bridge derivation: Newtonian space and time to Relativity's spacetime

[bhpayne]

bridging the assumption that "space and time are separate" and "spacetime"

[bhpayne]

prompt

In Newtonian physics, the separation of space and time refers to the concept that space and time are completely independent entities. They exist separately and neither is affected by the other. Both are considered as absolute and unchanging throughout the universe. Time flows at a constant rate regardless of where you are or how you're moving in space.

In contrast, Einstein's theory of relativity introduces the idea of spacetime where space and time are intertwined and can be affected by gravity and motion.

In mathematical Physics a derivation is comprised of steps and equations. Is there a derivation that bridges the Newtonian view of space and time with the relativity concept of spacetime? If yes, provide the steps of the derivation. Format the answer using Latex as a single file.

Latex is a software system for typesetting documents. LaTeX is widely used for the communication and publication of scientific documents and technical note-taking in many fields, owing partially to its support for complex mathematical notation.

Do not have equations inline with the text. Equations should be separate and inside a math environment for Latex:

\begin{equation}
\end{equation}

Each equation should have a left-hand side and right-hand side. Specify mathematical equations explicitly. Define the variables used in equations.

thoughts

The user is asking for a derivation that bridges Newtonian space and time with relativistic spacetime.
This is a tricky question because Newtonian physics is a limiting case of relativistic physics, specifically when velocities are much smaller than the speed of light and gravitational fields are weak.
A direct derivation bridging the two as if they are two independent theories to be unified is unlikely. Instead, we should look for how relativistic spacetime reduces to Newtonian space and time under specific conditions.

Here's a possible approach:

1. Start with relativistic concepts: Begin with the core principles of special and general relativity.
2. Identify the key differences: Pinpoint the crucial distinctions between Newtonian and relativistic views, such as the constancy of the speed of light, time dilation, length contraction, and the equivalence principle (gravity as spacetime curvature).
3. Introduce approximations: Consider scenarios where relativistic effects become negligible. This typically involves:
   • Low velocities compared to the speed of light (v << c).
   • Weak gravitational fields.
4. Show how relativistic equations reduce to Newtonian equations under these approximations.

Let's consider the key relativistic effects and their Newtonian limits:

• Time dilation: Relativistic time dilation formula:
  Δt = γ Δt₀, where γ = 1 / √(1 - v²/c²)
  For v << c, we can use Taylor expansion for γ:
  γ ≈ 1 + (1/2) v²/c² + O(v⁴/c⁴)
  As v/c → 0, γ → 1, so Δt ≈ Δt₀. This means time dilation becomes negligible, and time becomes absolute as in Newtonian physics.

• Length contraction: Relativistic length contraction formula:
  L = L₀ / γ
  For v << c, as γ → 1, L ≈ L₀. Length contraction becomes negligible, and space becomes absolute.

• Relativistic momentum and energy:
  Relativistic momentum p = γmv
  Relativistic energy E = γmc²
  For v << c, p ≈ mv (Newtonian momentum)
  E ≈ mc² + (1/2)mv² (Newtonian kinetic energy + rest energy mc²)

• Gravity: General relativity describes gravity as curvature of spacetime. In weak gravitational fields and low velocities, General Relativity should reduce to Newtonian Gravity.
  This reduction is more complex and involves the Einstein field equations and the Newtonian limit. Poisson's equation for Newtonian gravity should be derived from the field equations in the weak field limit. This is mathematically involved.

For a simpler "derivation" focused on space and time separation, we can focus on the time dilation and length contraction aspects and show how they disappear in the Newtonian limit.

Let's outline a simplified derivation concentrating on the time concept:

1. Start with Relativistic Time Dilation Equation: Write down the time dilation formula.
2. Consider the Newtonian Limit: State the condition for the Newtonian limit (v << c).
3. Approximate the Lorentz Factor: Expand the Lorentz factor γ for v << c using Taylor series.
4. Show Time Approaches Absolute: Show that as v/c approaches zero, relativistic time approaches Newtonian absolute time.

This will not be a derivation "bridging" in the sense of unifying two separate starting points. It will be demonstrating how relativistic spacetime contains Newtonian space and time as a limiting case. This is the more accurate interpretation of the relationship.

