Relational Quantum Gravity: <Einstein’s Law of Gravitation>

Revision [433]

Last edited on 2012-04-03 05:24:13 by CharlesFrancis

Additions:

""====""""====
======[[GTRTensors ←]] Einstein’s Law of Gravitation [[Gravity ↑]] [[LargeScaleStructure →]]======
It is shown that, for weak gravitational fields, the effect of gravitational redshift on geodesic motion gives an identical acceleration to that of a classical gravitational field. Einstein combined Newton’s law of gravity with his three laws of motion into a single tensor law. The Schwarzschild metric, describing gravity in the region of a star or a planet, is calculated. Black holes are introduced.
""The Weak Field Limit""
""Conservation of Mass-Energy""
""The Stress-Energy Tensor""
""Einstein’s Law of Gravitation""
""Stationary Matter""
""The Newtonian Approximation""
""The Schwarzschild Solution""
""Black Holes""
====""""The Weak Field Limit====
Using Cartesian space coordinates, the ""metric"" determined from the radar method is,
"" Gravitation-1

""
"" Using Gravitation-3

"", the index raising operators are
"" Gravitation-4

""
The ""geodesic equation"" is
"" Gravitation-5

""
where the ""Christoffel symbols"" are
"" Gravitation-6

""
For the Christoffel symbols to be non-zero, two indices must be the same. In a constant field, the other must not be zero. For non-relativistic velocities, terms in the order of velocity squared can be ignored, and we have "" Gravitation-7

"". Then, 3-acceleration is given by, for ""a ≠ 0"",
"" Gravitation-9

""
Writing ""k = 1 + z"", where redshift ""z"" is small, we have that acceleration is minus the gradient of ""z"",
""""
So, gravitational redshift can be identified with the [[http://en.wikipedia.org/wiki/Scalar_potential scalar potential]] in Newtonian gravity. The time component of the metric is
""""
This is called the //weak field limit//. ""mV"" has units of kinetic energy, ""½mv²"". To ""convert"" to conventional units, we must divide by ""c^{2"".

""""

Thus,

""""

and it is seen that very small redshifts are associated with normal accelerations in a gravitational field.

====""""Conservation of Mass-Energy====

Consider a particle of matter, using primed coordinates in which the particle is stationary. A fundamental observation is that its mass remains constant in time. We say that mass is conserved. This is expressed in the simple equation,

""""

This is not a covariant equation, as it depends on coordinates in which the particle is stationary. ""General covariance"" means this should be replaced by a vector or tensor equation, saying exactly the same thing, but in any coordinates. Replace ""m"" by a vector, ""p^i' = (m, 0, 0, 0)"", which expresses the fact that the particle is stationary, and replace ""d/dt'"" by the covariant derivate. Then the equation ,

""""

is a restatement of conservation of mass, and is expressed in terms of vectors so that, in any coordinate system, it has the same form. More generally, a particle at a point is replaced by a probability density function, reflecting the fact that we don’t know exact position. ""p"" is then a vector field, describing momentum density. We can then sum the densities for all particles to find the distribution of matter and net momentum within neighbourhood. By linearity, the equation of conservation of mass still holds.

In classical physics, this equation appears in a number of contexts, e.g, using, for ""i = 1, 2, 3"",

""""

the [[http://en.wikipedia.org/wiki/Navier-Stokes_equations/Derivation#Conservation_of_mass mass continuity equation]] in fluid dynamics,

""""

where ""ρ"" is mass density and ""v"" is 3-velocity, or the [[http://en.wikipedia.org/wiki/Continuity_equation continuity equation]] in electrodynamics

""""

where ""ρ"" is charge density and ""j"" is current density.

====""""The Stress-Energy Tensor====

We know from observation that matter is the source of the gravitational field. Einstein sought a law to express this. Quantitative science requires that laws are given in the form of equations. A tensor equation is required by ""general covariance"", so as to ensure the general principle of relativity, that local laws of physics are the same for all observers. We need a tensor equation relating curvature to mass density. A four index equation for the Riemann curvature tensor would be too restrictive — ""tidal forces"" show that curvature is not zero in empty space. Einstein started to look for an equation involving the ""Ricci tensor"". The Ricci tensor is a rank 2 tensor. To form an equation we need a rank 2 tensor describing the density of matter.

It is trivial to write down a rank 2 tensor describing a stationary particle of mass, ""m"". In primed, locally Minkowski, coordinates, this is

""""

The same tensor is described generally using coordinate transformation. Coordinate transformations for tensors act on both indices. For Minkowski coordinates, coordinate tranformations are Lorentz transformations, and represent a velocity boost such that proper velocity ""(1, 0, 0, 0)"" becomes ""v^a = (v⁰, v¹, v², v³)"". To transform the tensor ""m^{i'j'"", we must apply the transformation to both indices separately. The result is

""""

More generally, mass and velocity of a point particle are replaced by density functions. We then sum over all the particles of matter within a region. The resulting symmetrical tensor, ""T^ab"", is the [[http://en.wikipedia.org/wiki/Stress-energy_tensor stress energy tensor]]. The stress-energy tensor is defined at each point of a spacetime coordinate system (i.e. it is a ""field""). While the momentum vector, ""p"", for a body, has a simple intuitive meaning as the sum of the momenta of its component subparticles, it is less easy to see the meaning of the sum of products, ""v^av^{b"", in the stress-energy tensor. In statistics, similar sums of products appear in the study of [[http://en.wikipedia.org/wiki/Correlation correlation]]. The stress-energy tensor may be regarded as a measure of not just of total motion, but also of the correlation between the motions of component particles, as described in the term “stress”.

Observe that

""""

by ""conservation of mass-energy"" and by the ""geodesic equation"", which follows from Newton’s first law. The //law of local energy-momentum conservation// follows by linearity.

