equations, so ignored.

Now, let's proceed with the calculations without mentioning the common 13.8 billion years:

We want to find the coordinates for o₂ and C₂:

Let (x₁, y₁, z₁, t₁) be the coordinates of C₁ at o₁, where (x₁, y₁, z₁) represents the spatial position, and t₁ is the time coordinate relative to t₀.

Let (x₂, y₂, z₂, t₂) be the coordinates of C₂ at o₂, where (x₂, y₂, z₂) represents the spatial position, and t₂ is the time coordinate relative to t₀.

Given that C₂ is elevated by 403 km, we can calculate the coordinates of C₂ at o₂ as follows:

x₂ = x₁ (no change in horizontal position)

y₂ = y₁ (no change in horizontal position)

z₂ = z₁ + 403 km (accounting for the elevation)

Now, we can calculate the time coordinate t₂ for C₂ at o₂:

t₂ = t₁ + (t₂ - t₁) = t₁ + 00:51 Hrs.

So, the coordinates for o₂ and C₂, without mentioning the common 13.8 billion years, are:

Coordinates of o₂: (x₁, y₁, z₁ + 403 km, t₁ + 00:51 Hrs)

Coordinates of C₂: (x₁, y₁, z₁ + 403 km, t₁ + 00:51 Hrs)

This implies that, After elevating C₂ to a height of 403 km at t₂ relative to t₀, both C₂ and o₂ share the same spatial position in the (x, y, z) coordinate system at a specific spatial time (09-09-2023). Their temporal coordinates are also the same, with a difference of 00:51 Hrs from t₁, measured relative to the cosmic time origin t₀. This highlights the interrelationship between spatial and temporal dimensions, allowing us to describe the positions and movements of objects in a unified framework.

Example question (2) (Clock is in motion):

Clock C₁ is located at o₁ at (t1=10:30 Hrs) on 09-09-2023; 

Where: (t₁ - t₀) = 13.8 billion years, 

at (t2=11:21 Hrs) on 09-09-2023;

C₂ travelled to a distance (h) = 403 km;

Decide coordinates of o₂ and C₂; 

Solution:

Clock C₁ is located at o₁ at t₁ = 10:30 Hrs on 09-09-2023, where (t₁ - t₀) = 13.8 billion years.

Clock C₂ is at a spatial distance (h) = 403 km from o₁ at t₂ = 11:21 Hrs on 09-09-2023.

We will ignore the common 13.8 billion years as previously discussed, focusing only on the spatial and temporal coordinates.

First, let's calculate the spatial coordinates of o₂ and C₂:

Since C₂ is elevated to a height (h) of 403 km from o₁, the spatial coordinates of C₂ will be the same as o₁'s (x₁, y₁, z₁) with an additional 403 km in the z-axis direction:

Coordinates of o₂: (x₁, y₁, z₁)

Coordinates of C₂: (x₁, y₁, z₁ + 403 km)

Next, we need to calculate the temporal coordinates. Since we have (t₁ - t₀) = 13.8 billion years, and we want to find the temporal coordinates for o₂ and C₂ at t₂, we can calculate t₂ as follows:

t₂ = t₁ + (t₂ - t₁)

To calculate the time difference (t₂ - t₁) between t₂ and t₁, we can subtract the hours and minutes:

t₂ - t₁ = (11:21 Hrs) - (10:30 Hrs)

Now, calculate the difference in hours and minutes:

t₂ - t₁ = 0 Hrs 51 mins

Now, add this time difference to t₁ to get the temporal coordinates:

Temporal coordinate of o₂: t₁ + 0 Hrs 51 mins

Temporal coordinate of C₂: t₁ + 0 Hrs 51 mins

So, the coordinates of o₂ and C₂ are as follows:

Coordinates of o₂: (x₁, y₁, z₁)

Coordinates of C₂: (x₁, y₁, z₁ + 403 km)

Temporal coordinates for both o₂ and C₂: t₁ + 0 Hrs 51 mins

In this scenario, both o₂ and C₂ have the same spatial coordinates, indicating that C₂ moved a distance of 403 km vertically along the z-axis from its original position at o₁. Additionally, their temporal coordinates are the same, with a time difference of 51 minutes from t₁. This demonstrates how spatial and temporal dimensions are interrelated and can be used to describe the positions and movements of objects in a unified framework.

References: 

[1] Weber, Hans J.; Arfken, George B. (2003). Essential Mathematical Methods for Physicists, ISE. London: Academic Press.
[2] Thakur, S. N., Samal, P., & Bhattacharjee, D. (2023). Relativistic effects on phaseshift in frequencies invalidate time dilation II. TechRxiv Org. https://doi.org/10.36227/techrxiv.22492066.v2
[3] Lee, J. M. (2013). Introduction to smooth manifolds. Springer Science & Business Media.
[4] Goldstein, H. (1950). Classical Mechanics
[5] Szekeres, P. (2004). A course in modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry. Cambridge University Press.
[6] Bergmann, P. G. (1976). Introduction to the Theory of Relativity. Courier Corporation.

Cosmic Microwave Background Radiation (CMB), the Observable and Non-Observable Universes, and Their Respective Distances

Soumendra Nath Thakur* Priyanka Samal1 Onwuka Frederick2

Guided by the monumental Big Bang theory, this research paper embarks on a captivating exploration of cosmic vistas, delving into the enigmatic origins of the universe, the enigmatic cosmic microwave background radiation (CMBR), and the intricate interplay between observable and non- observable universes. Intricately woven into this cosmic fabric are the pioneering observations of the Cosmic Background Explorer (COBE), the Wilkinson Microwave Anisotropy Probe (WMAP), and the Planck satellite. These scientific sentinels have charted the CMBR's temperature fluctuations, offering windows into the universe's earliest epochs. COBE's revelation of temperature variations in 1992 solidified the Big Bang theory. WMAP, launched in 2001, delved deeper, unraveling the CMBR's anisotropies and refining cosmic parameter measurements. The Planck satellite, soaring into space in 2009, etched precision onto the cosmic canvas, encapsulating the universe's age, composition, andevolutionary trajectory.This narrative unfolds the epic tale of emergence from a minuscule singularity and the subsequent grand cosmic inflation. At the heart of this cosmic odyssey lies the CMBR, a whispered echo from the universe's primordial dawn that unfolds the saga of its early epochs and the daring journey of the first light. 

DOI http://dx.doi.org/10.13140/RG.2.2.11748.76165

The tapestry of time itself is probed, unveiling the temporal intricacies of these realms shaped by phenomena like redshift and the relentless cosmic expansion. Across a 13.8-billion-year narrative, our observational prowess extends to reveal galaxies within a 46.5 billion light-year radius—43% visible, 57% veiled. Anchored by elegant mathematical frameworks like Hubble's Law, this journey through cosmic mechanicsnurtures the growth of understanding and discovery. An expedition both poetic andscientific, this paper unearths the blueprint of the universe, resonating with themes ofcuriosity and evolution that span unfathomable eons. 

The Significance of Origins in Spacetime (v3). Integrating Local Time with Cosmic Time:

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.