Let's refine the steps for the Latex answer:

Title: Bridging Newtonian and Relativistic Views of Time: The Newtonian Limit of Time Dilation

Introduction: Briefly explain Newtonian absolute space and time and relativistic spacetime. State the goal of showing how relativistic time reduces to Newtonian time under certain conditions.

Derivation Steps:

1. Relativistic Time Dilation: Present the time dilation equation. Define variables.
2. Lorentz Factor: Define the Lorentz factor.
3. Newtonian Limit Condition: State the condition for the Newtonian limit (v << c).
4. Taylor Expansion of Lorentz Factor: Show the Taylor expansion of γ for small v/c.
5. Approximation of Lorentz Factor in Newtonian Limit: Simplify γ to approximately 1 for v << c.
6. Newtonian Time: Substitute the approximated γ back into the time dilation equation and show that Δt ≈ Δt₀, meaning time becomes absolute (Newtonian).

Conclusion: Summarize that in the limit of low velocities, relativistic time dilation becomes negligible, effectively recovering the Newtonian concept of absolute time. Mention that a similar argument can be made for space (length contraction). Conclude that Newtonian space and time are a valid approximation of spacetime at low velocities and weak gravity.

Let's start writing the Latex code based on these steps.

response

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\title{Bridging Newtonian and Relativistic Views of Time: The Newtonian Limit of Time Dilation}
\date{}
\begin{document}
\maketitle

\section{Introduction}

In Newtonian physics, space and time are considered absolute and independent entities. Time, in particular, is universal and flows uniformly for all observers, irrespective of their motion or gravitational environment. This is the concept of \textit{absolute time}.

Einstein's theory of relativity, however, revolutionized our understanding of space and time by introducing the concept of \textit{spacetime}. In relativity, space and time are intertwined into a single four-dimensional continuum. Furthermore, time is no longer absolute but is relative and can be affected by motion and gravity.  This leads to phenomena such as time dilation and length contraction, which are absent in Newtonian physics.

While relativity provides a more accurate description of the universe, Newtonian physics remains a highly successful approximation in scenarios where velocities are much smaller than the speed of light and gravitational fields are weak. This document aims to illustrate how the relativistic concept of time approaches the Newtonian concept of absolute time in such conditions, effectively showing a bridge through a limiting case. We will focus on the phenomenon of time dilation from special relativity and show its Newtonian limit.

\section{Derivation}

We start with the time dilation formula from special relativity, which describes how time intervals are measured differently by observers in relative motion.

\subsection{Relativistic Time Dilation}
The time dilation formula is given by:
\begin{equation}
\Delta t = \gamma \Delta t_0
\end{equation}
Where:
\begin{itemize}
    \item $\Delta t$ is the time interval measured by an observer in a reference frame in relative motion.
    \item $\Delta t_0$ is the proper time interval, measured by an observer at rest relative to the event.
    \item $\gamma$ is the Lorentz factor.
\end{itemize}

\subsection{Lorentz Factor}
The Lorentz factor, $\gamma$, is defined as:
\begin{equation}
\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}
\end{equation}
Where:
\begin{itemize}
    \item $v$ is the relative velocity between the two reference frames.
    \item $c$ is the speed of light in a vacuum.
\end{itemize}

\subsection{Newtonian Limit Condition}
The Newtonian limit is characterized by velocities much smaller than the speed of light, i.e., $v \ll c$. In this regime, the ratio $\frac{v^2}{c^2}$ becomes very small.

\subsection{Taylor Expansion of Lorentz Factor for Small Velocities}
We can use the Taylor series expansion for $(1-x)^{-1/2}$ around $x=0$, which is given by $(1-x)^{-1/2} \approx 1 + \frac{1}{2}x + O(x^2)$ for small $x$. In our case, $x = \frac{v^2}{c^2}$.  Applying this expansion to the Lorentz factor:
\begin{equation}
\gamma = \left(1 - \frac{v^2}{c^2}\right)^{-1/2} \approx 1 + \frac{1}{2} \frac{v^2}{c^2} + O\left(\left(\frac{v^2}{c^2}\right)^2\right)
\end{equation}
For $v \ll c$, the term $\frac{v^2}{c^2}$ is very small, and higher order terms $O\left(\left(\frac{v^2}{c^2}\right)^2\right)$ become negligible. Therefore, we can approximate the Lorentz factor as:
\begin{equation}
\gamma \approx 1 + \frac{1}{2} \frac{v^2}{c^2}
\end{equation}