<<""The law of local energy-momentum conservation: T^ab_;b = 0.""<<

More generally, local energy-momentum conservation still holds when particles interact, since momentum is conserved in interaction. This will be shown in [[QED Quantum Electrodynamics]]. For now, it is treated as a fundamental law with application in specific equations of physics. For example, in the non-relativistic approximation,

""""

reduces to, for ""a = 0"", ""v⁰ = 1"", to conservation of mass in fluid dynamics, and, for ""a ≠ 0"" it is the [[http://en.wikipedia.org/wiki/Navier-Stokes_equations#Derivation_and_description Navier-Stokes equations]] for a perfect fluid in the absence of pressure and external forces. ""T^ab_;b = 0"" can also be shown, from [[http://en.wikipedia.org/wiki/Maxwell%27s_equations Maxwell’s equations]], to express conservation of energy-momentum for a classical electromagnetic field, in which energy is given by

""""

and momentum is given by

""""

====""""Einstein’s Law of Gravitation====

In his early researches Einstein tried to relate the ""Ricci tensor"" to the stress energy tensor, but found that this did not work. If curvature is the consequence of stress-energy, then it must reflect the law of local energy-momentum conservation as an identity. The Einstein tensor, ""G^{ab"", is unique as a rank 2 tensor formed by contraction of the ""the Riemann curvature tensor"", which is symmetrical, and which satisfies an identity,

""""

the ""contracted Bianchi identity"". Einstein therefore wrote down the law of gravitation known as [[http://en.wikipedia.org/wiki/Einstein_field_equations Einstein’s field equation]]

""""

where ""κ"" is a constant of proportionality, to be determined by comparison with the Newtonian approximation. This law summarises the two principle tennets of Einstein’s theory of gravity, that matter is the cause of curvature, and, through the Bianchi identity, that energy-momentum is conserved and hence that inertial objects follow geodesics, thereby combining Newton’s three laws of motion and Newton’s law of universal gravitation into a single tensorial law.

====""""Stationary Matter====

In the case of a static body of uniform density, ""ρ"", ""T^{ab"" is,

""""

Einstein’s field equation, written in terms of the Ricci tensor is

""""

Contract the indices ""a"" and ""b"", noting from the summation convention that """",

""""

""""

So, the total scalar curvature in the presence of matter is given by

""""

Hence

""""

""""

====""""The Newtonian Approximation====

The ""Christoffel symbols"" are

"""".

The ""Ricci tensor"" is

""""

In the Newtonian approximation, the metric, ""g"", is slowly varying in space and constant in time. We may neglect terms of second order in derivatives of the metric, and set time derivatives to zero. Then

""""

In Cartesian, ""x-"", ""y-"", ""z-""coordinates

""""

where """" is the [[http://en.wikipedia.org/wiki/Laplacian Laplacian operator]]. Thus,

""""

[[http://en.wikipedia.org/wiki/Poisson's_equation Poisson’s equation]] for a Newtonian gravitational potential, ""k"", due to a mass distribution of density ""ρ"" is

""""

where ""G"" is Newton’s universal gravitational constant. Thus, ""κ = 8πG"", and Einstein’s field equation is

""""

====""""The Schwarzschild Solution====

The first solution to Einstein’s field equation was found, within a few months of the publication of general relativity, by [[http://en.wikipedia.org/wiki/Karl_Schwarzschild Karl Schwarzschild]], who was on active service in Russia. Schwarzschild had been known as a prodigy and the [[Schwarzschild calculation]] is quite involved. It shows that the metric outside of an isolated, spherically symmetric, non-rotating, gravitating body of mass ""M"" is

""""

This is the [[http://en.wikipedia.org/wiki/Schwarzschild_metric Schwarzschild metric]]. In the ""weak field limit"", the redshift factor

""""

is equivalent to a [[http://en.wikipedia.org/wiki/Potential_energy#Gravitational_potential_energy potential]],

""""

in accordance with Newton’s inverse square law of gravity.

====""""Black Holes====

>>""Definition: The Schwarzschild radius is r = 2GM.""

>> The Schwarzshild metric becomes singular at the [[http://en.wikipedia.org/wiki/Schwarzschild_radius Schwarzschild radius]]. For a normal star or planet, this is not important because Schwarzchild geometry applies only in empty space outside of the star. The radius of a normal star is greater than its Schwarzschild radius, but there is a theoretical possibility that a body could exist for which its mass is contained within its Schwarzschild radius. Such a body is called a [[http://en.wikipedia.org/wiki/Black_hole black hole]]. In practice, we know that when a star burns out it can become a [[http://en.wikipedia.org/wiki/White_dwarf white dwarf]]. Provided its mass is less than the [[http://en.wikipedia.org/wiki/Chandrasekhar_limit Chandrasekhar limit]] (about 1.4 solar masses) [[http://en.wikipedia.org/wiki/Electron_degeneracy_pressure electron degeneracy pressure]] due to the ""Pauli Exclusion Principle"" prevents further collapse against the force of gravity. For a burned out star greater than this mass, gravity overcomes electron degeneracy pressure and the star collapses. Energy released in collapse creates a [[http://en.wikipedia.org/wiki/Supernova supernova explosion]], electrons and protons combine to form neutrons, and a [[http://en.wikipedia.org/wiki/Neutron_star neutron star]] may form. If the neutron star has a mass greater than about 5 solar masses, gravity overcomes [[http://en.wikipedia.org/wiki/Degenerate_matter neutron degeneracy pressure]], and collapse continues indefinitely. In practice, many galaxies, including our own, appear to have supermassive black holes at their centres. [[http://www.journals.uchicago.edu/ApJ/journal/issues/ApJ/v616n2/60440/60440.html Observations]] show that [[http://en.wikipedia.org/wiki/Sgr_A%2A Sgr A*]], at the centre of the Milky way, is almost certainly a supermassive black hole.

From the pespective of an external observer, the redshift factor, ""k = (g₀₀)^{−½"", becomes zero at the Schwarzschild radius, showing that time, and all physical processes, slow down relative to time measured by a distant observer as matter approaches the black hole. The external observer would calculate that matter does not actually fall through the Schwarzschild radius, but stops at it. This is true also of light. The Schwarzschild radius is an [[http://en.wikipedia.org/wiki/Event_horizon event horizon]]. Processes inside an event horizon are hidden to the observer. An issue was raised as to whether a ""singularity"" could actually form according to the equations of relativity. [[http://en.wikipedia.org/wiki/Robert_Oppenheimer Oppenheimer]] & [[http://en.wikipedia.org/wiki/Hartland_Snyder Snyder]] published the first calculation of gravitational collapse in 1939, showing that one can.

In fact the ""singularity"" at the event horizon is only a singularity in a particular coordinate system. If instead of using coordinates defined by an external observer, stationary with respect to the hole, we use coordinates determined by an observer falling into it, it can be shown that no singularity arises at the Schwarzschild radius. According to general relativity, the observer simply falls through empty space at the Schwarzschild radius, into a region from which he can no longer communicate with the external observer. There is no singularity in coordinates defined by the falling observer, and physical processes proceed as normal until he meets the true singularity at ""r = 0"". That is the solution according to Einstein’s field equation, but the question still arises as to whether it is a real physical solution. The meaning of the singularity at ""r = 0"" is that known laws of physics break down. We cannot say, from classical general relativity, precisely at what point the laws of physics break down in the vicinity of the singularity. Relational quantum gravity will reexamine this issue in the light of a unification with quantum theory.

