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Black-Hole and Time machine
- Thread starter ra1
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yes, that is true, you can't have infinite enery,then all the ideal situations such as zero friction, zero gravity, zero momentum, all these to be realistic!
cj4ever said:
CONSULT A DOCTOR, FINGER PRINTS OF TWO IDENTICAL TWINS ARE 100% DIFFERENT AT ALL TIMES!
mutantlast said:
i dont think humans thirst for speed has limits. at 50's scientists believe dat da human could only survive up to 3 G force but now days pilots experiencing more dan 15 G's
and hav u heard of BIG BANG Machine in dat machine Beams of protons would be hurled together at 99.9999999% of the speed of light. its a big step isnt it? making something travel as fast as light
it may seem impossible now with these evidences i think its possible in da future
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THE PROBLEM IS THAT PAST DOESNOT EXSIST ANYMORE. ENTIRE ROMAN EMPIRE IS ON RUINS AND IF WE TRAVEL THROUGH TIME TO ROMAN AGE, WHERE THE GREAT EMPIRE EXSISTS? IT SHOULD BE ON EARTH AT SAME LOCATION, BUT IT IS NO MORE THERE, HOW CAN WE CLARIFY THIS?
ra1 said:
Black Hole යනු කලු හිල....
අභ්*යවකාශයේ පවතින හිල් වර්ගයකිTime machine යනු කාල යන්ත්*රය... මාදන්නා පරිදි මෙය ඔරලෝසුවයි
It's only a imagine bro..funkoluwa said:
I request you to study Quantum theories and Albert Einstein's theories about time and space. Then you will understand this.
If not so, your idea is a personal sense only..
suppa malinda said:Black Hole යනු කලු හිල....
Time machine යනු කාල යන්ත්*රය... මාදන්නා පරිදි මෙය ඔරලෝසුවයි
mara joke ekak machan.. bada palenda hina yanawa..
not even light,,,,,, "TIME" also drawn into the black hole as they say...........have u ever watch discovery channel about this????dinesh4u said:
mutantlast said:
R u thinking "Big Bang Machine" is a Imagination ohh bro just Google it n see
and also time traveling is a theory of Einstein it aint ma personal sense its all about positiveness
Presumably, Brian’s goal is to be able to travel nearly instantaneously between any two points in the galaxy. The problem is that there are physical laws we have to obey; we don’t have a choice. One of them is special relativity, which tells us that the maximum speed any object can move through space is c, or the speed of light in a vacuum, also known as 299,792,458 meters-per-second. Furthermore, only things with zero mass can ever travel even that fast (and even then only in a vacuum); everything with a mass always, by the laws of physics, has to travel slower than that.
First off, why is that the case? The easiest explanation is that light always travels at the speed of light. Well, duh, you might say, that’s by its definition! But this is actually profound. What it means is that, no matter how quickly you move, light will always move relative to you at the same speed! Let’s say I shine a light at BatmanTM, and Batman and I are both standing still. We can both measure the speed of that light, and we’ll both get the same value: c. On the other hand, what if I move towards Batman? If it were anything other than light, like a baseball, a bullet, or an electron (even electrons have mass), Batman and I would measure different velocities. Here’s why; let’s say I throw a baseball at 100 mi/hr, and I run towards Batman at 20 mi/hr. I would measure the baseball to be moving at 100 mi/hr, but Batman would measure the baseball at 120 mi/hr.
But this doesn’t happen the same way with light; regardless of how fast I’m moving or how fast Batman moves, we both always measure the light beam to be moving at the same speed, c, or 670,000,000 mi/hr. If I take something with a mass, like my baseball, into a rocket, and I fly at 400,000,000 mi/hr towards Batman and launch the baseball at 400,000,000 mi/hr, will he observe the baseball moving at 800,000,000 mi/hr? No. Special Relativity tells us that velocities don’t work like that close to the speed of light, and Batman would see the baseball actually moving at 580,000,000 mi/hr, or less than the speed of light in a vacuum. Lots of funny things happen close to the speed of light, but this really boils down to three things:
This last one explains why something with a mass can never move faster than c. No matter how much energy I put into something with a mass, trying to accelerate it faster and faster, I wind up making it heavier close to the speed of light, making it harder to accelerate. It gets so hard as I approach the speed of light, that to actually reach c, anything with a mass requires an infinite amount of energy! And that is why it’s impossible for anything with a mass to travel even as fast as the speed of light in a vacuum.
funkoluwa said:
I did not tell you that big bang machine is not a reality. I know about it..
