In other words, the earth is in the framework of the fixed stars and the spaceship is too but it is not within the framework of the earth as an enertial frame moving the opposite way. With this clarification, time has a different connotation. Since inertia (mass) is shown to increase at relativistic speeds, the ship and all that is in it is affected by the slower speed at which objects move in that local Space-Now ("Time-B"). Special Relativity does not exclude Space-Now.
Special Relativity Postulated Mass Increae Formula
Special Relativity Mass Increae Graph
Muon Decay and Distance Decrease:
From the muon's perspective, the distance to Earth is dramatically length-contracted, meaning it has a much shorter distance to travel in its own existence.
Because of symmetry in SR, the earth is moving towards the muon at the same speed and in it's frame, it sees a Lorentz contracted distance to the muon.
Decay Process: Muon decay is governed by the weak interaction, characterized by a decay constant. In special relativity, the proper time in the muon’s frame is dilated in the Earth frame (t = γt).
For v = 0.999cv = 0.999cv = 0.999c, γ ≈ 22.4 nanoseconds, so the half-life appears as 2.2 × 22.4 ≈ 49.3 μs, allowing more muons to reach Earth’s surface.
Experimental Evidence: Cosmic ray experiments (e.g., Rossi-Hall, 1941) and accelerator tests (e.g., CERN muon storage rings, 1970s) show muon lifetimes extended by exactly γ, matching time dilation predictions. Other particles (e.g., pions, kaons) show similar lifetime extensions at relativistic speeds, proportional to γ, regardless of their rest mass.
If mass increase altered decay, we’d expect different particles (e.g., muons vs. pions) to show different decay behaviors at the same velocity, due to their different rest masses. Instead, all unstable particles show lifetime extensions proportional to γ, consistent with time dilation.
Photons, as massless particles traveling at the speed of light, do not experience time or distance in the way observers with mass do. According to special relativity, for a photon moving at the speed of light (c), time dilation and length contraction reach their extreme limits:Time: From the perspective of a photon, time is effectively "frozen." In the reference frame of a photon (if such a frame could exist, which it technically cannot due to relativity), the proper time experienced is zero. This means that from the moment a photon is emitted to the moment it is absorbed, no time passes for the photon itself, regardless of how long it appears to take from an external observer’s perspective.
Distance: Similarly, due to length contraction, the distance a photon travels is contracted to zero in its own "perspective." For a photon, the starting point and endpoint of its journey are effectively the same point in space.
This is a consequence of Lorentz transformations in special relativity, where the spacetime interval for a light-like path (null geodesic) is zero. However, this is a theoretical limit, as we cannot define a rest frame for a photon since it always moves at (c) in all inertial frames.In summary, photons do not experience time or distance in any meaningful sense; their "experience" (if we anthropomorphize) is instantaneous and point-like, even across vast cosmic distances as perceived by us.
Explanation of Space Shrinkage (Lorentz Contraction) for a Fast Muon:Relativistic Speed of Muons:Muons are subatomic particles often produced in high-energy processes, like cosmic ray collisions in Earth's atmosphere, moving at speeds approaching c (e.g., 0.99c or higher).
Lorentz Contraction:The length of an object (or the space it moves through) in the direction of motion is contracted where: (L) is the contracted length observed by a stationary observer (symmetry that cancels time dilation conclusion) is the proper length (the length in the muon's rest frame).. This means the space in front of the muon appears significantly shortened
to an observer on Earth.
Space Shrinkage in Front of the Muon:From the perspective of an observer on Earth, the distance the muon travels (e.g., through the atmosphere to a detector) is contracted because of its high velocity. For example, a 10 km distance in the Earth's frame might appear much shorter (e.g., ~1 km or less) to the muon in its own frame due to Lorentz contraction. This is equivalent to saying the space in front of the muon "shrinks" in the direction of its motion, as observed by the stationary observer.
Why This Matters for Muons: Muons have a short mean lifetime (~2.2 µs in their rest frame) before decaying into other particles. At non-relativistic speeds, they wouldn't travel far enough to reach Earth's surface from the upper atmosphere. For example, a muon traveling at 0.99c sees the atmosphere's thickness contracted, so it perceives a shorter distance to travel, effectively "shrinking" the space in front of it. Intuitive Picture: Imagine the muon moving through a "compressed" space in the direction of its motion, as seen by a stationary observer. The faster the muon moves, the more the space in front of it appears squeezed, reducing the effective distance it needs to cover to reach a detector.