[[Gravitation Einstein’s Law of Gravitation ↑]] [[LargeScaleStructure Large Scale Structure →]]}}}}}}

Deletions:

""""====""""====
======[[GeneralRelativity ←]] Riemann Curvature [[Gravity ↑]] [[Gravitation →]]======
If index gymnastics were a physical sport, this page would be a training session for the fit and athletic. In it, the covariant derivative is established from local parallelism, the ""Riemann curvature tensor"" is found, properties are analysed, and the ""Einstein curvature tensor"" is found and shown to obey the ""contracted Bianchi identity"", which has importance in ""Einstein’s law of gravitation"". From a philosophical perspective, the important aspect is that manipulations in mathematics introduce no new physical principles, and merely express relationships which necessarily hold in a universe obeying the general principle of relativity and in which we can translate objects through small distances. If you are prepared to take that on faith, you can skip the calculations and move quickly on to the next section. That is quite reasonable. You may reflect that these calculations have been checked and rechecked by tens of thousands of mathematicians since their original formulation in the 19th century. Such is the requirement of reproducibility in a strict approach to science. If you take the strict approach, that nothing should be taken on faith, and require that logic, rather than authority, should be the final arbiter, there is no help for it; you have to do the training session. No sympathy can be afforded to those who decry authority and yet are too idle or unfit to do the training.
""Christoffel Symbols""
""Parallel Displacement""
""The Covariant Derivative""
""Geodesics""
""The Riemann Curvature Tensor""
""Symmetries of the Curvature Tensor""
""The Bianchi Identity""
""Contracted Curvature Tensors""
====""""Christoffel Symbols====
Consider a region in which the metric field is ""g_{ab"". [[http://en.wikipedia.org/wiki/Christoffel_symbol Christoffel symbols]] have a vital role in calculating the effect of parallel displacement.

<<""Definition: Christoffel symbols of the first kind:""

""""

<<

Christoffel symbols, defined using ""partial differentiation"", are not tensors, but indices can be raised and lowered in the usual way. It is common to raise the first index.

<<""Definition: Christoffel symbols of the second kind:""

""""

<<

Clearly Christoffel symbols are symmetric in the last two suffixes, and satisfy

""""

Christoffel symbols are sometimes said to define the connection in general relativity. From an empirical perspective, the connection is the physical prescription for comparing the coordinate system set up an observer with that of a second, nearby observer, using parallel displacement in tangent space. In pure mathematics there is no physical prescription, so a different definition of a connection is required. In my view, adopting a mathematical definition reduces the empirical foundations of general relativity to metaphysics, in conflict with Einstein’s approach to physics. In this empirically based account, the purpose of Christoffel symbols is to enable us to calculate the effect of parallel displacement without reference to a non-physical tangent space. A ""tangent chart"" at ""x"" is defined with non-physical metric ""h"" using primed coordinates. At ""x"",

""""

""""

""""

Observe that, from ""Clairaut’s Theorem"", the partial derivative of the transformation matrix is symmetrical in its lower indices,

""""

Interchange ""a ↔ c"" and ""b ↔ c"",

""""

""""

Then,

""""

====""""Parallel Displacement====

Parallel displacement of a vector ""p"" from ""x"" to ""x + dx"" in tangent space keeps the primed components constant, """". Multiply by """", to lower the index and convert to unprimed coordinates,

""""

""""

Then, ignoring terms ""O(max(dxⁱ))"",

""""

""""

Substituting the Christoffel symbol eliminates the dependency on tangent space,

""""

Raise and lower indices to find the standard formula for infinitesimal parallel displacement referring to covariant components,

<<""Infinitesimal parallel displacement: ""

""""

<<

For a second vector ""q"", ""p · q"" is invariant,

""""

""""

""""

Since this is true for all ""p_{a"", we find the standard formula for infinitesimal parallel displacement referring to contravariant components,

<<""Infinitesimal parallel displacement: ""

""""

<<

====""""The Covariant Derivative====

We have seen that the ""partial derivative"" of a vector field is not a tensor field because its definition requires a subtraction between vectors in different vector spaces. However, we may parallel displace a vector from ""x"" to a nearby point, ""x + dx"" and it becomes a vector defined at ""x + dx"". Then we can define the covariant derivative using vector addition, and the result will be a tensor.

<<""Definition: For contraviant and covariant vector fields, p_b and q^a, the covariant derivative, is:""

""""

""""

""where dxⁱ is a small vector along the i-axis.""

<<

One may see that ""i"" is a vector index from first principles, or by using the result that the ""partial derivative"" of a scalar field is a vector field for each value of ""a"" and ""b"". Evidently,

""""

""""

<<""Definition: For a contraviant or covariant vector field, p, the covariant derivative operator, is""

""""

<<

The covariant derivative of the tensor product ""p^aq^{b"" may be found from first principles by parallel displacing ""p^{a"" and ""q^{b"". One finds

""""

The product rule of differentiation holds,

""""

Since these rules apply to a ""basis"", they apply to all contravariant rank 2 tensor fields by linearity. Similarly,

""""

One may extend this rule to tensors with any number of up and downstairs indices, adding a ""Γ"" term for each superfix and subtracting one for each suffix (use induction). The rule applies also to scalar fields, and states also that the covariant derivative of a scalar field, ""S : x → S(x)"", is equal to the partial derivative,

""""

The second covariant derivative of a scalar field, ""S"", commutes,

""""

The metric behaves as a constant with respect to covariant differentiation;

""""

This simply states that vectors have constant magnitude under parallel displacement.

====""""Geodesics====

A ""geodesic"" is a curve found by parallel transport of a vector in the direction of that vector. To apply parallel displacement to the ""tangent vector"", """", of a curve, ""t → x^a(t)"" we put """" into the standard formula for infinitesimal parallel displacement referring to contravariant components,

""""

Thus, we obtain:

<<""The geodesic equation: ""

""""

<<

For a time-like geodesic, we may normalise the tangent vector. In this case, the parameter ""t"" is ""proper time"". We may rewrite the geodesic equation in terms of velocity, """",

""""

If a particle is represented by a density, rather than as a point, then velocity is a field, and we have,

""""

""""

""""

Then the geodesic equation takes the simple form:

<<""The geodesic equation (alternate form):""

""""

<<

In a reference frame in which the particle is not moving, ""v^b = (1, 0, 0, 0)"", and this simply states

""""

>>""Definition: Proper acceleration is rate of change of velocity with respect to proper time.""

>>i.e. //proper acceleration// is equal to zero. This holds in any frame, since proper acceleration is a vector (the derivative of a vector with respect to a scalar). Using ""τ"" for proper time,

""""

We can now make the distinction that an ""active force"" causes a proper acceleration according to ""Newton’s second law"", while an ""inertial force"" does not. An active force can thus be considered as a physical quantity represented by a vector, while an inertial force is frame dependent.

