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General Relativity

 

Einstein worked (from) Newton’s equations describing matter and gravity: And he also changed a lot of it too, describing gravity in whole new terms to physicists. Newton’s law of gravitation is given as:

 

F = G M_1M_2/d^2

 

Where G is the gravitational constant, m1 is mass one, and m2 is the second body with mass, and F was the force distributed between them. Also, Einstein worked with the already existing equation describing the laws between two masses m_1 and m_2, finding a square force that weakened over greater factors:

Charles Augustine de Coulomb in 1785 showed that the force of attraction and the force of repulsion between two electrically charged bodies and also between magnetic poles also obey an inverse square law. The force for two magnetic bodies are given as:

 

Fm = (1/µ)(p_1p_2/r^2)

 

With what Einstein had in mind for this universe, which was pure curvature, he was able to use these concepts to create a geometrical vision of our cosmos. It described that matter was actually an energy:

 

E=Mc^2

 

Which converts to the negative of E=Mc^2

 

E=-Mc^2+(E=Mc^2) = 1022KeV of gamma energy

 

Through:

 

M=E/c^2

 

Which is the reverse of E=Mc^2 and was first developed by Poncair. This scientist was also known for his remarkable work on relativity, and is often forgotten. In fact, General Relativity unifies the work of Poncair and Einstein in their theories describing Special Relativity with Newton’s law of Gravitation.

 

Matter warped space, and time told matter how to move. And it can be said that time warped matter, and that matter told space how to act. This is because of Einstein’s equivalence principle, which covers a massive scope in his mathematically-genius work.

 

This next equation is Einstein’s field tensor, and you will most certainly learn it in a standard course of physics at college or university:

 

G_ab=kT_ab

 

This equation is very important, where the Gab Einstein Tensor Factor, and the Stress Energy Tensor is given as and Tab and k is a constant. This equation relates to the curvature of space and time, saying that stress energy is what causes the disturbance of spacetime. As we have seen, Einstein used Newton’s law of Gravity in his Field Equations, then we find the constant of k to have a value of:

 

Where π is pi, and G is the gravitational constant and k is a coupling constant, which will be most familiar as

 

k=8 π G/c4

 

The following equation which is an extension of the above equation connects matter with energy with the geometry of spacetime (on the left):

 

Guv(-8 π G/c2)T^µv

 

But using the more well-known value of kappa, it gives the more recognized value of the above equation:

 

Guv= (8 π G/c4)T^µv

 

A major consequence of General Relativity, is that it describes that time is dilated round strong gravitational fields. Tim can also be warped traveling through spacetime at very high speeds in long distances. We will explore that soon.

 

t’=t√1-2GMc^2r

 

Where,

• t' = Time inside the gravitational field.

• t = Time outside the gravitational field.

• M= The mass causing the gravitational field.

• r = The distance from the center of the gravitational field.

• c = the speed of light in a vacuum 186,000 mps.

• G = The gravitational constant = 6.6742 x10^-11 N m^2 kg-2

 

From Pound and Rebka’s experiment in 1959 at the Harvard University, we know that the shift round in the gravitational distortions warp time only by infinitesimal standards. In order to measure the dilation with significant results, we have resorted to using atomic clocks. In other words, the life spans of particles could be experienced to be longer moving at very very high speeds. We can such particles, such as gold atoms to a fraction short of ‘c’, but we can never exceed that v>c because then we would require an infinite amount of energy… which is another successful prediction of relativity.

 

General Relativity was by far the more difficult to create, as it took Einstein so many more years to finish it. Special Relativity was also mathematically beautiful and so very straight forward when one began to appreciate it.

 

Special Relativity

 

1. Special Relativity and Flat Spacetime involve deep understanding into the spacetime interval — the metric — spacetime diagrams — vectors — the tangent space — dual vectors — tensors — tensor products — the Levi-Civita tensor —electromagnetism — Lorentz transformations — differential forms — worldlines — proper time — energy-momentum vector — energy momentum tensor — perfect fluids — energy-momentum conservation index manipulation, and more.

 

In these relativistic principles, we need to know some concepts.

 

Minkowski’s flat vector spacetime is best used in physics today. It is given as (-, -, -, +) or (+,+,+,-) and with a matrix of, and in special conditions, can be written as (0,0,0,0) in zero-dimensions:

 

…..1000

…..0100

N=0010

…..0001-

 

The following equations are called ‘’Cartesian’’ coordinated systems, and they describe the distance between two points:

 

s2 = (∆x)^2 + (∆y)^2

 

In a rotated system, we twist coordinates around in space, and we find them as a geometry of distance. The new coordinates are given as:

 

s2 = (∆x′)^2 + (∆y′)^2

 

Being almost identical math, they are easy to remember. In this case, we say that distance is an invariant of these equations. More interesting is that we learn that time is also an invariant of space.