Key Takeaway:Lorentz contraction causes the space in front of a fast-moving muon to appear significantly shortened in the direction of motion for a stationary observer.
Even if these subatomic phenomena prove the time SR dilation postulate, the entire subatomic matter has no effect on human existence which will do much better in understanding all events by seeing and believing that the world exhibits space and now and not time. Life's more real and free to do so. This muon effect can be compared to Einstein's view of "spooky action at a distance" for instantaneous quantum paired particles' effect on each other regardless of distance. The muon extended existence is "spooky" and it's interpretation negates living life in space and now for a better understanding of reality without faulty past, present and fure ideas and the anxiety it can create for no reason.
Hafele-Keating Experiment
The Hafele-Keating Experiment: The Airplane Test of Time with Cesium Clocks. The "airplane test of time" refers to the famous Hafele-Keating experiment conducted in 1971, which tested Albert Einstein's theories of special and general relativity using cesium atomic clocks aboard four commercial airliners plus one in the "proper frame" on earth to compare with. This experiment will be covered in detail below. Because of an arbitrary choice of one of two frames, making one the "proper frame', the exeriment simply exibits symmetry of exended relative motions in now. See discussion below.
Military GPS and civilian GPS adjust for time dilation caused by both special and general relativity, as this correction is crucial for accuracy; the adjustment is made by pre-launch clock frequency adjustments and ongoing recalibrations performed by ground control centers to compensate for the differing effects of speed and gravity on the satellites' atomic clocks. Without these adjustments, positioning errors would accumulate rapidly, rendering the system useless.
How Time Dilation Affects GPS:
Special Relativity):
The high speed of the GPS satellites causes their onboard clocks to tick slower relative to clocks on Earth.
• General Relativity
The weaker gravitational field experienced by the satellites at their high altitude causes their clocks to tick faster than ground-based clocks.
The Combined Effect:
• The gravitational effect is stronger than the speed effect, resulting in a net increase in time on the satellite clocks compared to Earth-based clocks.
• If left uncorrected, this difference would accumulate to several kilometers of positional error per day.
The Correction Process:
Pre-Launch Adjustment:
The atomic clocks on the satellites are designed to "tick" at a slightly slower frequency than ground reference clocks before launch.
In-Orbit Recalibration:
The GPS ground control center, located at Colorado Springs, regularly monitors the satellite clocks and uploads adjustments to the onboard oscillators to maintain synchronization.
Receiver Calculations:
The GPS receivers on the ground also incorporate data from the satellites to perform any necessary special relativistic timing calculations for positioning.
Both military and civilian GPS systems must adjust for time dilation, but the term "military GPS" doesn't refer to a separate system with different physics; rather, it's the same GPS network whose critical applications (like precision navigation) rely on robust relativistic corrections to function accurately. The adjustments involve pre-launch clock adjustments and ongoing corrections that account for both special relativistic time dilation (due to satellite speed) and general relativistic time dilation (due to gravity). Without these adjustments, GPS would produce errors large enough to be useless within a single day.
Commercial airlines
Commercial airlines do not take time dilation into consideration because the effect is too minuscule to be relevant for flight operations, though it can be measured with atomic clocks and is crucial for the accuracy of GPS systems. Passengers on a commercial flight age a fractionally slower (or faster, depending on direction) than someone on the ground, but the difference is measured in nanoseconds and accumulates to less than a thousandth of a second over a 70-year lifespan of constant flight.
Why Time Dilation Doesn't Matter for Airlines
• Negligible Speed:
The speed of commercial aircraft (around 500-600 mph) is very small compared to the speed of light. Time dilation effects only become significant at speeds approaching the speed of light.
• Minor Gravitational Effect:
While planes are at a higher altitude and experience a weaker gravitational pull, speeding up time, this effect is also tiny. The combination of speed and gravity effects results in a negligible difference for passengers.