====""""The Riemann Curvature Tensor====

In order to describe the ""curvature"" in spacetime we require a tensor field defined in terms of second derivatives, with four indices, and which vanishes in flat space. The second covariant derivative of a vector field, ""p_a : x → p_a(x)"", is

""""

""""

""""

If we ""antisymmetrise"" the second and third indices, by switching them and subtracting, a number of terms drop out.

""""

""""

This is a tensor equation for all vectors ""p_{c"". So, by the ""quotient theorem"",

""""

is a tensor field. """" is known as the [[http://en.wikipedia.org/wiki/Riemann_tensor Riemann curvature tensor]], and depends only on the metric and the connection. We have that the antisymmetrised second covariant derivative of a vector field is equal to curvature tensor contracted with that vector field,

""""

In a flat space we may choose rectilinear coordinates in which the Riemann curvature tensor vanishes. Contrariwise, if the Riemann curvature tensor vanishes, we may parallel displace a vector ""p_{a"" from ""x"" to ""x + dx"", and then to ""x + dx + dy"" and the result is the same as if we parallel displaced it from ""x"" to ""x + dy"" and then to ""x + dy + dx"". So, parallel transport is path independent. Rectilinear coordinates may then be defined by parallel transport of an orthonormal axes set to any point, showing that space is flat.

====""""Symmetries of the Curvature Tensor====

It is easily seen that

""""

and

""""

On lowering the top index,

""""

""""

Using

""""

we find

""""

""""

""""

Using

""""

we find

""""

""""

Then examination reveals the symmetries,

""""

and

""""

As a result of all these symmetries, one may calculate that the Riemann curvature tensor has only 20 independent components.

====""""The Bianchi Identity====

To find the second covariant derivative of a tensor field, first consider the outer product of two vector fields ""p_{a"" and ""q_{b"",

""""

""""

Interchange ""i"" and ""j"", and subtract,

""""

The antisymmetrised second covariant derivative of a vector field is equal to curvature tensor contracted with that vector field. So,

""""

So, by linearity, for a rank 2 tensor field ""T_{ab"",

""""

Now, if ""T"" is the covariant derivative of a vector field, ""T_ab = p_{a;b"",

""""

Make cyclic permutations of ""b"", ""i"", ""j"" and add, noting that """",

""""

""""

""""

""""

This is true for all vectors ""p_{c"". so we have the //Bianchi identity//,

""""

====""""Contracted Curvature Tensors====

If we contract the top index with the mid lower we find a symmetrical tensor,

""""

known as the [[http://en.wikipedia.org/wiki/Ricci_tensor Ricci tensor]]. (Older treatments usually contract with the last index. This is an opposite sign convention due to antisymmetry of the last two indices, and results in a minus sign which appeared in Einstein’s original formulation of the field equation, but which is not usual in more recent accounts). Contracting again gives the [[http://en.wikipedia.org/wiki/Ricci_scalar scalar curvature]], //total curvature//, or //Ricci scalar//,

""""

The Bianchi identity has five indices. If we contract it twice, we will have a relation with one suffix. Put ""c = j"" and multiply by ""g^{ab"",

""""

Because ""g"" behaves like a constant with respect to ""covariant differentiation"", we can use it to contract under the derivative. Since the Ricci tensor is symmetrical, it can be written with one index above the other. Thus,

""""

Now

""""

So,

""""

This is the Bianchi identity for the Ricci tensor. Raising the suffix ""i"",

""""

prompts the definition of the [[http://en.wikipedia.org/wiki/Einstein_tensor Einstein tensor]],

""""

Then the contracted Bianchi identity can be written more neatly:

<<""The contracted Bianchi identity: The Einstein curvature tensor satisfies""

""""

<<

[[GTRTensors Riemann Curvature ↑]] [[Gravitation Einstein’s Law of Gravitation →]]}}}}}}}}}}}}}

Revision [432]
Edited on 2012-04-03 05:23:39 by CharlesFrancis

Additions:

""""====""""====
======[[GeneralRelativity ←]] Riemann Curvature [[Gravity ↑]] [[Gravitation →]]======
If index gymnastics were a physical sport, this page would be a training session for the fit and athletic. In it, the covariant derivative is established from local parallelism, the ""Riemann curvature tensor"" is found, properties are analysed, and the ""Einstein curvature tensor"" is found and shown to obey the ""contracted Bianchi identity"", which has importance in ""Einstein’s law of gravitation"". From a philosophical perspective, the important aspect is that manipulations in mathematics introduce no new physical principles, and merely express relationships which necessarily hold in a universe obeying the general principle of relativity and in which we can translate objects through small distances. If you are prepared to take that on faith, you can skip the calculations and move quickly on to the next section. That is quite reasonable. You may reflect that these calculations have been checked and rechecked by tens of thousands of mathematicians since their original formulation in the 19th century. Such is the requirement of reproducibility in a strict approach to science. If you take the strict approach, that nothing should be taken on faith, and require that logic, rather than authority, should be the final arbiter, there is no help for it; you have to do the training session. No sympathy can be afforded to those who decry authority and yet are too idle or unfit to do the training.
""Christoffel Symbols""
""Parallel Displacement""
""The Covariant Derivative""
""Geodesics""
""The Riemann Curvature Tensor""
""Symmetries of the Curvature Tensor""
""The Bianchi Identity""
""Contracted Curvature Tensors""
====""""Christoffel Symbols====
Consider a region in which the metric field is ""g_{ab"". [[http://en.wikipedia.org/wiki/Christoffel_symbol Christoffel symbols]] have a vital role in calculating the effect of parallel displacement.

<<""Definition: Christoffel symbols of the first kind:""

""""

<<

Christoffel symbols, defined using ""partial differentiation"", are not tensors, but indices can be raised and lowered in the usual way. It is common to raise the first index.