I tolled you about being fast than light..
You can travel in light speed, Only if you have 0 mass. It is the theory..
PLS study more about time and space theories like this..
then you will understand about this..
eference frames, coordinates and the Lorentz transformation
Relativity theory depends on "reference frames". A reference frame is an observational perspective in space at rest, or in uniform motion, from which a position can be measured along 3 spatial axes. In addition, a reference frame has the ability to determine measurements of the time of events using a 'clock' (any reference device with uniform periodicity).
An event is an occurrence that can be assigned a single unique time and location in space relative to a reference frame: it is a "point" in space-time. Since the speed of light is constant in relativity in each and every reference frame, pulses of light can be used to unambiguously measure distances and refer back the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired.
For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four space-time coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let's call this reference frame S.
In relativity theory we often want to calculate the position of a point from a different reference point.
Suppose we have a second reference frame S', whose spatial axes and clock exactly coincide with that of S at time zero, but it is moving at a constant velocity
Since there is no absolute reference frame in relativity theory, a concept of 'moving' doesn't strictly exist, as everything is always moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be comoving. Therefore S and S' are not comoving.
Let's define the event to have space-time coordinates
The
A quantity invariant under Lorentz transformations is known as a Lorentz scalar.
[edit] Simultaneity
Event B is simultaneous with A in the green reference frame, but it occurred before in the blue frame, and will occur later in the red frame.
Main article: Relativity of simultaneity
From the first equation of the Lorentz transformation in terms of coordinate differences
[edit] Time dilation and length contraction
Writing the Lorentz transformation and its inverse in terms of coordinate differences we get
Similarly, suppose we have a measuring rod at rest in the unprimed system. In this system, the length of this rod is written as Δx. If we want to find the length of this rod as measured in the 'moving' system S', we must make sure to measure the distances x' to the end points of the rod simultaneously in the primed frame S'. In other words, the measurement is characterized by Δt' = 0, which we can combine with the fourth equation to find the relation between the lengths Δx and Δx':
These effects are not merely appearances; they are explicitly related to our way of measuring time intervals between events which occur at the same place in a given coordinate system (called "co-local" events). These time intervals will be different in another coordinate system moving with respect to the first, unless the events are also simultaneous. Similarly, these effects also relate to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spacial distance from each other when seen from another moving coordinate system.
See also the twin paradox.
[edit] Causality and prohibition of motion faster than light
See also: Causality
Diagram 2. Light cone
In diagram 2 the interval AB is 'time-like'; i.e., there is a frame of reference in which event A and event B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the cause and B the effect).
The interval AC in the diagram is 'space-like'; i.e., there is a frame of reference in which event A and event C occur simultaneously, separated only in space. However there are also frames in which A precedes C (as shown) and frames in which C precedes A. If it were possible for a cause-and-effect relationship to exist between events A and C, then paradoxes of causality would result. For example, if A was the cause, and C the effect, then there would be frames of reference in which the effect preceded the cause. Although this in itself won't give rise to a paradox, one can show[19][20] that faster than light signals can be sent back into one's own past. A causal paradox can then be constructed by sending the signal if and only if no signal was received previously.
Therefore, one of the consequences of special relativity is that (assuming causality is to be preserved), no information or material object can travel faster than light. On the other hand, the logical situation is not as clear in the case of general relativity, so it is an open question whether there is some fundamental principle that preserves causality (and therefore prevents motion faster than light) in general relativity.
Even without considerations of causality, there are other strong reasons why faster-than-light travel is forbidden by special relativity. For example, if a constant force is applied to an object for a limitless amount of time, then integrating F = dp/dt gives a momentum that grows without bound, but this is simply because p = mγv approaches infinity as v approaches c. To an observer who is not accelerating, it appears as though the object's inertia is increasing, so as to produce a smaller acceleration in response to the same force. This behavior is in fact observed in particle accelerators.
See also the Tachyonic Antitelephone.