Gravity and Curved Spacetime - General Relativity
Curved Spacetime Gravity is Not a Force as in Newtonian Classical Physics
Einstein's proposed gravity is not a Newtonian force but a manifestation of the curvature of #spacetime. Einstein used the mathematical framework of Riemann, Ricci's synthetic curvature (sub-Riemannian geometry) and Minkowski.
https://einstein.stanford.edu/SPACETIME/spacetime2.html
Curved "Spacetime" Gravity
But, in 1908, Minkowski presented the above light cone geometric interpretation of spacetime special #relativity into a single four-dimensional continuum now known as Minkowski spacetime in the absence of gravitation. Einstein initially dismissed Minkowski's interpretation as "superfluous learnedness".
Further, E=mc^2 existed before Einstein. Isaac Newton, S. Tolver Preston, Poincaré, De Pretto and F. Hasenöhrl are the philosophers and physicists who had previously put forth the idea of E=mc^2 before 1905.
https://physicsforums.com/threads/einstein-did-not-derive-e-mc2-first.28362/
David Hilbert was a leading mathematician who worked alongside and corresponded with Albert Einstein during the development of Einstein's General Theory of Relativity in 1915. While Einstein conceived the core physical ideas, Hilbert developed rigorous mathematical foundations, even publishing his version of the field equations around the same time as Einstein's final paper. Einstein acknowledged Hilbert's mathematical genius and the resulting priority dispute was resolved by Einstein's gracious letter and a shared understanding that both were vital contributors to the theory's development.
Collaboration and the "Relativity Race"
• Meeting in Göttingen:
In 1915, Hilbert invited Einstein to the University of Göttingen to lecture on his developing theory of general relativity.
• Mutual Influence:
During this visit and through subsequent correspondence, Einstein and Hilbert exchanged ideas and worked on the mathematical framework for gravity.
• The Priority Dispute:
Both men worked feverishly to find the final form of the field equations. Einstein submitted his final paper just days before Hilbert submitted his, leading to a brief period of controversy over who discovered the theory first.
Their Contributions to General Relativity
• Einstein's Physical Intuition: Hilbert's strength was in formal mathematics, and his contribution provided
the rigorous mathematical framework that solidified the theory. [5]
The Einstein-Hilbert Action: a mathematical formulation that yields the field equations of general relativity when applied to the principle of least action.
Resolution and Legacy [6]
Einstein's Graciousness:
Einstein wrote a letter to Hilbert expressing friendship and acknowledging their shared efforts, despite initial anger over the perceived plagiarism of ideas from Hilbert's paper.
• Shared Foundation:
The combined efforts of Einstein and Hilbert are seen as a testament to the synergistic relationship between physics and mathematics, with both providing essential pieces for the complete picture of general relativity.
German physicist Heinrich Hertz
The photoelectric effect
The photoelectric effect was discovered by German physicist Heinrich Hertz in 1887 when he observed that shining ultraviolet light on a metal could cause it to release sparks. While Hertz made the initial discovery, it was Albert Einstein who provided the theoretical explanation in 1905, introducing the concept of photons and earning a Nobel Prize for his work. Discovery of the Photoelectric Effect: • Who: Heinrich Hertz • When: 1887 • What: Hertz noticed that when ultraviolet light hit a metal plate, the metal emitted sparks. He observed that the energy of the light needed to be above a certain threshold frequency for this to happen, which was a puzzling observation that could not be explained by existing wave theory. Explanation of the Effect: • Who: Albert Einstein • When: 1905 • What: Einstein explained Hertz's findings by proposing that light energy is carried in discrete packets called photons. He demonstrated that these photons interact with electrons in the metal, kicking them out of the material, and this explanation became a foundational concept in quantum theory.
French physicist Jean Perrin
The person who helped confirm Albert Einstein's theory of
Brownian motion and used it to determine the size of atoms was French
physicist Jean Perrin. Here is how their work collaborated: • Einstein's Theory (1905): At a time when many scientists still debated the existence of atoms, Einstein published his mathematical theory on Brownian motion. He showed that the random jiggling of microscopic particles, observed by botanist Robert Brown, was caused by the particles being hit by invisible molecules of the surrounding fluid. Einstein's analysis provided a way to link this visible motion to the statistical behavior of the invisible molecules. • Perrin's Experiments (1908): Starting in about 1908, Perrin and his students conducted a series of experiments to test Einstein's predictions. He studied the motion of tiny, suspended particles, such as those from the gamboge plant, using a powerful ultramicroscope. By measuring the particles' movement and sedimentation, Perrin was able to verify Einstein's equations. • Confirming the Size of Atoms: Perrin's work experimentally confirmed the kinetic theory of matter and provided a reliable way to calculate the size of molecules and atoms. It also led to an accurate determination of Avogadro's number, which is the number of atoms or molecules in one mole of a substance. • End of the Atomic Debate: The conclusive experimental evidence provided by Perrin's work ended the long-standing scientific skepticism about the physical reality of atoms. For this achievement, Perrin was awarded the Nobel Prize in Physics in 1926.