<<""Definition: Christoffel symbols of the second kind:""

""""

<<

Clearly Christoffel symbols are symmetric in the last two suffixes, and satisfy

""""

Christoffel symbols are sometimes said to define the connection in general relativity. From an empirical perspective, the connection is the physical prescription for comparing the coordinate system set up an observer with that of a second, nearby observer, using parallel displacement in tangent space. In pure mathematics there is no physical prescription, so a different definition of a connection is required. In my view, adopting a mathematical definition reduces the empirical foundations of general relativity to metaphysics, in conflict with Einstein’s approach to physics. In this empirically based account, the purpose of Christoffel symbols is to enable us to calculate the effect of parallel displacement without reference to a non-physical tangent space. A ""tangent chart"" at ""x"" is defined with non-physical metric ""h"" using primed coordinates. At ""x"",

""""

""""

""""

Observe that, from ""Clairaut’s Theorem"", the partial derivative of the transformation matrix is symmetrical in its lower indices,

""""

Interchange ""a ↔ c"" and ""b ↔ c"",

""""

""""

Then,

""""

====""""Parallel Displacement====

Parallel displacement of a vector ""p"" from ""x"" to ""x + dx"" in tangent space keeps the primed components constant, """". Multiply by """", to lower the index and convert to unprimed coordinates,

""""

""""

Then, ignoring terms ""O(max(dxⁱ))"",

""""

""""

Substituting the Christoffel symbol eliminates the dependency on tangent space,

""""

Raise and lower indices to find the standard formula for infinitesimal parallel displacement referring to covariant components,

<<""Infinitesimal parallel displacement: ""

""""

<<

For a second vector ""q"", ""p · q"" is invariant,

""""

""""

""""

Since this is true for all ""p_{a"", we find the standard formula for infinitesimal parallel displacement referring to contravariant components,

<<""Infinitesimal parallel displacement: ""

""""

<<

====""""The Covariant Derivative====

We have seen that the ""partial derivative"" of a vector field is not a tensor field because its definition requires a subtraction between vectors in different vector spaces. However, we may parallel displace a vector from ""x"" to a nearby point, ""x + dx"" and it becomes a vector defined at ""x + dx"". Then we can define the covariant derivative using vector addition, and the result will be a tensor.

<<""Definition: For contraviant and covariant vector fields, p_b and q^a, the covariant derivative, is:""

""""

""""

""where dxⁱ is a small vector along the i-axis.""

<<

One may see that ""i"" is a vector index from first principles, or by using the result that the ""partial derivative"" of a scalar field is a vector field for each value of ""a"" and ""b"". Evidently,

""""

""""

<<""Definition: For a contraviant or covariant vector field, p, the covariant derivative operator, is""

""""

<<

The covariant derivative of the tensor product ""p^aq^{b"" may be found from first principles by parallel displacing ""p^{a"" and ""q^{b"". One finds

""""

The product rule of differentiation holds,

""""

Since these rules apply to a ""basis"", they apply to all contravariant rank 2 tensor fields by linearity. Similarly,

""""

One may extend this rule to tensors with any number of up and downstairs indices, adding a ""Γ"" term for each superfix and subtracting one for each suffix (use induction). The rule applies also to scalar fields, and states also that the covariant derivative of a scalar field, ""S : x → S(x)"", is equal to the partial derivative,

""""

The second covariant derivative of a scalar field, ""S"", commutes,

""""

The metric behaves as a constant with respect to covariant differentiation;

""""

This simply states that vectors have constant magnitude under parallel displacement.

====""""Geodesics====

A ""geodesic"" is a curve found by parallel transport of a vector in the direction of that vector. To apply parallel displacement to the ""tangent vector"", """", of a curve, ""t → x^a(t)"" we put """" into the standard formula for infinitesimal parallel displacement referring to contravariant components,

""""

Thus, we obtain:

<<""The geodesic equation: ""

""""

<<

For a time-like geodesic, we may normalise the tangent vector. In this case, the parameter ""t"" is ""proper time"". We may rewrite the geodesic equation in terms of velocity, """",

""""

If a particle is represented by a density, rather than as a point, then velocity is a field, and we have,

""""

""""

""""

Then the geodesic equation takes the simple form:

<<""The geodesic equation (alternate form):""

""""

<<

In a reference frame in which the particle is not moving, ""v^b = (1, 0, 0, 0)"", and this simply states

""""

>>""Definition: Proper acceleration is rate of change of velocity with respect to proper time.""

>>i.e. //proper acceleration// is equal to zero. This holds in any frame, since proper acceleration is a vector (the derivative of a vector with respect to a scalar). Using ""τ"" for proper time,

""""

We can now make the distinction that an ""active force"" causes a proper acceleration according to ""Newton’s second law"", while an ""inertial force"" does not. An active force can thus be considered as a physical quantity represented by a vector, while an inertial force is frame dependent.

====""""The Riemann Curvature Tensor====

In order to describe the ""curvature"" in spacetime we require a tensor field defined in terms of second derivatives, with four indices, and which vanishes in flat space. The second covariant derivative of a vector field, ""p_a : x → p_a(x)"", is

""""

""""

""""

If we ""antisymmetrise"" the second and third indices, by switching them and subtracting, a number of terms drop out.

""""

""""

This is a tensor equation for all vectors ""p_{c"". So, by the ""quotient theorem"",

""""

is a tensor field. """" is known as the [[http://en.wikipedia.org/wiki/Riemann_tensor Riemann curvature tensor]], and depends only on the metric and the connection. We have that the antisymmetrised second covariant derivative of a vector field is equal to curvature tensor contracted with that vector field,

""""

In a flat space we may choose rectilinear coordinates in which the Riemann curvature tensor vanishes. Contrariwise, if the Riemann curvature tensor vanishes, we may parallel displace a vector ""p_{a"" from ""x"" to ""x + dx"", and then to ""x + dx + dy"" and the result is the same as if we parallel displaced it from ""x"" to ""x + dy"" and then to ""x + dy + dx"". So, parallel transport is path independent. Rectilinear coordinates may then be defined by parallel transport of an orthonormal axes set to any point, showing that space is flat.

====""""Symmetries of the Curvature Tensor====

It is easily seen that

""""

and

""""

On lowering the top index,

""""

""""

Using

""""

we find

""""

""""

""""

Using

""""

we find

""""

""""

Then examination reveals the symmetries,

""""

and

""""

As a result of all these symmetries, one may calculate that the Riemann curvature tensor has only 20 independent components.

====""""The Bianchi Identity====

To find the second covariant derivative of a tensor field, first consider the outer product of two vector fields ""p_{a"" and ""q_{b"",

""""

""""

Interchange ""i"" and ""j"", and subtract,

""""

The antisymmetrised second covariant derivative of a vector field is equal to curvature tensor contracted with that vector field. So,

""""

So, by linearity, for a rank 2 tensor field ""T_{ab"",

""""

Now, if ""T"" is the covariant derivative of a vector field, ""T_ab = p_{a;b"",

""""

Make cyclic permutations of ""b"", ""i"", ""j"" and add, noting that """",

""""

""""

""""

""""

This is true for all vectors ""p_{c"". so we have the //Bianchi identity//,

""""

====""""Contracted Curvature Tensors====

If we contract the top index with the mid lower we find a symmetrical tensor,

""""

known as the [[http://en.wikipedia.org/wiki/Ricci_tensor Ricci tensor]]. (Older treatments usually contract with the last index. This is an opposite sign convention due to antisymmetry of the last two indices, and results in a minus sign which appeared in Einstein’s original formulation of the field equation, but which is not usual in more recent accounts). Contracting again gives the [[http://en.wikipedia.org/wiki/Ricci_scalar scalar curvature]], //total curvature//, or //Ricci scalar//,