 

Because of this, we can therefore find the following equation describing a spacetime interval:

 

s^2 = −(c∆t)^2 + (∆x)^2 + (∆y)^2 + (∆z)^2

 

Where (t, x, y and z) are the coordinates of spacetime, because we can rotate space, and find a corresponding value with time, and this is why we say that space and time are one thing. All these equations lead to many more equations, just as Lorentz Boosts which derive from the mathematics described by Galileo, and his coordinates are given through the variables:

 

x' = x − vt

y' = y

z' = z

t' = t

 

We therefore give the spacetime metric a 4x4 matrix. Time coordinates are found as being invariant to the system. We then have the formula,

 

s^2 =η_μ_v∆x^u∆x^v

 

We can simplify the transformation in spacetime into a more arbitrary equation,

 

x^u → x^u ’ = x^u + a^u

 

Where aμ is a set of four fixed numbers. Translations leave the differences ∆xμ unchanged, so it is not remarkable that the interval is unchanged. The only other kind of linear transformation is to multiply xμ by a (spacetime-independent) matrix:

 

x_μv’ = η^u_v x^v

 

We should now start with rotational boosts, such as found in an x and y plain.

 

We are always taught that the rotational value θ is in fact a periodic invariant with a period with a value corresponding to 2π.

 

Now, the Boosts are in fact nothing but rotations in time and space. It can be put into a matrix, but i very much doubt my variables would show up here, so i shall continue just explaining.

 

The Boost parameter φ (found in the matrix), is in fact defined to infinity -00 to 00, totally in this direction. The rotation angles need not abide by such a rule, if my memory serves me correctly.

 

t′ = t cosh φ − x sinh φ

x′ = −t sinh φ + x cosh φ .

 

Here we have a set of Boosts, where the transformed coordinated are t' and x', and this transformation has a moving value of x'. And therego, we say such a Boost has a velocity of

 

v = x/t= sinh ang/cosh ang = tanh ang

 

To cut them down we can replace φ = tanh−1 v to obtain,

 

t′ = ang (t − vx)

x′ = ang (x − vt)

 

Where gamma = 1/√1 − v2. This satisfies the expressions for Lorentz transformations. Applying these formulae leads to time dilation, length contraction, ect. In Lorentzian spacetime, their components are unbalanced.

 

ω_μ = (−ω_0,ω_1,ω_2,ω_3)

 

But, even in a more complicated form, a curved spacetime, seems to solve the problems.

 

Before relativity, we never considered time as a vector of space. Now we cannot remove either, as it has been shown that time is a distance as well in space. To move through space, is to take a massive journey in time for us. In fact, time moves at the speed of light, and because we live in such a slow part of the universe, we cannot help but flow along with it.

 

Relativity was also developed to answer for optical phenomenon. One of these the Length Contraction of a physical body accelerating through spacetime.

 

The Length Contraction formula is given as:

 

L=L_0 (1-v^2/c^2) 1/2

 

Here, L0 is the proper length, and v is for velocity and c is for the speed of light. This equation shows that an object moving through spacetime is found to contract in length to the observer. Such paradoxes like, the pole and barn paradox are prime examples of this optical phenomenon.

 

So in short notation, we say that space contracts and time dilates by a factor of:

 

√(1-v^2/c^2)

 

As I explained, one example of Length Contraction is given by the pole-barn paradox. This is where a pole is traveling through space, and is physically contracted. If the pole is larger than the barn to begin with, and now it is shorter because of length contraction it can fit in the barn. Paradox is, how can a pole larger than the barn be length contracted so that it fits as it passes by? In this next set of equations, we work with a pole traveling through space which has a proper length of 20 meters. An observer moving at a speed v = 0.98 c will experience a contraction as shown:

 

L_0 = 20 m

L = L_0 (1 - v^2/c^2)1/2

= 20 [ 1 - (0.98)^2 ]1/2

= 3.98 m

 

If an object is accelerating through spacetime, it will experience a time warp. This is also been known to be called time dilation. If we experience time warps, then according to relativity this must also mean space warps.

 

We don’t experience space warps so much because we move so fast through time. In fact, we spend more time in the time dimension than we do in space. The time dilation formula is given as:

 

∆t=∆t_0/(1-v^2/c^2) 1/2

 

In Einstein’s paradox, a moving spacetime traveler (twin one) is going at speeds short of ‘c’ arrives home having only aged a year or so, and on Earth his twin has aged considerably. In relativity, we learn that E=Mc^2, and that energy can be transferred through angular momentum of a system, and the conservation of physics states that the loss of energy is equal to the gain of energy given as:

 

∆E=∆Mc^2

 

Where mass gained is a loss in energy. And we also find that energy gained or lost, is equivalent to a gain in mass:

 

∆M=∆E/c^2

 