Quantum Entanglemant Casts Doubt on Light Being an Insurmoutable Speed Limit
The loophole-free Bell test conducted by TU Delft using diamonds demonstrated quantum nonlocality. The experiment showed that a measurement on one entangled particle can influence its partner instantaneously, regardless of distance, which refutes local realism—the worldview that objects are only influenced by their immediate surroundings. The experiment and its key findings In 2015, a team led by Professor Ronald Hanson at Delft University of Technology conducted a groundbreaking Bell test using electron spins in diamonds separated by 1.3 kilometers.
Closed all loopholes: The experiment was the first to simultaneously close the three major "loopholes" that could allow for alternative, non-quantum explanations of Bell test results.
Locality loophole:
By separating the labs by 1.3 km, the experiment ensured the measurement settings were chosen and results recorded too quickly for any signal traveling at or below the speed of light to pass between them.
Detection loophole:
Using nitrogen-vacancy (NV) centers in diamonds allowed the researchers to achieve a nearly 100% detection rate for the entangled electron spins, unlike photon-based experiments that often lose particles. ◦ Freedom-of-choice loophole: Fast, random measurement choices were used to ensure the settings were not predetermined, preventing a local-realist theory from correlating the settings with the particles' properties.
Confirmed nonlocality: The experiment's results clearly violated the Bell inequality, confirming the existence of the instantaneous quantum connection that Einstein famously called "spooky action at a distance". Entanglement vs. faster-than-light communication While the Delft experiment confirmed that quantum effects are nonlocal..
No controllable signal: The instantaneous "spooky action" is random and cannot be controlled to send a message. When a measurement is made on one entangled particle, the outcome is fundamentally probabilistic.
Correlations revealed later: An observer at one location cannot know the measurement outcome at the other location until the results are compared classically, which must happen at or below the speed of light. The instantaneous effect only establishes the correlation, not a usable information channel, unless your local particle reacts to the affect on the farther particle to detect a disturbance there, instantaneously here, as a sort of radar warning.
Delft University of Technology finally performed what is seen as the ultimate test against Einstein’s worldview: the loophole-free Bell test. The scientists found that two electrons, separated 1.3 km from each other on the Delft University campus, can indeed have an invisible and
instantaneous connection: the spooky action is real.
Quantum computers
Quantum computers do operate, but they are highly specialized, sensitive machines that require extreme conditions to function, such as near-absolute-zero temperatures and insulation from environmental interference. They leverage quantum mechanical phenomena like superposition using qubits to perform calculations far beyond the capabilities of classical computers, though they are still in early stages of development and not yet ready for widespread, practical applications.
How they operate:
Quantum Mechanics: Quantum computers exploit the principles of quantum mechanics to process information in ways impossible for classical computers. Instead of classical bits, they use qubits, which can represent 0, 1, or both states simultaneously through superposition. • Superposition: This property allows a qubit to exist in multiple states at once, exponentially increasing computational power for certain problems.
Entanglement
:
Multiple qubits can be entangled, meaning their fates are linked, allowing for complex correlations and significantly higher computational power.
Quantum Gates:
Quantum operations are performed by manipulating qubits with quantum gates, similar to how logic gates operate on bits in a classical computer.
Operating conditions:
Extreme Cold:
Many quantum computers require extremely low temperatures, close to absolute zero, to maintain the delicate quantum states of their qubits
.
Isolation:
They must be carefully insulated from environmental factors like vibrations, electromagnetic fields, and atmospheric pressure to prevent "measurement errors" and the loss of information.
Controlled Environment:
These precise conditions are achieved through complex machinery with specialized components like vacuum chambers and shielding.
Current Status: Early Development:
While operational quantum computers exist, they are in their infancy and face significant hurdles in scalability and error correction.
