Curvature and General Relativity

[kurious]

Did Einstein think that the stress-energy tensor in GR was the cause of the
curved paths that particles follow through three dimensional space,or did he
think that the curved paths were caused by something more fundamental,
given that he searched for a unified field theory?

 

[pmb_phy]

Did Einstein think that the stress-energy tensor in GR was the cause of the
curved paths that particles follow through three dimensional space,or did he
think that the curved paths were caused by something more fundamental,
given that he searched for a unified field theory?

Einstein held that mass generates a gravitational field. Since mass is related to energy then the mathematical quantity which represents the total desrciption of mass is the stress-energy-momentum tensor Einstein use that tensor into his field equations. Or as Einstein said

The special theory of relativity has led to the conclusion that inert mass is nothing more or less than energy, which finds its complete mathematical expression in a symmetrical tensor of second rank, the energy-tensor.

Pete

 

[DW]

Did Einstein think that the stress-energy tensor in GR was the cause of the
curved paths that particles follow through three dimensional space,or did he
think that the curved paths were caused by something more fundamental,
given that he searched for a unified field theory?

In Einstein's unification theory the electromagnetic field was an anti symmetric part of the metric. In otherwords gravitation and electromagnetism were both described as a part of the same thing, the differential spacetime geometry. For general relativity, not just mass, but also electromagnetism itself contributes to the stress energy tensor which is the source term in Einstein's field equations. In his unification theory the electromagnetic field is not a part of the stress energy tensor, but is built into the other side of the equation. In both theories the stress energy tensor acting as the source term ends up determining the differential geometry of the spacetime through second order nonlinear differential equations and that geometry described by the metric determines what paths are geodesics or the "curved paths" that you are referring through which things follow. The difference is that in general relativity the charge follows a nongeodesic path in reaction to the field whereas in his unification theory the charge follows a geodesic as that is determined in part by the electromagnetic field portion of the spacetime geometry.
See-
http://www.geocities.com/zcphysicsms/chap12.htm#BM12_5

 

[kurious]

DW:
In his unification theory the electromagnetic field is not a part of the stress energy tensor, but is built into the other side of the equation

Kurious:
So "all sources of energy contribute to curvature" becomes "all curvature
contributes to total energy?"

 

[DW]

DW:
In his unification theory the electromagnetic field is not a part of the stress energy tensor, but is built into the other side of the equation

Kurious:
So "all sources of energy contribute to curvature" becomes "all curvature
contributes to total energy?"

Not exactly. The electromagnetic field would be a part of the metric, but the "sources" of the electromagnetic field would be a part of the stress energy tensor.

 

[kurious]

As three dimensional space expands does the stress-energy tensor, for a large mass made of lots of particles, stay constant, or does it change? For example photons in the Sun would redshift as space expands, so I would expect the energy density component of the stress-energy tensor for the Sun to decrease.And is dark energy included in the stress-energy tensor
for a star like the Sun?

 

[pmb_phy]

As three dimensional space expands does the stress-energy tensor, for a large mass made of lots of particles, stay constant, or does it change?

That depends on the matter in question. For example: If the matter is bound together (like a atoms in a steel plate rather than a gas of dust particles) then the mass density will remain constant. However the gravitational tidal forces (aka "spacetime curvature) will induce stresses in the matter. That stress is a part of the stress-energy-momentum tensor (SEM) which describes the matter. So in this case some components would change and other components would remain the same. If you're speaking of dust in an expanding universe then the mass density would change etc.

And is dark energy included in the stress-energy tensor for a star like the Sun?

Since nobody knows what dark energy is that question cannot yet be answered. Its concievable that the SEM tensor describes part of it and its also possible that dark energy would not be any part of the SEM tensor as is the case when there is a non-zero cosmological constant.

Pete

 

[DW]

As three dimensional space expands does the stress-energy tensor, for a large mass made of lots of particles, stay constant, or does it change? For example photons in the Sun would redshift as space expands, so I would expect the energy density component of the stress-energy tensor for the Sun to decrease.And is dark energy included in the stress-energy tensor
for a star like the Sun?

What people are referring to as "dark energy" is known to be nothing more or less than the cosmological constant and no that is not included in the stress-energy tensor, but is a separate term in Einstein's field equations. It acts as a source term much like the stress-energy tensor which is why a few have chosen to call it dark energy. The particles comprising the sun like any star are kept from expanding with the space by gravitational attraction so in rigid ruler coordinates, no the energy density does not decrease for the sun over time. It is the density of galaxies that can be approximated as comoving that would decrease according to rigid coordinates. Compositions of things that extend to significantly smaller scales tend not to expand with the space due to gravitational or electromagnetic attractions.

 

[kurious]

DW:

What people are referring to as "dark energy" is known to be nothing more or less than the cosmological constant and no that is not included in the stress-energy tensor, but is a separate term in Einstein's field equations

Kurious:
Why didn't general relativity,when Einstein first formulated it, predict the existence of dark energy/ the cosmological constant? Surely an examination of the original field equations of Einstein would show that something needed to be added to them.
Is there something missing from the foundations of general relativity i.e. is there something missing from the theory of special relativity?
Is special relativity a theory that should be built from another theory
that has the cosmological constant at its core?
What if the cosmological constant is the cause of the inertia of masses and not the Higgs field?

 

[jcsd]

Why didn't general relativity,when Einstein first formulated it, predict the existence of dark energy/ the cosmological constant? Surely an examination of the original field equations of Einstein would show that something needed to be added to them.
Is there something missing from the foundations of general relativity i.e. is there something missing from the theory of special relativity?
Is special relativity a theory that should be built from another theory
that has the cosmological constant at its core?
What if the cosmological constant is the cause of the inertia of masses and not the Higgs field?

It DID in a way predict dark energy, but until very recently there seemed no good reason to suspect that it actually conformed to reality. It was orignally included to enable a staedy-state model of the universe (which it fails to do as there are no stable relastic solutions without contraction or expansion) which was the prevailing view in cosmology at the time.

Though Einstein called it his greatest mistake it never went a way completely models with a non-zero cosmological constant, though assumed by many to be redundant, were still studied. Lemaitre was quite content to include the cosmological constant in his cosmological models.

 

[Chronos]

Current theory says, the 'cosmological constant', is very much alive. Call it 'dark energy' or whatever. Einstein called it 'the greatest mistake of his career'. It now appears he was right all along. His humility gave an example of how science should be conducted. He questioned everything and arrived at explanations that gave us the abilitiy to explain everything.

 

[pmb_phy]

Current theory says, the 'cosmological constant', is very much alive. Call it 'dark energy' or whatever. Einstein called it 'the greatest mistake of his career'. It now appears he was right all along. His humility gave an example of how science should be conducted. He questioned everything and arrived at explanations that gave us the abilitiy to explain everything.

The cosmological constant also plays a role in the infationary theory of the universe.