\subsection{Approximation of Time Dilation in Newtonian Limit}
Substituting the approximated Lorentz factor back into the time dilation formula (1):
\begin{equation}
\Delta t \approx \left(1 + \frac{1}{2} \frac{v^2}{c^2}\right) \Delta t_0
\end{equation}
\begin{equation}
\Delta t \approx \Delta t_0 + \frac{1}{2} \frac{v^2}{c^2} \Delta t_0
\end{equation}
In the Newtonian limit where $v \ll c$, the term $\frac{1}{2} \frac{v^2}{c^2} \Delta t_0$ is extremely small compared to $\Delta t_0$.  As $v/c \rightarrow 0$, this term approaches zero. Thus, in the Newtonian limit, we have:
\begin{equation}
\Delta t \approx \Delta t_0
\end{equation}
This result indicates that in the limit of low velocities ($v \ll c$), the time interval measured by a moving observer ($\Delta t$) is approximately equal to the proper time interval ($\Delta t_0$). This means that time dilation becomes negligible, and time is measured in essentially the same way by all observers, regardless of their relative motion. This is consistent with the Newtonian concept of absolute time, where time is universal and unaffected by motion.

\section{Conclusion}

We have shown that in the limit of low velocities ($v \ll c$), the relativistic time dilation effect becomes negligible, and the time interval measured by moving observers approaches the proper time interval. This demonstrates how the relativistic concept of time, as described by special relativity, reduces to the Newtonian concept of absolute time under Newtonian conditions.  While this derivation focuses on time, a similar argument can be made for spatial dimensions and length contraction, showing that relativistic space also approaches Newtonian absolute space in the limit of low velocities. Therefore, Newtonian physics can be seen as a low-velocity approximation of relativistic physics, where the intertwining of space and time becomes less apparent, and space and time appear to be separate and absolute entities.

\end{document}

[bhpayne]

prompt

For the following Latex file, write out the file and insert a label for each equation. The label that is added should be descriptive of the equation and contain no spaces, only underscores.

Labels for equations should be unique within the document.

The following Latex file contains a derivation with many steps. Sometimes a previous expression is referenced implicitly. Edit the following text to manually insert references where appropriate. For example replace text like "substitute the forces" with "substitute Eq.~\ref{} to get". Be sure to reference the relevant equation labels so that Latex can compile the document.

Just write the revised Latex file content as your answer.

\documentclass{article}

\usepackage{amsmath}

\title{Newtonian space and time to Relativity's spacetime}
\author{Ben Payne, Gemini 2.0 Flash Thinking}
\date{\today}

% margins of 1 inch:
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{9in}
\setlength{\oddsidemargin}{0in}
\setlength{\textwidth}{6.5in}

\usepackage[pdftex]{hyperref} % hyperlink equation and bibliographic citations


\begin{document}
\maketitle

\section{Introduction}

In Newtonian physics, space and time are considered absolute and independent entities. Time, in particular, is universal and flows uniformly for all observers, irrespective of their motion or gravitational environment. This is the concept of \textit{absolute time}.

Einstein's theory of relativity, however, revolutionized our understanding of space and time by introducing the concept of \textit{spacetime}. In relativity, space and time are intertwined into a single four-dimensional continuum. Furthermore, time is no longer absolute but is relative and can be affected by motion and gravity.  This leads to phenomena such as time dilation and length contraction, which are absent in Newtonian physics.

While relativity provides a more accurate description of the universe, Newtonian physics remains a highly successful approximation in scenarios where velocities are much smaller than the speed of light and gravitational fields are weak. This document aims to illustrate how the relativistic concept of time approaches the Newtonian concept of absolute time in such conditions, effectively showing a bridge through a limiting case. We will focus on the phenomenon of time dilation from special relativity and show its Newtonian limit.

\section{Derivation}

We start with the time dilation formula from special relativity, which describes how time intervals are measured differently by observers in relative motion.