Specialized Tasks:
They are not suitable for all computational tasks but are being developed for specific, complex problems, such as drug discovery, materials science, and cryptography.
Ongoing Research:
Major companies and research institutions are investing heavily in advancing quantum computing hardware, algorithms, and error correction techniques to make them more practical.
Current Challenges Facing Quantum Computers
While quantum computers exist in a basic experimental form, several challenges must be addressed before they can be widely used in industry:
1. Scalability: Current quantum computers have only a limited number of qubits. For practical use, quantum computers will need to scale up to thousands or even millions of qubits.
2. Quantum Decoherence: Qubits lose their quantum state due to interactions with their environment, a phenomenon known as decoherence. Maintaining qubit coherence for longer periods is a critical hurdle.
3. Error Rates: Quantum systems are prone to errors, and managing these errors remains a significant challenge. Quantum error correction, as mentioned, is an area of intense research.
4. Temperature Requirements: Many quantum computers, especially those using superconducting qubits, require extremely low temperatures to function, which makes them expensive to build and maintain
.
5. Practical Algorithms: While there are a few quantum algorithms that show promise, many others remain theoretical or require a much more advanced quantum computer to be useful.
Quantum computers do exist, but they are not yet fully ready for mainstream applications. We are still in the early phases of developing quantum hardware, algorithms, and error-correction techniques. Researchers and companies are making significant strides toward creating functional, scalable quantum computers, but much work remains to be done.
The Einstein field Equation
The Einstein field equation (EFE) may be written in the form:
where
Rμν is the
Ricci curvature tensor,
R is the
scalar curvature,
gμν is the
metric tensor,
Λ is the
cosmological constant,
G is
Newton's gravitational constant,
c is the
speed of light in vacuum, and
Tμν is the
stress–energy tensor.
Despite the fact that it's over a century old,
Einstein's theory of general relativity
is our current understanding (physicists' consensus) of how gravity operates. In this view,
space and time are merged together into a unified framework known as (no
surprises here) space-time. This space-time isn't just a fixed stage
but bends and flexes in response to the presence of matter and energy.
That bending, warping and flexing of space-time then goes on to tell matter
how to move. In general relativity, everything from bits of light to
speeding bullets to blasting spaceships want to travel in straight
lines. But the space around them is warped, forcing them all to
follow curved trajectories — like trying to cross a mountain pass in a
straight line, but following the peaks and valleys of the topography. So-called gravitational time dialation (slowing relative to an outside observer somewhere else) in a strong gravitaional field is as follows:
T = T'([1-2GM/rc^2])^1/2
Spacetime curvature accounts for tidal accelerations of objects
It is the same as when a weighted object sits on a stretched rubber trampoline-like surface and its weight (mass) indents the surface causing mass objects far away to start moving through this curvature in three dimensions. The space around the earth is just like the rubber surface and is "indented" by the earth's mass, you could say as if in an additional dimension not readily perceptible to humans. The earth is basically lying in space as if space were the rubber surface holding it in place.
Space is Something not Nothing. Einstein's equation postulates that curvature and changing distance act just like moving mass - "mass creates geometry, and geometry acts like mass".
Spacetime curvature is demonstrated by change in separation of two originally parallel worldlines
Einstein postulates gravitation as action at a distance; curvature of spacetime and nothing more is all that is required to describe the millimeter or two change in separation in 8 seconds of two ball bearings, originally 20 meters apart in space above Earth, and endowed at the start with zero relative velocity. Moreover, this curvature completely accounts for gravitation.
Acceleration toward Earth: Totalized effect of relative accelerations, each particle toward its neighbor, in a chain of particles that girdles globe
Many local reference frames, fitted together, make up the global structure of spacetime. Each local Lorentz frame can be regarded as having one of the ball bearings at its center. The ball bearings all simultaneously approach their neighbors (curvature). Then the large-scale structure of spacetime bends and pulls nearer to Earth (Figure 9.6). In this way many local manifestations of curvature add up to give the appearance of long-range gravitation originating from Earth as a whole.
In brief, the geometry used to describe motion in any local free-float frame is the flat-spacetime geometry of Lorentz (special relativity). Relative to such a local freefloat frame, every nearby electrically neutral test particle moves in a straight line with constant velocity. Slightly more remote particles are detected as slowly changing their velocities, or the directions of their worldlines in spacetime. These changes are described as tidal effects of gravitation. They are understood as originating in the local curvature of spacetime.