""""

The Bianchi identity has five indices. If we contract it twice, we will have a relation with one suffix. Put ""c = j"" and multiply by ""g^{ab"",

""""

Because ""g"" behaves like a constant with respect to ""covariant differentiation"", we can use it to contract under the derivative. Since the Ricci tensor is symmetrical, it can be written with one index above the other. Thus,

""""

Now

""""

So,

""""

This is the Bianchi identity for the Ricci tensor. Raising the suffix ""i"",

""""

prompts the definition of the [[http://en.wikipedia.org/wiki/Einstein_tensor Einstein tensor]],

""""

Then the contracted Bianchi identity can be written more neatly:

<<""The contracted Bianchi identity: The Einstein curvature tensor satisfies""

""""

<<

[[GTRTensors Riemann Curvature ↑]] [[Gravitation Einstein’s Law of Gravitation →]]}}}}}}}}}}}}}

Deletions:

""""====""""====
======[[GTRTensors ←]] Einstein’s Law of Gravitation [[Gravity ↑]] [[LargeScaleStructure →]]======
It is shown that, for weak gravitational fields, the effect of gravitational redshift on geodesic motion gives an identical acceleration to that of a classical gravitational field. Einstein combined Newton’s law of gravity with his three laws of motion into a single tensor law. The Schwarzschild metric, describing gravity in the region of a star or a planet, is calculated. Black holes are introduced.
""The Weak Field Limit""
""Conservation of Mass-Energy""
""The Stress-Energy Tensor""
""Einstein’s Law of Gravitation""
""Stationary Matter""
""The Newtonian Approximation""
""The Schwarzschild Solution""
""Black Holes""
====""""The Weak Field Limit====
Using Cartesian space coordinates, the ""metric"" determined from the radar method is,
""""
"" Using "", the index raising operators are
""""
The ""geodesic equation"" is
""""
where the ""Christoffel symbols"" are
""""
For the Christoffel symbols to be non-zero, two indices must be the same. In a constant field, the other must not be zero. For non-relativistic velocities, terms in the order of velocity squared can be ignored, and we have """". Then, 3-acceleration is given by, for ""a ≠ 0"",
""""
Writing ""k = 1 + z"", where redshift ""z"" is small, we have that acceleration is minus the gradient of ""z"",
""""
So, gravitational redshift can be identified with the [[http://en.wikipedia.org/wiki/Scalar_potential scalar potential]] in Newtonian gravity. The time component of the metric is
""""
This is called the //weak field limit//. ""mV"" has units of kinetic energy, ""½mv²"". To ""convert"" to conventional units, we must divide by ""c^{2"".

""""

Thus,

""""

and it is seen that very small redshifts are associated with normal accelerations in a gravitational field.

====""""Conservation of Mass-Energy====

Consider a particle of matter, using primed coordinates in which the particle is stationary. A fundamental observation is that its mass remains constant in time. We say that mass is conserved. This is expressed in the simple equation,

""""

This is not a covariant equation, as it depends on coordinates in which the particle is stationary. ""General covariance"" means this should be replaced by a vector or tensor equation, saying exactly the same thing, but in any coordinates. Replace ""m"" by a vector, ""p^i' =; (m, 0, 0, 0)"", which expresses the fact that the particle is stationary, and replace ""d/dt'"" by the covariant derivate. Then the equation ,

""""

is a restatement of conservation of mass, and is expressed in terms of vectors so that, in any coordinate system, it has the same form. More generally, a particle at a point is replaced by a probability density function, reflecting the fact that we don’t know exact position. ""p"" is then a vector field, describing momentum density. We can then sum the densities for all particles to find the distribution of matter and net momentum within neighbourhood. By linearity, the equation of conservation of mass still holds.

In classical physics, this equation appears in a number of contexts, e.g, using, for ""i = 1, 2, 3"",

""""

the [[http://en.wikipedia.org/wiki/Navier-Stokes_equations/Derivation#Conservation_of_mass mass continuity equation]] in fluid dynamics,

""""

where ""ρ"" is mass density and ""v"" is 3-velocity, or the [[http://en.wikipedia.org/wiki/Continuity_equation continuity equation]] in electrodynamics

""""

where ""ρ"" is charge density and ""j"" is current density.

====""""The Stress-Energy Tensor====

We know from observation that matter is the source of the gravitational field. Einstein sought a law to express this. Quantitative science requires that laws are given in the form of equations. A tensor equation is required by ""general covariance"", so as to ensure the general principle of relativity, that local laws of physics are the same for all observers. We need a tensor equation relating curvature to mass density. A four index equation for the Riemann curvature tensor would be too restrictive — ""tidal forces"" show that curvature is not zero in empty space. Einstein started to look for an equation involving the ""Ricci tensor"". The Ricci tensor is a rank 2 tensor. To form an equation we need a rank 2 tensor describing the density of matter.

It is trivial to write down a rank 2 tensor describing a stationary particle of mass, ""m"". In primed, locally Minkowski, coordinates, this is

""""

The same tensor is described generally using coordinate transformation. Coordinate transformations for tensors act on both indices. For Minkowski coordinates, coordinate tranformations are Lorentz transformations, and represent a velocity boost such that proper velocity ""(1, 0, 0, 0)"" becomes ""v^a = (v⁰, v¹, v², v³)"". To transform the tensor ""m^{i'j'"", we must apply the transformation to both indices separately. The result is

""""

More generally, mass and velocity of a point particle are replaced by density functions. We then sum over all the particles of matter within a region. The resulting symmetrical tensor, ""T^ab"", is the [[http://en.wikipedia.org/wiki/Stress-energy_tensor stress energy tensor]]. The stress-energy tensor is defined at each point of a spacetime coordinate system (i.e. it is a ""field""). While the momentum vector, ""p"", for a body, has a simple intuitive meaning as the sum of the momenta of its component subparticles, it is less easy to see the meaning of the sum of products, ""v^av^{b"", in the stress-energy tensor. In statistics, similar sums of products appear in the study of [[http://en.wikipedia.org/wiki/Correlation correlation]]. The stress-energy tensor may be regarded as a measure of not just of total motion, but also of the correlation between the motions of component particles, as described in the term “stress”.

Observe that

""""

by ""conservation of mass-energy"" and by the ""geodesic equation"", which follows from Newton’s first law. The //law of local energy-momentum conservation// follows by linearity.

