On Fri, 21 Feb 2025 19:48:16 +0000, Paul B. Andersen wrote:
Den 21.02.2025 02:45, skrev rhertz:
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18.02.2025 o 21:59, Paul B. Andersen wrote:
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In physics, an "event" is a point in space at a time,
or a point in spacetime.
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The metric can be used to find the spacetime interval between
two events, or the spacetime interval along a path between two events.
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It is quite common to use s² as the interval, but it is more 'natural'
to call the interval s, so that's what I will do.
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's' consists of two components, a temporal and a spatial.
If we call the temporal component cT and the spatial component D,
we have: s² = −c²T² + D²
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If D > cT then S is spacelike (s² > 0) D/T > c
If D = cT then S is lightlike (s² = 0) D/T = c
If D < cT then S is timelike (s² < 0) D/T < c
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Two events on the worldline of a massive object will always be
separated by a timelike interval, because the object's speed D/T
is always less than c, and D < cT.
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In the latter case it is common to set s = -cτ, and
the Schwarzschild metric becomes:
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c²dτ² = (1 - 2GM/c²r)c²dt² - 1/(1 - 2GM/c²r)dr² - r² dɸ²
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You can see this metric applied on satellites here:
https://paulba.no/pdf/Clock_rate.pdf
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(I know I am an idiot who bother to try to teach you
what you never will learn.)
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Watch what you asserted. A string of idiocies, scrambling everything
and
introducing your own terminology (arrogant asshole): :
>
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It is quite common to use s² as the interval, but it is more 'natural'
to call the interval s, so that's what I will do.
>
's' consists of two components, a temporal and a spatial.
If we call the temporal component cT and the spatial component D,
we have: s² = −c²T² + D²
>
If D > cT then S is spacelike (s² > 0) D/T > c
If D = cT then S is lightlike (s² = 0) D/T = c
If D < cT then S is timelike (s² < 0) D/T < c
>
Two events on the worldline of a massive object will always be
separated by a timelike interval, because the object's speed D/T
is always less than c, and D < cT.
*************************************************************************
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Pretentious asshole: Using s instead of ds? You are a mental case.
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ROFL
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You are yet again making a fool of yourself.
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Instead of your ignorant expression: s² = −c²T² + D²
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Minkowski's metric for spacetime is universally represented as:
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ds² = c² dτ² = -(c dx⁰)² + (dx¹)² + (dx²)² + (dx³)²
or
ds² = c² dτ² = -(c dt)² + dx² + dy² + dz² = -(c dt)² + dr²
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ds² = - c²dτ²
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ds² = -(c dt)² + dr²
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In this FLAT metric, ds²>0 doesn't mean FTL events.
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You are right about that, because ds²>0 is meaningless nonsense.
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It only takes two
events
to be SIMULTANEOUS in the same worldline to define dt=0.
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But in a curved spacetime defined by Schwarzschild's metric (around a
massive body), the line element is much more complex and subtle.
Spacelike events don't require FTL occurrence. It's just mathematical
common sense.
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"Spacelike events don't require FTL occurrence."
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:-D
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Being
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ds² = -(1-2GM/c²r) c²dt² + 1/(1-2GM/c²r) dr² + r²(dθ² + sin² θ dϕ²)
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For events around a single massive object (what was Schwarzschild's
metric
conceived for), the equation for different examples, being ds²>0 are:
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1.Two Events at the Same Time but Different Radial Coordinates
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So let's specify the events:
E(t,r,θ,ϕ) E₁ = (0,r₁,0,0) E₂ = (0,r₂,0,0)
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Consider two events occurring at the same coordinate time t but at
different radial coordinates r1 and r2. The spacetime interval between
these events is:
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ds² = dr²/(1-2GM/c²r) + r²(dθ² + sin² θ dϕ²)
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If the angular separation is zero (dθ = dϕ = 0), the interval simplifies
to:
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ds² = dr²/(1-2GM/c²r)
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OK
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ds = (1/√(1-2GM/c²r))dr
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r₂
s = ∫(1/√(1-2GM/c²r))dr = D, a distance
r₁
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s² = D²>0 the interval is spacelike , D/T = ∞ > c
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Obviously! Same time and separate position => space like.
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Since 1/(1-2GM/c²r) > 0 outside the event horizon (c²rv>v2GM), ds²>0,
and the events are spacelike separated.