[edit] Composition of velocities
Main article: Velocity-addition formula
If the observer in S sees an object moving along the x axis at velocity w, then the observer in the S' system, a frame of reference moving at velocity v in the x direction with respect to S, will see the object moving with velocity w' where
[edit] Mass, momentum, and energy
Main article: Mass in special relativity
Main article: Conservation of energy
In addition to modifying notions of space and time, special relativity forces one to reconsider the concepts of mass, momentum, and energy, all of which are important constructs in Newtonian mechanics. Special relativity shows, in fact, that these concepts are all different aspects of the same physical quantity in much the same way that it shows space and time to be interrelated.
There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses conservation laws. If these laws are to remain valid in SR they must be true in every possible reference frame. However, if one does some simple thought experiments using the Newtonian definitions of momentum and energy one sees that these quantities are not conserved in SR. One can rescue the idea of conservation by making some small modifications to the definitions to account for relativistic velocities. It is these new definitions which are taken as the correct ones for momentum and energy in SR.
Given an object of invariant mass m traveling at velocity v the energy and momentum are given (and even defined) by
Relativistic energy and momentum can be related through the formula
For velocities much smaller than those of light, γ can be approximated using a Taylor series expansion and one finds that
Looking at the above formulas for energy, one sees that when an object is at rest (v = 0 and γ = 1) there is a non-zero energy remaining:
Taking this formula at face value, we see that in relativity, mass is simply another form of energy. In 1927 Einstein remarked about special relativity:
Under this theory mass is not an unalterable magnitude, but a magnitude dependent on (and, indeed, identical with) the amount of energy.[21]
This formula becomes important when one measures the masses of different atomic nuclei. By looking at the difference in masses, one can predict which nuclei have extra stored energy that can be released by nuclear reactions, providing important information which was useful in the development of nuclear energy and, consequently, the nuclear bomb. The implications of this formula on 20th-century life have made it one of the most famous equations in all of science.
[edit] Relativistic mass
Introductory physics courses and some older textbooks on special relativity sometimes define a relativistic mass which increases as the velocity of a body increases. According to the geometric interpretation of special relativity, this is often deprecated and the term 'mass' is reserved to mean invariant mass and is thus independent of the inertial frame, i.e., invariant.
Using the relativistic mass definition, the mass of an object may vary depending on the observer's inertial frame in the same way that other properties such as its length may do so. Defining such a quantity may sometimes be useful in that doing so simplifies a calculation by restricting it to a specific frame. For example, consider a body with an invariant mass m moving at some velocity relative to an observer's reference system. That observer defines the relativistic mass of that body as:
Physics textbooks sometimes use the relativistic mass as it allows the students to utilize their knowledge of Newtonian physics to gain some intuitive grasp of relativity in their frame of choice (usually their own!). "Relativistic mass" is also consistent with the concepts "time dilation" and "length contraction".
[edit] Force
The classical definition of ordinary force f is given by Newton's Second Law in its original form:
and this is valid in relativity.
Many modern textbooks rewrite Newton's Second Law as
This form is not valid in relativity or in other situations where the relativistic mass M is varying.
This formula can be replaced in the relativistic case by
As seen from the equation, ordinary force and acceleration vectors are not necessarily parallel in relativity.
However the four-vector expression relating four-force
to invariant mass m and four-acceleration
restores the same equation form
[edit] The geometry of space-time
Main article: Minkowski space
SR uses a 'flat' 4-dimensional Minkowski space, which is an example of a space-time. This space, however, is very similar to the standard 3 dimensional Euclidean space, and fortunately by that fact, very easy to work with.
The differential of distance (ds) in cartesian 3D space is defined as:
where (dx1,dx2,dx3) are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension is added, derived from time, so that the equation for the differential of distance becomes:
If we wished to make the time coordinate look like the space coordinates, we could treat time as imaginary: x4 = ict . In this case the above equation becomes symmetric:
This suggests what is in fact a profound theoretical insight as it shows that special relativity is simply a rotational symmetry of our space-time, very similar to rotational symmetry of Euclidean space. Just as Euclidean space uses a Euclidean metric, so space-time uses a Minkowski metric. Basically, SR can be stated in terms of the invariance of space-time interval (between any two events) as seen from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group) of Minkowski space-time. According to Misner (1971 §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) rather than a "disguised" Euclidean metric using ict as the time coordinate.
