Relativistic Coordination of Spatial and Temporal Dimensions

Soumendra Nath Thakur¹
¹Tagore's Electronic Lab. India
¹postmasterenator@gmail.com
¹postmasterenator@telitnetwork.in 
12 September 2023
@ResearchGate

Chapter Abstract: 

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.  

The list of coordinate system entities used in this paper is described below under the heading ’10.0 Entities in Coordinate Systems’.

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

Mathematical Presentation:

1.1 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.

1.2 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:

1.3 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.

1.4 Temporal Coordinate:

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

2.0. 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.'

3.1. 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.'

3.2. 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.'

4.1. 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.'

4.2. 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.

5.0 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.

6.1 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)

6.2 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.

7.1 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 minutes.

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

Temporal coordinate of o₂: t₁ + 0 Hrs 51 minutes

Temporal coordinate of C₂: t₁ + 0 Hrs 51 minutes

7.2 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.

8.0 Conclusion:

In conclusion, this paper 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.

9.0 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.

10.0 Entities in Coordinate Systems:

• 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.