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Your blunder again. ds can't be assigned a value.
Basic calculus error!
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2.Two Events at the Same Radius but Different Times (conflicting views):
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E(t,r,θ,ϕ) E₁ = (t₁,r,0,0) E₂ = (t₂,r,0,0)
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Note that the events are at the same spatial position.
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Two events occurring at the same radial coordinate r but at different
times t1 and t2. The spacetime interval between these events is:
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ds² = −(1-2GM/c²r) dt²
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ds² = −(1-2GM/c²r)c²dt²
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t₂
s = √(-(1-2GM/c²r))c∫dt = √(-(1-2GM/c²r))c(t₂-t₁) an imaginary number
t₁
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s² = -(1-2GM/c²r)c²(t₂-t₁) < 0, the interval is timelike
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In this case it would be better to use the metric:
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c²dτ² = (1-2GM/c²r)c²dt²
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t₂
τ = √(1-2GM/c²r)∫dt = (1-2GM/c²r))(t₂-t₁) = T, a time interval
t₁
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s² = - c²τ² = - c²T² < 0, the interval is timelike, D/T = 0 < c
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Obviously! Same position and separate time => timelike.
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Since (1-2GM/c²r) > 0 outside the event horizon, ds² < 0, and the events
are timelike separated.
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Your blunder yet again.
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HOWEVER, if the events occur at the same radius
but are separated by a large angular distance (e.g., Δϕ or Δθ), the
interval becomes:
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Another one with same time and separate position is too boring.
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Let's have separate position and separate times.
A more interesting case.
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I choose to let ϕ = 0
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E(t,r,θ,ϕ) E₁ = (t₁,r,θ₁,0) E₂ = (t₂,r,θ₂,0)
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ds² = -(1-2GM/c²r)c²dt² + r²dθ²
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ds² = ds₁² + ds₂²
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ds₁² = -(1-2GM/c²r)c²dt² or τ² = (1-2GM/c²r)dt²
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t₂
τ = √(1-2GM/c²r)∫dt = (1-2GM/c²r))(t₂-t₁) = T, a proper time interval
t₁
s₁² = −c²T²
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ds₂² = r²dθ²
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θ₂
s₂ = r∫dθ = r(θ₂-θ₁) = D, a distance, the length of an arc
θ₁
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s² = s₁² + s₂² = −c²T² + D²
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Why did you think that this equation meant that spacetime was flat?
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If cT > D or (1-2GM/c²r))(ct₂-ct₁) > r(θ₂-θ₁), the interval is timelike
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If cT < D or (1-2GM/c²r))(ct₂-ct₁) < r(θ₂-θ₁), the interval is spacelike
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If cT = D or (1-2GM/c²r))(ct₂-ct₁) = r(θ₂-θ₁), the interval is lightlike
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ds² = r²(Δθ² + sin² θ Δϕ²) r
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If the angular separation is large enough, ds²>0, and the events are
SPACELIKE separated.
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In this case you have same time and separate positions and
the interval is _always_ spacelike irrespective of what the distance
between the positions might be.
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But for the umpteenth time: you can't assign a specific value to ds,
and if you could ds² would always be positive and according to you,
all intervals should be spacelike. This is nonsense!
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Why are you so ignorant of basic calculus?
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We stop here.
I'm bored with your stupid replies.
Spacelike events using Schwarzschild metric are seriously analyzed and
used by relativists in several fields.
Before I post a wall of words with examples, I want to clarify your
stupid comment "ds can't be assigned a value" and having the stoneface
to attribute to me such interpretation. This is far from true, norwegian
wood.
Now, EAT THIS EXTRACT ABOUT USE OF SPACELIKE EVENTS IN CURRENT PHYSICS
(Hint: It doesn't imply AT ALL FTL events).
THIS EXTRACT SHOWS HOW FUCKED UP RELATIVISTIC PHYSICS IS, TRYING TO
EXPLAIN
THE WEIRDEST THINGS ACCOMODATING GENERAL RELATIVITY AT EVERY CORNER.
AND THAT'S WHY MODERN SCIENCE BASED ON RELATIVITY IS SO FUCKED UP.
AS I SAID, YOU CAN EAT WHAT FOLLOWS, EXTRACTED FROM THE CORE OF GR
MODERN
APPLICATIONS. IT'S PURE SHIT. A PSEUDOSCIENCE.