Gravitation shows itself not at all in the motion of one particle but only in the change of separation of two or more nearby particles. "Rather than have one global frame with gravitational forces we have many local frames without gravitational forces." However, these local dimension changes add up to an effect on the global spacetime structure that one interprets as "gravitation" in its everyday manifestations.
Local curvature adding up to the appearance of long-range gravitation. The shortening of distance between any one pair, "A" and "B", of ball bearings is small when the distance itself is small. However, small separation between each ball bearing and its partner demands many pairs to encompass Earth. The totalized shortening of the circumference in any given time - the shortening of one separation times the number of separations - is independent of the fineness of the subdivision. That totalized pulling in of the circumference carries the whole necklace of masses inward. This is free fall, this is gravity, this is a large scale motion interpreted as a consequence of local curvature. Example:
Original separation between A and B
-and every other pair: 20 meters
Time of observation: 8 seconds
Shortening of separation in that time: 1 millimeter
Fractional shortening: 1 millimeter/20 meters =1/20,000
Circumference of Earth (length of airy necklace of ball bearings): 4.0030×107 meters
Shrinkage of this circumference in 8 seconds: 1/20,000×4.0030×107 meters = 2001.5 meters
Decrease in the distance from the center of Earth (drops by the same factor 1/20,000):
1/20,000×6.371×106 meters = 315 meters.
This apparently large-scale effect is caused - in Einstein’s picture - by the addition of a multitude of small-scale effects: the changes in the local dimensions associated with the curvature of geometry (failure of "B" to remain at rest as observed in the free-float frame associated with "A").
What is the source of the curvature of spacetime? Momenergy is the source. In Chapter 8 we saw the primacy of momenergy in governing interactions between particles.
The curvature in its character is totally “tideproducing,” totally “ noncontractile.”
Matter and Energy
Einstein's equations can be loosely summarized as the main relation between matter and the geometry of spacetime (describing gravitational motion). On the right hand side of the equation, the most important thing is the appearance of the energy-momentum tensor Tμν
It encodes exactly how the matter---understood in a broad sense, i.e. any energy (or mass or momentum or pressure) carrying medium---is distributed in the universe. For understanding how to interpret the subscript indices of the T, see explanation of the metric tensor below.
It is multiplied by some fundamental constants of nature (the factor 8πGc4)
but this isn't of any crucial importance: One can view them as
book-keeping tools that keep track of the units of the quantities that
are related by the equation. In fact, professional physicists typically
take the liberty to redefine our units of measurements in order to
simplify the look of our expressions by getting rid of pesky constants
such as this. One particular option would be to choose "reduced Planck
units", in which 8πG=1 and c=1, so that the factor becomes 1.
On the left hand side of Einstein's equations, we find a few
different terms, which together describe the geometry of space.
General relativity is a theory which uses the mathematical framework
known as (semi-)Riemannian geometry. In this branch of mathematics, one studies spaces which are in a certain sense smooth, and that are equipped with a metric. Let us first try to understand what these two things mean.
The smoothness property can be illustrated by the intuitive (and
historically important!) example of a smooth (two-dimensional) surface
in ordinary three-dimensional space. Imagine, for instance, the surface
of an idealized football, i.e. a 2-sphere. Now, if one focuses ones
attention to a very small patch of the surface (hold the ball up to your
own face), it seems like the ball is pretty much flat. However, it is
obviously not globally flat. Without regards for mathematical rigor, we can say that spaces that have this property of appearing locally flat are smooth
in some sense. Mathematically, one calls them manifolds. Of course, a
globally flat surface such as an infinite sheet of paper is the simplest
example of such a space.
In Riemannian geometry (and differential geometry
more generally) one studies such smooth spaces (manifolds) of arbitrary
dimension. One important thing to realize is that they can be studied without
imagining them to be embedded in a higher-dimensional space, i.e.
without the visualization we were able to use with the football, or any
other reference to what may or may not be "outside" the space itself.
One says that one can study them, and their geometry, intrinsically.
The metric
When it comes to intrinsically studying the geometry of manifolds,
the main object of study is the metric (tensor). Physicists typically
denote it by gμν
.
In some sense, it endows us with a notion of distance on the manifold.