\subsection{Relativistic Time Dilation}
The time dilation formula is given by:
\begin{equation}
\Delta t = \gamma \Delta t_0
\end{equation}
Where:
\begin{itemize}
    \item $\Delta t$ is the time interval measured by an observer in a reference frame in relative motion.
    \item $\Delta t_0$ is the proper time interval, measured by an observer at rest relative to the event.
    \item $\gamma$ is the Lorentz factor.
\end{itemize}

\subsection{Lorentz Factor}
The Lorentz factor, $\gamma$, is defined as:
\begin{equation}
\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}
\end{equation}
Where:
\begin{itemize}
    \item $v$ is the relative velocity between the two reference frames.
    \item $c$ is the speed of light in a vacuum.
\end{itemize}

\subsection{Newtonian Limit Condition}
The Newtonian limit is characterized by velocities much smaller than the speed of light, i.e., $v \ll c$. In this regime, the ratio $\frac{v^2}{c^2}$ becomes very small.

\subsection{Taylor Expansion of Lorentz Factor for Small Velocities}
We can use the Taylor series expansion for $(1-x)^{-1/2}$ around $x=0$, which is given by 
\begin{equation}
    (1-x)^{-1/2} \approx 1 + \frac{1}{2}x + O(x^2)
\end{equation}
  for small $x$. In our case, 
  \begin{equation}
  x = \frac{v^2}{c^2}     
  \end{equation}
Applying this expansion to the Lorentz factor:
\begin{equation}
\gamma = \left(1 - \frac{v^2}{c^2}\right)^{-1/2} \approx 1 + \frac{1}{2} \frac{v^2}{c^2} + O\left(\left(\frac{v^2}{c^2}\right)^2\right)
\end{equation}
For $v \ll c$, the term $\frac{v^2}{c^2}$ is very small, and higher order terms $O\left(\left(\frac{v^2}{c^2}\right)^2\right)$ become negligible. Therefore, we can approximate the Lorentz factor as:
\begin{equation}
\gamma \approx 1 + \frac{1}{2} \frac{v^2}{c^2}
\end{equation}

\subsection{Approximation of Time Dilation in Newtonian Limit}
Substituting the approximated Lorentz factor back into the time dilation formula (1):
\begin{equation}
\Delta t \approx \left(1 + \frac{1}{2} \frac{v^2}{c^2}\right) \Delta t_0
\end{equation}
\begin{equation}
\Delta t \approx \Delta t_0 + \frac{1}{2} \frac{v^2}{c^2} \Delta t_0
\end{equation}
In the Newtonian limit where $v \ll c$, the term $\frac{1}{2} \frac{v^2}{c^2} \Delta t_0$ is extremely small compared to $\Delta t_0$.  As $v/c \rightarrow 0$, this term approaches zero. Thus, in the Newtonian limit, we have:
\begin{equation}
\Delta t \approx \Delta t_0
\end{equation}
This result indicates that in the limit of low velocities ($v \ll c$), the time interval measured by a moving observer ($\Delta t$) is approximately equal to the proper time interval ($\Delta t_0$). This means that time dilation becomes negligible, and time is measured in essentially the same way by all observers, regardless of their relative motion. This is consistent with the Newtonian concept of absolute time, where time is universal and unaffected by motion.

\section{Conclusion}

We have shown that in the limit of low velocities ($v \ll c$), the relativistic time dilation effect becomes negligible, and the time interval measured by moving observers approaches the proper time interval. This demonstrates how the relativistic concept of time, as described by special relativity, reduces to the Newtonian concept of absolute time under Newtonian conditions.  While this derivation focuses on time, a similar argument can be made for spatial dimensions and length contraction, showing that relativistic space also approaches Newtonian absolute space in the limit of low velocities. Therefore, Newtonian physics can be seen as a low-velocity approximation of relativistic physics, where the intertwining of space and time becomes less apparent, and space and time appear to be separate and absolute entities.

\end{document}

thoughts

The user wants me to modify a Latex file.
I need to perform two main tasks:

1. Add labels to each equation: The labels should be descriptive, unique, and use underscores instead of spaces.
2. Insert references to equations: I need to identify implicit references to previous equations and replace them with explicit references using Eq.~\ref{}.