Where L is the angular momentum, r is the position of the particle, x is the cross product and p is the linear momentum:

 

P=Mv

 

This shows, as it does in any text book, that momentum p is related to mass multiplied by the velocity v. The importance of this equation is that energy can only ever be transferred through angular momentum. So the particle is said to be given as:

 

L=r x p

 

And between two objects, one finds that one gains momentum and the other looses momentum, given as:

 

∆p_1=-∆p_2

 

Because everything is conserved, the state of momenta found prior to the transfer is equal to the momenta afterwards:

 

m_1u_1+m_2u_2=m_1v_1+m_2v_2

 

These equations are beautiful. I have also equated that if an end is desired by time and the cosmos, then somehow mass and energy is an illusion:

 

E=Mc^2+E

 

And if ‘’c’’ is not equal to zero, which we know it isn’t, then we find through algebra that:

 

Mc^2=0

 

This can be found to be true about this universe. All matter comes to zero when added with the energy in the vacuum:

 

(E=Mc^2)+(E=-Mc^2) = 0

 

So the converse can be accumulated:

 

M=E/c^2+M

 

Then all energy comes to zero as well:

 

E/c^2=0

 

The reason why this happens, is because we are adding all the matter and energy, about 1080 particles ‘’pop’’ into existence, and when added to the negative energy of the vacuum produces a zero-total. The negative reservoir is called The Dirac Sea, and it is filled with negative spinning particles. In fact, Dirac postulated this sea using relativity. Virtual particles, like the kind found in this sea, don’t share the same properties as real energy:

 

E^2 = m^2c^4

 

And is found to reduce to this instead of the normal energy and momentum formula:

 

E^2 = p^2c^2 + m^2c^4

 

… for when a particle is at rest p=0. Where p is momentum and c is the speed of light. This new relativistic outlook on the electron allowed Dirac to formulate his famous equations describing antimatter.

 

These relativistic formulae show that every time an electron ‘’pops’’ into existence, it leaves behind a hole. This hole is found to be its antipartner, the positron.

 

To measure the Kinetic energy of something, you must calculate it with the formula:

 

KE=E-E_0

 

And to measure the Kinetic energy of a high speed particle, we use the equation:

 

KE=Mc^2 - M_0c^2

 

And the total energy is found as:

 

E=Mc^2 (1-v^2/c^2) 1/2

 

And the rest energy is given in relativity as:

 

E_0=Mc^2

 

Pure simplicity at its best. These equations have been the most influential in the world. These equations can describe the motions of little particles as well.

 

The complications of this are amazing. Time dilation in relativity means that a twin traveling off into the far reaches of spacetime a fraction short of c would return to earth hardly aged a year, whilst his twin is an old man. Einstein has shown the world of physics that time is not a fixed clock in the sky, and two events that are relative to each other will experience different times traveling at different speeds.

 

‘’Spacetime,’’ brought into one continuum, could be changed into what are called ‘’timelike’’ conditions. To explain what it is, a system in spacelike conditions experience space and time moving in the correct directions along their wordline. But in a timelike condition, spacetime switch roles and space becomes timelike in character. That would mean that you would begin to move through time like you had in space.

 

In spacelike paths, we define the path length given as,

 

∆s = ∫(√ η μv)(dxμ/dλ x dxν/dλ)(dλ)

 

In timelike conditions, we define the paths in real time,

 

∆t = ∫(√ - η μv)(dxμ/dλ x dxν/dλ)(dλ)

 

Which will be positive. Usually, paths taken by physical particles seldom change, so we find that in normal situations, massive particles move on timelike paths and null paths.

 

Vectors AND Scalars

 

In relativity, we find vector and scalar equations that describe in their own terms coordinates and fields.

 

For Vectors, we find that they describe:

 

Length, Area, Speed, Volume, Time, Mass, Energy Density, Pressure, Power, Temperature, Electric Charge and the Electric Potential.

 

For scalars, we find:

 

Displacement, Acceleration, Velocities, Force, Weight, Momentum, Torque, Electric Currents, Electric Field Strengths and Magnetic Field Strengths.

 

Even though they are described by different names, they are actually connected sensitively that one could say they are the same thing, because of magnetic and electric curl equations.

The electric constant (ε_0) has a value of 8.854*10-12Fm-1, and the magnetic Constant (μ_0) has a value of 4π*10-7Hm-1.

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Hello Graviphoton,

 

I have some understanding of relativity and a year of college physics (from more than 20 years ago, I am independently studying again now...).

 

Could you please help me answer and calculate the following?

 

1. If a high energy cosmic ray of 1 proton strikes 1 relatively stationary proton (equal mass) on Earth and the cosmic ray generates 10^20 eV, what percentage of the speed of light was the particle traveling?