Consider a two-dimensional manifold with metric, and put a "coordinate
grid" on it, i.e. assign to each point a set of two numbers, (x,y). Then, the metric can be viewed as a 2×2 matrix with 22=4 entries. These entries are labeled by the subscripts μ,ν, which can each be picked to equal x or y. The metric can then be understood as simply an array of numbers:
(gxxgyxgxygyy)
We should also say that the metric is defined such that gμν=gνμ
, i.e. it is symmetric with respect to its indices. This implies that, in our example, gxy=gyx.
Now, consider two points that are nearby, such that the difference in coordinates between the two is (dx,dy). We can denote this in shorthand notation as dlμ where μ is either x or y, and dlx=dx and dly=dy. Then we define the square of the distance between the two points, called ds, as
ds2=gxxdx2+gyydy2+2gxydxdy=∑μ,ν∈{x,y}gμνdlμdlν
To get some idea of how this works in practice, let's look at an
infinite two-dimensional flat space (i.e. the above-mentioned sheet of
paper), with two "standard" plane coordinates x,y
defined on it by a square grid. Then, we all know from Pythagoras' theorem that
ds2=dx2+dy2=∑μ,ν∈{x,y}gμνdlμdlν
This shows that, in this case, the natural metric on flat two-dimensional space is given by
gμν=(gxxgxygxygyy)=(1001)
Now that we known how to "measure" distances between nearby points, we can use a typical technique from basic physics and integrate small segments to obtain the distance between points that are further removed:
L=∫ds=∫∑μ,ν∈{x,y}gμνdlμdlν−−−−−−−−−−−−−√
The generalization to higher dimensions is straightforward.
Curvature tensors
As explained above, the metric tensor defines the
geometry of our manifold (or spacetime, in the physical case). In
particular, we should be able to extract all the relevant information
about the curvature of the manifold from it. This is done by
constructing the
Riemann (curvature) tensor Rμ νρσ
, which is a very complicated object that may, in analogy with the array visualization of the metric, be regarded as a four-dimensional array, with each index being able to take on
N values if there are
N coordinates
{x1,…xN} on the manifold (i.e. if we're dealing with an
N-dimensional
space). It is defined purely in terms of the metric in a complicated
way that is not all too important for now. This tensor holds pretty much
all the information about the curvature of the manifold---and much more
than us physicists are typically interested in. However, sometimes it
is useful to take a good look at the Riemann tensor if one
really wants to know what's going on. For instance, an everywhere vanishing Riemann tensor (
Rμ νρσ=0)
guarantees that the spacetime is flat. One famous case where such a thing is useful is in the
Schwarzschild metric describing a black hole, which seems to be singular at the Schwarzschild radius
r=rs≠0.
Upon inspection of the Riemann tensor, it becomes apparent that the
curvature is actually finite here, so one is dealing with a
coordinate singularity rather than a "real" gravitational singularity.
By taking certain "parts of" the Riemann tensor, we can discard some
of the information it contains in return for having to only deal with a
simpler object, the Ricci tensor:
Rνσ:=∑μ∈{x1,…xN}Rμ νμσ
This is one of the tensors that appears in the Einstein field equations. the second term of the equations features the Ricci
scalar R
, which is defined by once again
contracting (a fancy word for "summing over all possible index values of some indices") the Ricci tensor, this time with the
inverse metric
gμν which can be constructed from the usual metric by the equation
∑ν∈{x1,…,xN}gμνgνρ=1 if μ=ρ and 0 otherwise
As promised, the Ricci scalar is the contraction of the Ricci tensor and inverse metric:
R:=∑μ,ν∈{x1,…xN}gμνRμν
Of course, the Ricci scalar once again contains less information than
the Ricci tensor, but it's even easier to handle. Simply multiplying it
by
gμνonce again results in a two-dimensional array, just like
Rμν and
Tμν are. The particular combination of curvature tensors that appears in the Einstein field equations is known as the
Einstein tensor
Gμν:=Rμν−12Rgμν
The cosmological constant
There is one term that we have left out so far: The cosmological constant term
Λgμν
. As the name suggests,
Λ is simply a constant which multiplies the metric. This term is sometimes put on the other side of the equation, as
Λ
can be seen as some kind of "energy content" of the universe, which may
be more appropriately grouped with the rest of the matter that is
codified by
Tμν.