<<""The law of local energy-momentum conservation: T^ab_;b = 0.""<<

More generally, local energy-momentum conservation still holds when particles interact, since momentum is conserved in interaction. This will be shown in [[QED Quantum Electrodynamics]]. For now, it is treated as a fundamental law with application in specific equations of physics. For example, in the non-relativistic approximation,

""""

reduces to, for ""a = 0"", ""v⁰ = 1"", to conservation of mass in fluid dynamics, and, for ""a ≠ 0"" it is the [[http://en.wikipedia.org/wiki/Navier-Stokes_equations#Derivation_and_description Navier-Stokes equations]] for a perfect fluid in the absence of pressure and external forces. ""T^ab_;b = 0"" can also be shown, from [[http://en.wikipedia.org/wiki/Maxwell%27s_equations Maxwell’s equations]], to express conservation of energy-momentum for a classical electromagnetic field, in which energy is given by

""""

and momentum is given by

""""

====""""Einstein’s Law of Gravitation====

In his early researches Einstein tried to relate the ""Ricci tensor"" to the stress energy tensor, but found that this did not work. If curvature is the consequence of stress-energy, then it must reflect the law of local energy-momentum conservation as an identity. The Einstein tensor, ""G^{ab"", is unique as a rank 2 tensor formed by contraction of the ""the Riemann curvature tensor"", which is symmetrical, and which satisfies an identity,

""""

the ""contracted Bianchi identity"". Einstein therefore wrote down the law of gravitation known as [[http://en.wikipedia.org/wiki/Einstein_field_equations Einstein’s field equation]]

""""

where ""κ"" is a constant of proportionality, to be determined by comparison with the Newtonian approximation. This law summarises the two principle tennets of Einstein’s theory of gravity, that matter is the cause of curvature, and, through the Bianchi identity, that energy-momentum is conserved and hence that inertial objects follow geodesics, thereby combining Newton’s three laws of motion and Newton’s law of universal gravitation into a single tensorial law.

====""""Stationary Matter====

In the case of a static body of uniform density, ""ρ"", ""T^{ab"" is,

""""

Einstein’s field equation, written in terms of the Ricci tensor is

""""

Contract the indices ""a"" and ""b"", noting from the summation convention that """",

""""

""""

So, the total scalar curvature in the presence of matter is given by

""""

Hence

""""

""""

====""""The Newtonian Approximation====

The ""Christoffel symbols"" are

"""".

The ""Ricci tensor"" is

""""

In the Newtonian approximation, the metric, ""g"", is slowly varying in space and constant in time. We may neglect terms of second order in derivatives of the metric, and set time derivatives to zero. Then

""""

In Cartesian, ""x-"", ""y-"", ""z-""coordinates

""""

where """" is the [[http://en.wikipedia.org/wiki/Laplacian Laplacian operator]]. Thus,

""""

[[http://en.wikipedia.org/wiki/Poisson's_equation Poisson’s equation]] for a Newtonian gravitational potential, ""k"", due to a mass distribution of density ""ρ"" is

""""

where ""G"" is Newton’s universal gravitational constant. Thus, ""κ = 8πG"", and Einstein’s field equation is

""""

====""""The Schwarzschild Solution====

The first solution to Einstein’s field equation was found, within a few months of the publication of general relativity, by [[http://en.wikipedia.org/wiki/Karl_Schwarzschild Karl Schwarzschild]], who was on active service in Russia. Schwarzschild had been known as a prodigy and the [[Schwarzschild calculation]] is quite involved. It shows that the metric outside of an isolated, spherically symmetric, non-rotating, gravitating body of mass ""M"" is

""""

This is the [[http://en.wikipedia.org/wiki/Schwarzschild_metric Schwarzschild metric]]. In the ""weak field limit"", the redshift factor

""""

is equivalent to a [[http://en.wikipedia.org/wiki/Potential_energy#Gravitational_potential_energy potential]],

""""

in accordance with Newton’s inverse square law of gravity.

====""""Black Holes====

>>""Definition: The Schwarzschild radius is r = 2GM.""

>> The Schwarzshild metric becomes singular at the [[http://en.wikipedia.org/wiki/Schwarzschild_radius Schwarzschild radius]]. For a normal star or planet, this is not important because Schwarzchild geometry applies only in empty space outside of the star. The radius of a normal star is greater than its Schwarzschild radius, but there is a theoretical possibility that a body could exist for which its mass is contained within its Schwarzschild radius. Such a body is called a [[http://en.wikipedia.org/wiki/Black_hole black hole]]. In practice, we know that when a star burns out it can become a [[http://en.wikipedia.org/wiki/White_dwarf white dwarf]]. Provided its mass is less than the [[http://en.wikipedia.org/wiki/Chandrasekhar_limit Chandrasekhar limit]] (about 1.4 solar masses) [[http://en.wikipedia.org/wiki/Electron_degeneracy_pressure electron degeneracy pressure]] due to the ""Pauli Exclusion Principle"" prevents further collapse against the force of gravity. For a burned out star greater than this mass, gravity overcomes electron degeneracy pressure and the star collapses. Energy released in collapse creates a [[http://en.wikipedia.org/wiki/Supernova supernova explosion]], electrons and protons combine to form neutrons, and a [[http://en.wikipedia.org/wiki/Neutron_star neutron star]] may form. If the neutron star has a mass greater than about 5 solar masses, gravity overcomes [[http://en.wikipedia.org/wiki/Degenerate_matter neutron degeneracy pressure]], and collapse continues indefinitely. In practice, many galaxies, including our own, appear to have supermassive black holes at their centres. [[http://www.journals.uchicago.edu/ApJ/journal/issues/ApJ/v616n2/60440/60440.html Observations]] show that [[http://en.wikipedia.org/wiki/Sgr_A%2A Sgr A*]], at the centre of the Milky way, is almost certainly a supermassive black hole.

From the pespective of an external observer, the redshift factor, ""k = (g₀₀)^{−½"", becomes zero at the Schwarzschild radius, showing that time, and all physical processes, slow down relative to time measured by a distant observer as matter approaches the black hole. The external observer would calculate that matter does not actually fall through the Schwarzschild radius, but stops at it. This is true also of light. The Schwarzschild radius is an [[http://en.wikipedia.org/wiki/Event_horizon event horizon]]. Processes inside an event horizon are hidden to the observer. An issue was raised as to whether a ""singularity"" could actually form according to the equations of relativity. [[http://en.wikipedia.org/wiki/Robert_Oppenheimer Oppenheimer]] & [[http://en.wikipedia.org/wiki/Hartland_Snyder Snyder]] published the first calculation of gravitational collapse in 1939, showing that one can.