If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space
We see that the null geodesics lie along a dual-cone:
defined by the equation
or
Which is the equation of a circle with r=c×dt. If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone:
This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an event
meters away and d/c seconds in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the picture below represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)
The cone in the -t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'.
The geometry of Minkowski space can be depicted using Minkowski diagrams, which are also useful in understanding many of the thought-experiments in special relativity.
[edit] Physics in spacetime
Here, we see how to write the equations of special relativity in a manifestly Lorentz covariant form. The position of an event in spacetime is given by a contravariant four vector whose components are:
That is, x0 = t and x1 = x and x2 = y and x3 = z. Superscripts are contravariant indices in this section rather than exponents except when they indicate a square. Subscripts are covariant indices which also range from zero to three as with the spacetime gradient of a field φ:
[edit] Metric and transformations of coordinates
Having recognised the four-dimensional nature of spacetime, we are driven to employ the Minkowski metric, η, given in components (valid in any inertial reference frame) as:
Its reciprocal is:
Then we recognize that coordinate transformations between inertial reference frames are given by the Lorentz transformation tensor Λ. For the special case of motion along the x-axis, we have:
which is simply the matrix of a boost (like a rotation) between the x and t coordinates. Where μ' indicates the row and ν indicates the column. Also, β and γ are defined as:
More generally, a transformation from one inertial frame (ignoring translations for simplicity) to another must satisfy:
where there is an implied summation of
and
from 0 to 3 on the right-hand side in accordance with the Einstein summation convention. The Poincaré group is the most general group of transformations which preserves the Minkowski metric and this is the physical symmetry underlying special relativity.
All proper physical quantities are given by tensors. So to transform from one frame to another, we use the well-known tensor transformation law
Where
is the reciprocal matrix of
.
To see how this is useful, we transform the position of an event from an unprimed coordinate system S to a primed system S', we calculate
which is the Lorentz transformation given above. All tensors transform by the same rule.
The squared length of the differential of the position four-vector
constructed using
is an invariant. Being invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no Λ appears in its trivial transformation. Notice that when the line element
is negative that
is the differential of proper time, while when
is positive,
is differential of the proper distance.
The primary value of expressing the equations of physics in a tensor form is that they are then manifestly invariant under the Poincaré group, so that we do not have to do a special and tedious calculation to check that fact. Also in constructing such equations we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation.
[edit] Velocity and acceleration in 4D
Recognising other physical quantities as tensors also simplifies their transformation laws. First note that the velocity four-vector Uμ is given by
Recognising this, we can turn the awkward looking law about composition of velocities into a simple statement about transforming the velocity four-vector of one particle from one frame to another. Uμ also has an invariant form:
So all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. The acceleration 4-vector is given by
. Given this, differentiating the above equation by τ produces
So in relativity, the acceleration four-vector and the velocity four-vector are orthogonal.
[edit] Momentum in 4D
The momentum and energy combine into a covariant 4-vector:
where m is the invariant mass.
The invariant magnitude of the momentum 4-vector is:
We can work out what this invariant is by first arguing that, since it is a scalar, it doesn't matter which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.
We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero.
The rest energy is related to the mass according to the celebrated equation discussed above:
Note that the mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames.
[edit] Force in 4D
To use Newton's third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components.
If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the four-force. It is the rate of change of the above energy momentum four-vector with respect to proper time. The covariant version of the four-force is:
where
is the proper time.
In the rest frame of the object, the time component of the four force is zero unless the "invariant mass" of the object is changing in which case it is the negative of that rate of change times c2. In general, though, the components of the four force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, i.e.
while the four force is defined by the rate of change of momentum with respect to proper time, i.e.
.
In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is the negative of the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.
[edit] Relativity and unifying electromagnetism
Main article: Classical electromagnetism and special relativity
Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation-speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the Liénard–Wiechert potential, which is a step towards special relativity.
The Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.