Einstein's equation relates the matter content (right side of the
equation) to the geometry (the left side) of the system. It can be
summed up with "mass creates geometry, and geometry acts like mass".
For more detail, let's consider what a tensor is. A two-index tensor
(which is what we have in Einstein's equation), can be thought of as a
map which takes one vector into another vector. For example, the
stress-energy tensor takes a position vector and returns a momentum
vector, mathematically
pν=Tνμxμ
The interpretation is that the right side of
Einstein's equation tells us the momentum which is passing through a
surface defined by the position vector.
The left side can be interpreted in this manner as well. The Ricci curvature
Rμν
takes a position vector and returns a vector telling us how much the curvature is changing through the surface defined by
x⃗ . The second and third terms, both having factors of the metric
gμν,
tell us how much distance measurements are changed when traveling along
the vector. There are two contributions to this change in distance -
the scalar curvature
R and the
Λ. If
Rμν is "curvature in a single direction", than
R is the "total curvature".
Λ is a constant which tells us how much innate energy empty space has, making all distances get larger for
Λ>0.
So, reading the equation right to left, "Einstein's equation tells us
that momentum (moving mass) causes both curvature and a change in how
distances are measured." Reading left to right, "Einstein's equation
tells us that curvature and changing distance acts just like moving
mass."
Newton's Third Law of Motion: To every action there is a corresponding reaction.
Einstein: Spacetime acts on "momenergy", telling it how to move; "momenergy" reacts back on spacetime, telling it how to curve.
Although Relativity does not consider gravity as a "Force," the Einstein equation is consistent with how Classical Newtonian Gravitaional "Force" may be calculated:
The acceleration of gravity depends on the mass of an object and the distance from its center, according to Newton's law of universal gravitation.
Acceleration Due to Gravity on the Moon
The acceleration due to gravity at the Moon is 1.62 m/s^2. This is approximately 1/6 that of the acceleration due to gravity on Earth, 9.81 m/s^2. The acceleration of an object depends on the mass of the object and the amount of force applied. Newton's law of universal gravitation describes a universal force of attraction between any two objects, where the force is equal in magnitude and opposite in direction for both objects, as stated by his Third Law of Motion. The perceived "less gravity" isn't a difference in the gravitational force but in the resulting acceleration, because the Second Law of Motion
(
F = ma) shows that a smaller mass experiences a much larger acceleration from the same force. Therefore, a large object like Earth exerts the same gravitational pull on a small object as the small object exerts on it, but the Earth's immense mass means its resulting acceleration is unnoticeable, while the smaller object's acceleration is significant.
Despite its contradictions and intricacy, relativity
remains the most "peer" accepted way to account for physical phenomena. Yet scientists know that their models are incomplete because (they say)
relativity is still not fully reconciled with quantum mechanics. which explains the properties of subatomic particles with extreme precision but does not incorporate the force of gravity.
Gravity as the curviture of space was experimentally verified in 1919 during a solar eclipse, where stars behind the sun appeared to be aside the sun.
In conclusion, the idea that there are three separate dimensions involving the misnomer Time is a fiction to deal with the misconceptions involved in the use of the word "Time." In reality, these are three different phenomena fictionally subsumed under the rubric of the common notion of "Time" which only applies in quantum mechanical experiments of relitivistic "Time Dilation", otherwise humans only live the Now of Space.
Postscript
Copernicus, Guido, Socrates, DaVinci, and Einstein were not peer reviewed or understood by the propagandized scientific community. That's because the peer review process is just a box to keep "science" in the hands of the aristocracy, religion and the Military Industrial Complex. The eternal propaganda machine is used to stifle original ideas and disparage and discourage actual thinkers if they don’t conform to “established” peer review by pay through the nose publications; the very people who have the old ideas you want to change. Think for yourself like the greatest thinkers did. That's the only way to discover new and innovative ideas. Don’t follow the shepards like a sheep. If you disagree or don't understand any part of this paper, don't blame the fact that it is not peer reviewed. Review it yourself or disprove it yourself. Don't claim that it's not generally accepted because that is a pathetic excuse to avoid confronting the ideas and proofs in this paper. The world of science would be greatly enhanced and freed if they dealt with the universal misconceptions they have about time, space and now, or disprove anything in this paper, in writing.
References
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