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.