Let's go through the Latex file step by step and address each task.

Task 1: Add labels to equations

I will identify each equation environment in the Latex file and add a \label{} command after each \begin{equation}. I will come up with descriptive labels using underscores.

• Equation 1: \begin{equation} \Delta t = \gamma \Delta t_0 \end{equation}. Label: time_dilation_formula
• Equation 2: \begin{equation} \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \end{equation}. Label: lorentz_factor_definition
• Equation 3: \begin{equation} (1-x)^{-1/2} \approx 1 + \frac{1}{2}x + O(x^2) \end{equation}. Label: taylor_expansion_lorentz
• Equation 4: \begin{equation} x = \frac{v^2}{c^2} \end{equation}. Label: x_substitution
• Equation 5: \begin{equation} \gamma = \left(1 - \frac{v^2}{c^2}\right)^{-1/2} \approx 1 + \frac{1}{2} \frac{v^2}{c^2} + O\left(\left(\frac{v^2}{c^2}\right)^2\right) \end{equation}. Label: lorentz_factor_taylor_expanded
• Equation 6: \begin{equation} \gamma \approx 1 + \frac{1}{2} \frac{v^2}{c^2} \end{equation}. Label: lorentz_factor_approximation
• Equation 7: \begin{equation} \Delta t \approx \left(1 + \frac{1}{2} \frac{v^2}{c^2}\right) \Delta t_0 \end{equation}. Label: time_dilation_approx_lorentz_substituted
• Equation 8: \begin{equation} \Delta t \approx \Delta t_0 + \frac{1}{2} \frac{v^2}{c^2} \Delta t_0 \end{equation}. Label: time_dilation_expanded
• Equation 9: \begin{equation} \Delta t \approx \Delta t_0 \end{equation}. Label: newtonian_time_limit

Task 2: Insert references to equations

I will read through the text and identify phrases that implicitly refer to equations. I will then replace these phrases with explicit references using Eq.~\ref{label}.

• "Substituting the approximated Lorentz factor back into the time dilation formula (1)": Replace "(1)" with "Eq.~\ref{time_dilation_formula}".
• "time dilation formula (1)": Already handled in the previous step.

Let's double-check if I missed any implicit references. It seems like the only explicit reference needed is for the time dilation formula. The rest seem to be logical steps in the derivation.

Now, I will combine the original Latex code with the labels and references I have identified. I will present the final Latex code as the answer.

response

\documentclass{article}

\usepackage{amsmath}

\title{Newtonian space and time to Relativity's spacetime}
\author{Ben Payne, Gemini 2.0 Flash Thinking}
\date{\today}

% margins of 1 inch:
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{9in}
\setlength{\oddsidemargin}{0in}
\setlength{\textwidth}{6.5in}

\usepackage[pdftex]{hyperref} % hyperlink equation and bibliographic citations


\begin{document}
\maketitle

\section{Introduction}

In Newtonian physics, space and time are considered absolute and independent entities. Time, in particular, is universal and flows uniformly for all observers, irrespective of their motion or gravitational environment. This is the concept of \textit{absolute time}.

Einstein's theory of relativity, however, revolutionized our understanding of space and time by introducing the concept of \textit{spacetime}. In relativity, space and time are intertwined into a single four-dimensional continuum. Furthermore, time is no longer absolute but is relative and can be affected by motion and gravity.  This leads to phenomena such as time dilation and length contraction, which are absent in Newtonian physics.

While relativity provides a more accurate description of the universe, Newtonian physics remains a highly successful approximation in scenarios where velocities are much smaller than the speed of light and gravitational fields are weak. This document aims to illustrate how the relativistic concept of time approaches the Newtonian concept of absolute time in such conditions, effectively showing a bridge through a limiting case. We will focus on the phenomenon of time dilation from special relativity and show its Newtonian limit.

\section{Derivation}

We start with the time dilation formula from special relativity, which describes how time intervals are measured differently by observers in relative motion.