 

2. If the speed of the particle was c' (99.999...?% the speed of light), then if you observer the same collision in a reference frame where the Earth and the cosmic ray are traveling at the same speed (c'/2 and -c'/2). Then what energy will each particle generate? I know that the value would be exponentially smaller, not just half, but what is this value? (This reference frame would be that of a space ship traveling a c'/2 toward Earth coming from the same direction as the cosmic ray...)

 

Thank you,

JTankers

jim_tank@hotamil.com

 

(Calculations above appear to also help graphically illustrate the preferred reference frames of special relativity... http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Newton_bucket.html: Absolute space-time is a feature of special relativity which, contrary to popular belief, does not claim that everything is relative.)

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Question; Why are the Lorentz transformations only linear?

 

Let us see if I can show it to be true. I use modern notation and basic elements of differential geometry. I try to write things out in local coordinates where it is useful to do so.

 

First, let [math]M = (\mathbb{R}^{4},\eta)[/math] be 4-dimensional Minkowski space-time. That is, [math]\eta[/math] can always be globally diagonalised by a change of coordinates, so [math]\eta = diag(-1,1,1,1)[/math] (thought of as a matrix).

 

The first statement is that [math](X,Y) = \eta(X,Y)[/math], with [math]X,Y \in \mathfrak{X}(M)[/math], is independent of any coordinate system. Or we can say that the inner product is invariant under changes of coordinates.

 

Locally we have [math](X,Y) = \eta_{\mu \nu}X^{\mu}Y^{\nu}[/math].

 

So, we cannot get the Lorentz transformations by considering the passive representation of Diff(M). This is a good thing, as any sensible physical theory should not depend on how you describe it.

 

Ok, to get at the Lorentz transformations we need to consider active diffeomorphisms.

 

Let [math]\phi : M \rightarrow M[/math] be a diffeomorphism such that locally we have

[math]x^{\mu} \mapsto \overline{x}^{\mu}[/math] or more usefully, we define [math]\phi^{*}\overline{x}^{\mu} = \overline{x}^{\mu}(x)[/math].

 

Definition; The Poincare transformations are (linear) diffeomorphisms [math]\phi[/math] such that

 

[math]\eta = \phi^{*}\overline{\eta}[/math].

 

More concretely this means

 

[math]\eta(X,Y) = \overline{\eta}(\phi_{*}X, \phi_{*}Y)[/math]

 

 

 

Thus as [math](\phi_{*}X)^{\mu} = \frac{\partial \overline{x}^{\mu}(x)}{\partial x^{\nu}}X^{\nu} = \Lambda^{\mu}_{\:\: \nu}(x) X^{\nu}[/math] we must have for linear transformations,

 

[math]\phi^{*}\overline{x}^{\mu} = \overline{x}^{\mu}(x) = \Lambda^{\mu}_{\:\: \nu} x^{\nu} + a^{\mu}[/math],

 

we must also have

 

[math]\eta_{\mu \nu} = \Lambda^{\rho}_{\:\: \mu}\overline{\eta}_{\rho \sigma}\Lambda^{\sigma}_{\:\: \nu}[/math],

 

that is we need to find the matrices such that the components of [math]\eta_{\mu \nu}[/math] and [math]\eta_{\rho \sigma}[/math] agree. These are the Lorentz transformtions.

 

But why are they linear? I think the answer is that we are only interested in transformations that map lines to lines. In particular we don't want to destroy the light-cone structure. Or put another way, we require that the speed of light be the same in all frames.

 

Or more mathematically, we assume that space-time is homogeneous. That is the laws of physics are the same at all points on space-time. This means that any transformation must not depend on "where it is". Meaning that the transformations must be linear. For example, lets consider a quadratic change

 

[math]\overline{x}^{\mu}(x) = \Lambda^{\mu}_{\:\: \nu} x^{\nu} + \frac{1}{2}C^{\mu}_{\nu \rho}x^{\mu}x^{\rho} + a^{\mu}[/math].

 

Then, [math]\frac{\partial \overline{x}^{\mu}(x)}{\partial x^{\nu}} = \Lambda^{\mu}_{\:\: \nu} + C^{\mu}_{\nu \rho}x^{\rho}[/math] which has an explicit dependants on the coordinate [math]x[/math] (and so the points of space-time). Thus they must be linear.

 

Despite the fact that I think the Lorentz transformations are really derived from an active point of view, you can of course also use them passively.

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Its refreshing to see the math displayed correctly. Thank you. :)

 

Tankers

 

I would suggest you might want to get information or help from someone who knows about how to correctly predict the energies of cosmic collisions. I am certainly not one of them ;)

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  • 2 weeks later...
Or more mathematically, we assume that space-time is homogeneous.
It isn't. Let me repeat a quote from Einstein:

 

“Mach’s idea finds its full development in the ether of the general theory of relativity. According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that ‘empty space’ in its physical relation is neither homogeneous nor isotropic....”
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