[edit] Electromagnetism in 4D
Main article: Covariant formulation of classical electromagnetism
Maxwell's equations in the 3D form are already consistent with the physical content of special relativity. But we must rewrite them to make them manifestly invariant.[22]
The charge density
and current density
are unified into the current-charge 4-vector:
The law of charge conservation,
, becomes:
The electric field
and the magnetic induction
are now unified into the (rank 2 antisymmetric covariant) electromagnetic field tensor:
The density,
, of the Lorentz force,
, exerted on matter by the electromagnetic field becomes:
Faraday's law of induction,
, and Gauss's law for magnetism,
, combine to form:
Although there appear to be 64 equations here, it actually reduces to just four independent equations. Using the antisymmetry of the electromagnetic field one can either reduce to an identity (0=0) or render redundant all the equations except for those with λ,μ,ν = either 1,2,3 or 2,3,0 or 3,0,1 or 0,1,2.
The electric displacement
and the magnetic field
are now unified into the (rank 2 antisymmetric contravariant) electromagnetic displacement tensor:
Ampère's law,
, and Gauss's law,
, combine to form:
In a vacuum, the constitutive equations are:
Antisymmetry reduces these 16 equations to just six independent equations.
The energy density of the electromagnetic field combines with Poynting vector and the Maxwell stress tensor to form the 4D electromagnetic stress-energy tensor. It is the flux (density) of the momentum 4-vector and as a rank 2 mixed tensor it is:
where
is the Kronecker delta. When upper index is lowered with η, it becomes symmetric and is part of the source of the gravitational field.
The conservation of linear momentum and energy by the electromagnetic field is expressed by:
where
is again the density of the Lorentz force. This equation can be deduced from the equations above (with considerable effort).
The classical definition of ordinary force f is given by Newton's Second Law in its original form:
Many modern textbooks rewrite Newton's Second Law as
This formula can be replaced in the relativistic case by
However the four-vector expression relating four-force
[edit] The geometry of space-time
Main article: Minkowski space
SR uses a 'flat' 4-dimensional Minkowski space, which is an example of a space-time. This space, however, is very similar to the standard 3 dimensional Euclidean space, and fortunately by that fact, very easy to work with.
The differential of distance (ds) in cartesian 3D space is defined as:
If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space
The geometry of Minkowski space can be depicted using Minkowski diagrams, which are also useful in understanding many of the thought-experiments in special relativity.
[edit] Physics in spacetime
Here, we see how to write the equations of special relativity in a manifestly Lorentz covariant form. The position of an event in spacetime is given by a contravariant four vector whose components are:
[edit] Metric and transformations of coordinates
Having recognised the four-dimensional nature of spacetime, we are driven to employ the Minkowski metric, η, given in components (valid in any inertial reference frame) as:
All proper physical quantities are given by tensors. So to transform from one frame to another, we use the well-known tensor transformation law
To see how this is useful, we transform the position of an event from an unprimed coordinate system S to a primed system S', we calculate
The squared length of the differential of the position four-vector
The primary value of expressing the equations of physics in a tensor form is that they are then manifestly invariant under the Poincaré group, so that we do not have to do a special and tedious calculation to check that fact. Also in constructing such equations we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation.
[edit] Velocity and acceleration in 4D
Recognising other physical quantities as tensors also simplifies their transformation laws. First note that the velocity four-vector Uμ is given by
[edit] Momentum in 4D
The momentum and energy combine into a covariant 4-vector:
The invariant magnitude of the momentum 4-vector is:
The rest energy is related to the mass according to the celebrated equation discussed above:
[edit] Force in 4D
To use Newton's third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components.
If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the four-force. It is the rate of change of the above energy momentum four-vector with respect to proper time. The covariant version of the four-force is:
In the rest frame of the object, the time component of the four force is zero unless the "invariant mass" of the object is changing in which case it is the negative of that rate of change times c2. In general, though, the components of the four force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, i.e.
In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is the negative of the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.
[edit] Relativity and unifying electromagnetism
Main article: Classical electromagnetism and special relativity
Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation-speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the Liénard–Wiechert potential, which is a step towards special relativity.
The Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.
[edit] Electromagnetism in 4D
Main article: Covariant formulation of classical electromagnetism
Maxwell's equations in the 3D form are already consistent with the physical content of special relativity. But we must rewrite them to make them manifestly invariant.[22]
The charge density
The electric displacement
The energy density of the electromagnetic field combines with Poynting vector and the Maxwell stress tensor to form the 4D electromagnetic stress-energy tensor. It is the flux (density) of the momentum 4-vector and as a rank 2 mixed tensor it is:
The conservation of linear momentum and energy by the electromagnetic field is expressed by:
mutantlast said:[edit] Force
The classical definition of ordinary force f is given by Newton's Second Law in its original form:
and this is valid in relativity.