\subsection{Relativistic Time Dilation}
The time dilation formula is given by:
\begin{equation}
\Delta t = \gamma \Delta t_0
\label{time_dilation_formula}
\end{equation}
Where:
\begin{itemize}
    \item $\Delta t$ is the time interval measured by an observer in a reference frame in relative motion.
    \item $\Delta t_0$ is the proper time interval, measured by an observer at rest relative to the event.
    \item $\gamma$ is the Lorentz factor.
\end{itemize}

\subsection{Lorentz Factor}
The Lorentz factor, $\gamma$, is defined as:
\begin{equation}
\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}
\label{lorentz_factor_definition}
\end{equation}
Where:
\begin{itemize}
    \item $v$ is the relative velocity between the two reference frames.
    \item $c$ is the speed of light in a vacuum.
\end{itemize}

\subsection{Newtonian Limit Condition}
The Newtonian limit is characterized by velocities much smaller than the speed of light, i.e., $v \ll c$. In this regime, the ratio $\frac{v^2}{c^2}$ becomes very small.

\subsection{Taylor Expansion of Lorentz Factor for Small Velocities}
We can use the Taylor series expansion for $(1-x)^{-1/2}$ around $x=0$, which is given by
\begin{equation}
    (1-x)^{-1/2} \approx 1 + \frac{1}{2}x + O(x^2)
    \label{taylor_expansion_lorentz}
\end{equation}
  for small $x$. In our case,
  \begin{equation}
  x = \frac{v^2}{c^2}
  \label{x_substitution}
  \end{equation}
Applying this expansion to the Lorentz factor:
\begin{equation}
\gamma = \left(1 - \frac{v^2}{c^2}\right)^{-1/2} \approx 1 + \frac{1}{2} \frac{v^2}{c^2} + O\left(\left(\frac{v^2}{c^2}\right)^2\right)
\label{lorentz_factor_taylor_expanded}
\end{equation}
For $v \ll c$, the term $\frac{v^2}{c^2}$ is very small, and higher order terms $O\left(\left(\frac{v^2}{c^2}\right)^2\right)$ become negligible. Therefore, we can approximate the Lorentz factor as:
\begin{equation}
\gamma \approx 1 + \frac{1}{2} \frac{v^2}{c^2}
\label{lorentz_factor_approximation}
\end{equation}

\subsection{Approximation of Time Dilation in Newtonian Limit}
Substituting the approximated Lorentz factor back into the time dilation formula Eq.~\ref{time_dilation_formula}:
\begin{equation}
\Delta t \approx \left(1 + \frac{1}{2} \frac{v^2}{c^2}\right) \Delta t_0
\label{time_dilation_approx_lorentz_substituted}
\end{equation}
\begin{equation}
\Delta t \approx \Delta t_0 + \frac{1}{2} \frac{v^2}{c^2} \Delta t_0
\label{time_dilation_expanded}
\end{equation}
In the Newtonian limit where $v \ll c$, the term $\frac{1}{2} \frac{v^2}{c^2} \Delta t_0$ is extremely small compared to $\Delta t_0$.  As $v/c \rightarrow 0$, this term approaches zero. Thus, in the Newtonian limit, we have:
\begin{equation}
\Delta t \approx \Delta t_0
\label{newtonian_time_limit}
\end{equation}
This result indicates that in the limit of low velocities ($v \ll c$), the time interval measured by a moving observer ($\Delta t$) is approximately equal to the proper time interval ($\Delta t_0$). This means that time dilation becomes negligible, and time is measured in essentially the same way by all observers, regardless of their relative motion. This is consistent with the Newtonian concept of absolute time, where time is universal and unaffected by motion.

\section{Conclusion}

We have shown that in the limit of low velocities ($v \ll c$), the relativistic time dilation effect becomes negligible, and the time interval measured by moving observers approaches the proper time interval. This demonstrates how the relativistic concept of time, as described by special relativity, reduces to the Newtonian concept of absolute time under Newtonian conditions.  While this derivation focuses on time, a similar argument can be made for spatial dimensions and length contraction, showing that relativistic space also approaches Newtonian absolute space in the limit of low velocities. Therefore, Newtonian physics can be seen as a low-velocity approximation of relativistic physics, where the intertwining of space and time becomes less apparent, and space and time appear to be separate and absolute entities.

\end{document}