In fact the ""singularity"" at the event horizon is only a singularity in a particular coordinate system. If instead of using coordinates defined by an external observer, stationary with respect to the hole, we use coordinates determined by an observer falling into it, it can be shown that no singularity arises at the Schwarzschild radius. According to general relativity, the observer simply falls through empty space at the Schwarzschild radius, into a region from which he can no longer communicate with the external observer. There is no singularity in coordinates defined by the falling observer, and physical processes proceed as normal until he meets the true singularity at ""r = 0"". That is the solution according to Einstein’s field equation, but the question still arises as to whether it is a real physical solution. The meaning of the singularity at ""r = 0"" is that known laws of physics break down. We cannot say, from classical general relativity, precisely at what point the laws of physics break down in the vicinity of the singularity. Relational quantum gravity will reexamine this issue in the light of a unification with quantum theory.

[[Gravitation Einstein’s Law of Gravitation ↑]] [[LargeScaleStructure Large Scale Structure →]]}}}}}}

Revision [431]
Edited on 2012-04-03 05:14:24 by CharlesFrancis

Additions:

Writing ""k = 1 + z"", where redshift ""z"" is small, we have that acceleration is minus the gradient of ""z"",

Deletions:

Writing ""k^-1 = 1 + z"", where redshift ""z"" is small, we have that acceleration is minus the gradient of ""z"",

Revision [430]
Edited on 2012-04-03 05:11:36 by CharlesFrancis

Additions:

For the Christoffel symbols to be non-zero, two indices must be the same. In a constant field, the other must not be zero. For non-relativistic velocities, terms in the order of velocity squared can be ignored, and we have """". Then, 3-acceleration is given by, for ""a ≠ 0"",
Writing ""k^-1 = 1 + z"", where redshift ""z"" is small, we have that acceleration is minus the gradient of ""z"",

Deletions:

The partial derivatives of the metric are
""""
For the Christoffel symbols to be non-zero, two indices must be the same, and, in a constant field, the other must not be zero. For non-relativistic velocities, terms in the order of velocity squared can be ignored, and we have """". Then, 3-acceleration is given by, for ""a ≠ 0"",
Writing ""k^-1 = 1 + V"", where ""V"" is small, we have that acceleration is minus the gradient of ""V"",

Revision [429]
Edited on 2012-03-30 03:54:33 by CharlesFrancis

Additions:

Writing ""k^-1 = 1 + V"", where ""V"" is small, we have that acceleration is minus the gradient of ""V"",

Deletions:

Writing ""k^-1 = 1 + V"", where ""V"" is small, we have that acceleration is minus the gradient of ""V"",

Revision [428]
Edited on 2012-03-30 03:52:40 by CharlesFrancis

Additions:

Writing ""k^-1 = 1 + V"", where ""V"" is small, we have that acceleration is minus the gradient of ""V"",

Deletions:

Writing ""k = 1 + V"", where ""V"" is small, we have that acceleration is minus the gradient of ""V"",

Revision [427]
Edited on 2012-03-30 02:30:47 by CharlesFrancis

Additions:

Writing ""k = 1 + V"", where ""V"" is small, we have that acceleration is minus the gradient of ""V"",

Deletions:

Writing ""k = 1 + V"", where ""V = z"" is small, we have that acceleration is minus the gradient of ""V"",

Revision [313]
Edited on 2010-07-09 02:26:52 by CharlesFrancis

Additions:

""The Weak Field Limit""
""Conservation of Mass-Energy""
""The Stress-Energy Tensor""
""Einstein’s Law of Gravitation""
""Stationary Matter""
""The Newtonian Approximation""
""The Schwarzschild Solution""
""Black Holes""
From the pespective of an external observer, the redshift factor, ""k = (g₀₀)^{−½"", becomes zero at the Schwarzschild radius, showing that time, and all physical processes, slow down relative to time measured by a distant observer as matter approaches the black hole. The external observer would calculate that matter does not actually fall through the Schwarzschild radius, but stops at it. This is true also of light. The Schwarzschild radius is an [[http://en.wikipedia.org/wiki/Event_horizon event horizon]]. Processes inside an event horizon are hidden to the observer. An issue was raised as to whether a ""singularity"" could actually form according to the equations of relativity. [[http://en.wikipedia.org/wiki/Robert_Oppenheimer Oppenheimer]] & [[http://en.wikipedia.org/wiki/Hartland_Snyder Snyder]] published the first calculation of gravitational collapse in 1939, showing that one can.

In fact the ""singularity"" at the event horizon is only a singularity in a particular coordinate system. If instead of using coordinates defined by an external observer, stationary with respect to the hole, we use coordinates determined by an observer falling into it, it can be shown that no singularity arises at the Schwarzschild radius. According to general relativity, the observer simply falls through empty space at the Schwarzschild radius, into a region from which he can no longer communicate with the external observer. There is no singularity in coordinates defined by the falling observer, and physical processes proceed as normal until he meets the true singularity at ""r = 0"". That is the solution according to Einstein’s field equation, but the question still arises as to whether it is a real physical solution. The meaning of the singularity at ""r = 0"" is that known laws of physics break down. We cannot say, from classical general relativity, precisely at what point the laws of physics break down in the vicinity of the singularity. Relational quantum gravity will reexamine this issue in the light of a unification with quantum theory.}

Deletions:

From the pespective of an external observer, the redshift factor, ""k = (g₀₀)^{−½"", becomes zero at the Schwarzschild radius, showing that time, and all physical processes, slow down relative to time measured by a distant observer as matter approaches the black hole. The external observer would calculate that matter does not actually fall through the Schwarzschild radius, but stops at it. This is true also of light. The Schwarzschild radius is an [[http://en.wikipedia.org/wiki/Event_horizon event horizon]]. Processes inside an event horizon are hidden to the observer. An issue was raised as to whether a singularity could actually form according to the equations of relativity. [[http://en.wikipedia.org/wiki/Robert_Oppenheimer Oppenheimer]] & [[http://en.wikipedia.org/wiki/Hartland_Snyder Snyder]] published the first calculation of gravitational collapse in 1939, showing that one can.

In fact the singularity at the event horizon is only a singularity in a particular coordinate system. If instead of using coordinates defined by an external observer, stationary with respect to the hole, we use coordinates determined by an observer falling into it, it can be shown that no singularity arises at the Schwarzschild radius. According to general relativity, the observer simply falls through empty space at the Schwarzschild radius, into a region from which he can no longer communicate with the external observer. There is no singularity in coordinates defined by the falling observer, and physical processes proceed as normal until he meets the true singularity at ""r = 0"". That is the solution according to Einstein’s field equation, but the question still arises as to whether it is a real physical solution. The meaning of the singularity at ""r = 0"" is that known laws of physics break down. We cannot say, from classical general relativity, precisely at what point the laws of physics break down in the vicinity of the singularity. Relational quantum gravity will reexamine this issue in the light of a unification with quantum theory.}

Revision [64]
The oldest known version of this page was created on 2009-04-25 06:56:53 by CharlesFrancis