Many modern textbooks rewrite Newton's Second Law as
This form is not valid in relativity or in other situations where the relativistic mass M is varying.
This formula can be replaced in the relativistic case by
As seen from the equation, ordinary force and acceleration vectors are not necessarily parallel in relativity.
However the four-vector expression relating four-forceto invariant mass m and four-accelerationrestores the same equation form
[edit] The geometry of space-time
Main article: Minkowski space
SR uses a 'flat' 4-dimensional Minkowski space, which is an example of a space-time. This space, however, is very similar to the standard 3 dimensional Euclidean space, and fortunately by that fact, very easy to work with.
The differential of distance (ds) in cartesian 3D space is defined as:
where (dx1,dx2,dx3) are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension is added, derived from time, so that the equation for the differential of distance becomes:
If we wished to make the time coordinate look like the space coordinates, we could treat time as imaginary: x4 = ict . In this case the above equation becomes symmetric:
This suggests what is in fact a profound theoretical insight as it shows that special relativity is simply a rotational symmetry of our space-time, very similar to rotational symmetry of Euclidean space. Just as Euclidean space uses a Euclidean metric, so space-time uses a Minkowski metric. Basically, SR can be stated in terms of the invariance of space-time interval (between any two events) as seen from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group) of Minkowski space-time. According to Misner (1971 §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) rather than a "disguised" Euclidean metric using ict as the time coordinate.
If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space
We see that the null geodesics lie along a dual-cone:
defined by the equation
or
Which is the equation of a circle with r=c×dt. If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone:
This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an eventmeters away and d/c seconds in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the picture below represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)
The cone in the -t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'.
The geometry of Minkowski space can be depicted using Minkowski diagrams, which are also useful in understanding many of the thought-experiments in special relativity.
[edit] Physics in spacetime
Here, we see how to write the equations of special relativity in a manifestly Lorentz covariant form. The position of an event in spacetime is given by a contravariant four vector whose components are:
That is, x0 = t and x1 = x and x2 = y and x3 = z. Superscripts are contravariant indices in this section rather than exponents except when they indicate a square. Subscripts are covariant indices which also range from zero to three as with the spacetime gradient of a field φ:
[edit] Metric and transformations of coordinates
Having recognised the four-dimensional nature of spacetime, we are driven to employ the Minkowski metric, η, given in components (valid in any inertial reference frame) as:
Its reciprocal is:
Then we recognize that coordinate transformations between inertial reference frames are given by the Lorentz transformation tensor Λ. For the special case of motion along the x-axis, we have:
which is simply the matrix of a boost (like a rotation) between the x and t coordinates. Where μ' indicates the row and ν indicates the column. Also, β and γ are defined as:
More generally, a transformation from one inertial frame (ignoring translations for simplicity) to another must satisfy:
where there is an implied summation ofandfrom 0 to 3 on the right-hand side in accordance with the Einstein summation convention. The Poincaré group is the most general group of transformations which preserves the Minkowski metric and this is the physical symmetry underlying special relativity.
All proper physical quantities are given by tensors. So to transform from one frame to another, we use the well-known tensor transformation law
Whereis the reciprocal matrix of.
To see how this is useful, we transform the position of an event from an unprimed coordinate system S to a primed system S', we calculate
which is the Lorentz transformation given above. All tensors transform by the same rule.
The squared length of the differential of the position four-vectorconstructed using
is an invariant. Being invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no Λ appears in its trivial transformation. Notice that when the line elementis negative thatis the differential of proper time, while whenis positive,is differential of the proper distance.
The primary value of expressing the equations of physics in a tensor form is that they are then manifestly invariant under the Poincaré group, so that we do not have to do a special and tedious calculation to check that fact. Also in constructing such equations we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation.
