Relativistic space-time is described as a four-dimensional continuum comprising three dimensions of space and one dimension of time. In this framework, space and time are interwoven, forming an integrated space-time fabric. As time dilates due to relativistic effects, does this interconnected nature imply a dilation of space-time as a whole?
For context:
Cosmic Expansion: Describes how the distance between cosmic objects increases over time, which can be represented as:
t₀ < (t₀+Δt) = t₁ → (x₀,y₀,z₀,t₀) < (x₁,y₁,z₁,t₁)
Where (t₁ - t₀) = elapsed time.
Space-Time Dilation: Reflects how time dilation in relativistic contexts affects space-time coordinates:
t < t′ → (x,y,z,t) < (x′,y′,z′,t′)
Where t′ is dilated time
Given these representations, can the concept of space-time dilation be viewed as a form of space-time expansion in terms of their consequences?
No, one doesn’t have anything to do with the other. Time dilation and spatial contraction are coordinate artifacts when invariance under Lorentz transformations is a global symmetry; spacetime expansion is a property of a certain class of solutions to Einstein's equations, that express invariance under local Lorentz transformations.
It would be useful to study general relativity, for instance, from https://arxiv.org/abs/gr-qc/9712019
Stam Nicolis - I like to further reinforce your answer, which is spot on;
The concepts of space-time dilation and space-time expansion, while related to the fabric of space-time, are distinct and describe different phenomena.
Space-Time Dilation
Space-time dilation is a relativistic effect observed when there is a significant difference in velocity between two observers or when one observer is in a strong gravitational field. According to Einstein's theory of relativity, time dilates (slows down) for an observer moving at a significant fraction of the speed of light relative to another observer or in a strong gravitational field compared to a weaker one. This time dilation directly affects the time coordinate in space-time, leading to a change in the perception of time between different observers. However, it doesn't imply a stretching or expansion of the spatial dimensions themselves; it's more about how time is experienced differently depending on the observer's state of motion or gravitational influence.
Space-Time Expansion
The concepts of space-time dilation and space-time expansion, while related to the fabric of space-time, are distinct and describe different phenomena.
Space-Time Dilation
Space-time dilation is a relativistic effect observed when there is a significant difference in velocity between two observers or when one observer is in a strong gravitational field. According to Einstein's theory of relativity, time dilates (slows down) for an observer moving at a significant fraction of the speed of light relative to another observer or in a strong gravitational field compared to a weaker one. This time dilation directly affects the time coordinate in space-time, leading to a change in the perception of time between different observers. However, it doesn't imply a stretching or expansion of the spatial dimensions themselves; it's more about how time is experienced differently depending on the observer's state of motion or gravitational influence.
Space-Time Expansion
Space-time expansion, on the other hand, is a concept rooted in cosmology and is generally associated with the large-scale structure of the universe. It refers to the increase in the distance between two points in the universe over time due to the expansion of space itself, as described by the metric expansion of space in cosmology. This is what we observe as the universe expanding since the Big Bang. The spatial coordinates (x, y, z) are literally stretching, and this expansion is uniform on large scales.
The Key Differences
Scope and Scale
Space-time dilation is a localized effect that occurs due to relative motion or gravitational fields, whereas space-time expansion is a global phenomenon observed across the entire universe.
Nature of Effect
Space-time dilation affects the time coordinate relative to spatial coordinates, without stretching space. Space-time expansion, however, directly involves the stretching of the spatial dimensions, with time acting as the parameter that drives this expansion.
Relation to Observers
Time dilation depends on the relative state of motion or gravitational potential between observers. Space-time expansion occurs universally and is the same for all observers (on large scales).
Conceptual Equivalence
While both involve the fabric of space-time, space-time dilation and space-time expansion are not conceptually equivalent. Space-time dilation is about differences in the experience of time, while space-time expansion is about the growth of distances between points in space over time. They are different manifestations of the nature of space-time as described by General Relativity but arise from different causes and have different implications.
So, while space-time dilation impacts the experience of time in a specific reference frame, space-time expansion refers to the overall growth of the universe's spatial dimensions. Therefore, space-time dilation cannot be viewed as a form of space-time expansion in terms of their consequences; they describe different aspects of the universe's structure and dynamics.Space-time expansion, on the other hand, is a concept rooted in cosmology and is generally associated with the large-scale structure of the universe. It refers to the increase in the distance between two points in the universe over time due to the expansion of space itself, as described by the metric expansion of space in cosmology. This is what we observe as the universe expanding since the Big Bang. The spatial coordinates (x, y, z) are literally stretching, and this expansion is uniform on large scales.
Key Differences
Scope and Scale
Space-time dilation is a localized effect that occurs due to relative motion or gravitational fields, whereas space-time expansion is a global phenomenon observed across the entire universe.
Space-time dilation
This affects the time coordinate relative to spatial coordinates, without stretching space. Space-time expansion, however, directly involves the stretching of the spatial dimensions, with time acting as the parameter that drives this expansion.
Time dilation depends on the relative state of motion or gravitational potential between observers. Space-time expansion occurs universally and is the same for all observers (on large scales).
Further to that we have the following the issue of Conceptual Equivalence.
While both involve the fabric of space-time, space-time dilation and space-time expansion are not conceptually equivalent. Space-time dilation is about differences in the experience of time, while space-time expansion is about the growth of distances between points in space over time. They are different manifestations of the nature of space-time as described by General Relativity but arise from different causes and have different implications.
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