BUT, AS YOU LIKE IT, GO AHEAD AND EAT IT.
I, MEANWHILE, LAUGH AND DISPISE ALL THIS CRAPPY SHIT.
ENJOY.
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Spacelike events do have real use in science, especially in the context
of theoretical physics and general relativity. While spacelike events
themselves aren't directly observable (since they can’t be connected
causally through any physical signal, like light or matter), they are
crucial in understanding the structure of spacetime and the behavior of
the universe.
Some ways spacelike events are relevant:
1. Defining Causality and Spacetime Structure
Spacelike events help establish the causal structure of spacetime.
Understanding which events can influence each other and which cannot
(based on whether they are timelike, lightlike, or spacelike separated)
is essential in general relativity. It allows physicists to:
• Determine the behavior of particles and light in curved spacetime.
• Define the boundaries of possible information flow in the universe.
• Make predictions about the future evolution of systems (like stars,
black holes, or cosmological models).
In general relativity, the concept of spacelike separation plays an
important role in causal diagrams and spacetime diagrams, which are
used to visually understand the structure of events in spacetime.
In these diagrams, spacelike events are those that lie outside of
the light cones of each other and cannot be causally connected.
2. Relativity and the Speed of Light
The speed of light acts as a strict upper limit on the speed at which
information can travel through spacetime. Spacelike separation
represents situations where the distance between events is so large that
no signal,
even light, could travel from one to the other within any reasonable
time frame.
3. Quantum Field Theory and Entanglement
In the realm of quantum mechanics, spacelike separation also plays a
role,
especially in the context of quantum entanglement. Although two
particles
may be spacelike-separated (meaning no signal could travel between
them),
they can still exhibit correlated behavior due to their quantum
entanglement.
This phenomena, often discussed in terms of spooky action at a distance
(a phrase coined by Einstein), raises intriguing questions about
non-locality
in quantum mechanics.
While this doesn’t imply any physical causal connection in the classical
sense, the concept of spacelike separation helps to clarify the limits
of classical causality versus quantum entanglement, which is relevant in
quantum field theory.
4. Cosmology and the Early Universe
In cosmology, spacelike events are sometimes used to describe the
relationship between different points in the universe’s history or
different regions of spacetime that are not causally connected. For
example:
• During the early universe, before the universe became transparent
(before
the cosmic microwave background formed), different regions were causally
disconnected because they were separated by large distances, meaning
they
were spacelike-separated.
• Events that happened in distant regions of the universe at the same
time
can be spacelike-separated if no signal could travel between them due to
the
finite speed of light and the expansion of the universe.
• Understanding spacelike separation in this context helps to understand
cosmological models like the inflationary universe, where certain
regions
of space may have never interacted but are still part of the same
overall
structure.
5. Black Hole Information Paradox
In the context of black holes, the concept of spacelike-separated events
becomes important when analyzing phenomena like the information paradox.
When one part of the spacetime near a black hole’s event horizon might
be
spacelike-separated from another part inside the event horizon, and it's
wanted to understand how information might be preserved or lost. This
plays a role in understanding the fundamental principles behind quantum
gravity and the interaction between quantum mechanics and general
relativity.
6. Relativity in High-Energy Physics
In high-energy physics, such as in particle collisions or accelerator
experiments, spacelike separation come into play when dealing with
relativistic particles. For example:
• High-energy collisions can produce spacelike-separated events if the
particles created are far apart enough, and their separation exceeds
what can be connected by the speed of light during the time of the
experiment.
• These kinds of interactions can be analyzed using tools like Feynman
diagrams in quantum field theory, where spacelike-separated events might
represent interactions that cannot influence one another directly but
are part of a broader process.
Summary
While spacelike events themselves are not "observable" in the
traditional
sense (since they cannot influence each other causally), they are an
essential part of the theoretical understanding of spacetime, causality,
and the limits of information transfer. They help to understand:
• Causal structure in general relativity.
• The limits of communication in spacetime.
• Quantum phenomena like entanglement.
• Cosmological models of the early universe and inflation.
• High-energy particle interactions.
Spacelike-separated events play a significant role in the
theoretical framework that governs relativistic physics, helping to
understand the universe at both large and small scales.
Challenge for Paul; Probe that with Mercury ds^2>0 and the solution is spacelike
Re: Challenge for Paul; Probe that with Mercury ds^2>0 and the solution is spacelike