[edit] Velocity and acceleration in 4D
Recognising other physical quantities as tensors also simplifies their transformation laws. First note that the velocity four-vector Uμ is given by
Recognising this, we can turn the awkward looking law about composition of velocities into a simple statement about transforming the velocity four-vector of one particle from one frame to another. Uμ also has an invariant form:
So all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. The acceleration 4-vector is given by. Given this, differentiating the above equation by τ produces
So in relativity, the acceleration four-vector and the velocity four-vector are orthogonal.
[edit] Momentum in 4D
The momentum and energy combine into a covariant 4-vector:
where m is the invariant mass.
The invariant magnitude of the momentum 4-vector is:
We can work out what this invariant is by first arguing that, since it is a scalar, it doesn't matter which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.
We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero.
The rest energy is related to the mass according to the celebrated equation discussed above:
Note that the mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames.
[edit] Force in 4D
To use Newton's third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components.
If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the four-force. It is the rate of change of the above energy momentum four-vector with respect to proper time. The covariant version of the four-force is:
whereis the proper time.
In the rest frame of the object, the time component of the four force is zero unless the "invariant mass" of the object is changing in which case it is the negative of that rate of change times c2. In general, though, the components of the four force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, i.e.while the four force is defined by the rate of change of momentum with respect to proper time, i.e..
In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is the negative of the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.
[edit] Relativity and unifying electromagnetism
Main article: Classical electromagnetism and special relativity
Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation-speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the Liénard–Wiechert potential, which is a step towards special relativity.
The Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.
[edit] Electromagnetism in 4D
Main article: Covariant formulation of classical electromagnetism
Maxwell's equations in the 3D form are already consistent with the physical content of special relativity. But we must rewrite them to make them manifestly invariant.[22]
The charge densityand current densityare unified into the current-charge 4-vector:
The law of charge conservation,, becomes:
The electric fieldand the magnetic inductionare now unified into the (rank 2 antisymmetric covariant) electromagnetic field tensor:
The density,, of the Lorentz force,, exerted on matter by the electromagnetic field becomes:
Faraday's law of induction,, and Gauss's law for magnetism,, combine to form:
Although there appear to be 64 equations here, it actually reduces to just four independent equations. Using the antisymmetry of the electromagnetic field one can either reduce to an identity (0=0) or render redundant all the equations except for those with λ,μ,ν = either 1,2,3 or 2,3,0 or 3,0,1 or 0,1,2.
The electric displacementand the magnetic fieldare now unified into the (rank 2 antisymmetric contravariant) electromagnetic displacement tensor:
Ampère's law,, and Gauss's law,, combine to form:
In a vacuum, the constitutive equations are:
Antisymmetry reduces these 16 equations to just six independent equations.
The energy density of the electromagnetic field combines with Poynting vector and the Maxwell stress tensor to form the 4D electromagnetic stress-energy tensor. It is the flux (density) of the momentum 4-vector and as a rank 2 mixed tensor it is:
whereis the Kronecker delta. When upper index is lowered with η, it becomes symmetric and is part of the source of the gravitational field.
The conservation of linear momentum and energy by the electromagnetic field is expressed by:
whereis again the density of the Lorentz force. This equation can be deduced from the equations above (with considerable effort).
හොදටම තේරැණා
This is very simple...
Anything that can travel faster then light would have to be infinity which has not proven possible yet.
M = Mo / sqrt(1 - (v^2 / c^2) )
M is mass moving at speed v, Mo is rest mass (mass of body when it is at rest), c is speed of light.
When v becomes equal to c then the denominator in the equation will become 0,
so Mass M will become infinity.
Anything that can travel faster then light would have to be infinity which has not proven possible yet.
M = Mo / sqrt(1 - (v^2 / c^2) )
M is mass moving at speed v, Mo is rest mass (mass of body when it is at rest), c is speed of light.
When v becomes equal to c then the denominator in the equation will become 0,
so Mass M will become infinity.
funkoluwa said:
Unfortunately,
"not proven possible yet" this expression has been misunderstand by you.
Albert Einstein elaborated that faster-than-light travel is impossible.
If Albert Einstein Elaborated that, how can you say about a conception which is developed by using his theories.
Fried I have to say that again and again, PLS study about the root of this conception well.. then you will clearly understand, about this.
If you let me I will learn you the basic conceptions of the theory of Relativity, if your maths knowledge is well.
If not there is no use of taking about this, because it is only a personal sense..
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