Parameters of Theory of everything.

[MJ kihara]

Sometimes am having a problem accessing the thread..what a glitch...

17 hours ago, Markus Hanke said:

What do you mean by this, exactly?

Spacetime is not embedded in anything else, so when we are talking of geometry in this context, what we are referring to is intrinsic geometry on the manifold.

Riemann tensor has 256 components in 4d, Ricci tensor gets the averages/proportionality from Riemann tensor that tend to be more surface topological,Ricci tensor is more critical in constructing Einstein monifold..hope am not wrong on this.

What I meant by interior of the monifold is the other aspects that may have not been captured while getting Ricci  tensor.

[KJW]

7 minutes ago, MJ kihara said:

Riemann tensor has 256 components in 4d

Only 20 components are independent. The Ricci tensor has 10 independent components leaving 10 independent components for the Weyl conformal tensor.

 

Edited October 19 by KJW

[MJ kihara]

On 10/17/2024 at 4:12 PM, MigL said:

Energy is  determined by the configuration of a system ( think potential energy ), so different configurations will have differing energies, even though the systems are comprised of the same individual parts.

Is mass linear or nonlinear i.e don't we add up together mass?...E=mc^2 , individual parts with their masses doesn't their total mass give mass of total of the configuration....the discussion on thread page 5 is spinning off my head ...

[Markus Hanke]

40 minutes ago, MJ kihara said:

Riemann tensor has 256 components in 4d

That’s true - but it also has many symmetries in GR, hence only 20 of those are functionally independent.

42 minutes ago, MJ kihara said:

Ricci tensor gets the averages/proportionality from Riemann tensor that tend to be more surface topological,Ricci tensor is more critical in constructing Einstein monifold

The Riemann tensor can be decomposed into two parts - the Ricci tensor, and the Weyl tensor.

Suppose you have a small ball of test particles freely falling in a spacetime. The Ricci tensor tells you how fast the volume of this ball changes with time, whereas the Weyl tensor measures how the shape of the ball gets distorted as it falls.

In vacuum, shapes get distorted, but volumes are preserved during free fall (hence the Ricci tensor vanishes, but the Weyl tensor doesn’t). In the interior of sources, both shapes and volumes may change.

32 minutes ago, MJ kihara said:

Is mass linear or nonlinear i.e don't we add up together mass?

Mass itself adds linearly, but the source term in the field equations is not mass, but the energy-momentum tensor. 

[Mordred]

Your looking at Migl's statement wrong.

Let's use an simple everyday world example.

Let's use an electrical circuit.

Take a multimeter on a wire conducting some current. If you take the leads to two point is on the wire itself you cannot measure a voltage.

However if you add some load via say a resistor you can. That is an everyday example of potential difference.

Now let's apply that to our spacetime field. If every coordinate on that field has precisely the same potential energy (potential energy is energy due to location under field treatment) then gravity effectively is zero.

However if there is potential energy differences between coordinate A and coordinate B such as due to a center of mass then you have a gravity term in Newton terms the gradient.

 Now under GR using the full equation \[E^2=(pc^2+m_o c^2)^2\] 

When applied to every coordinate of a field you immediately recognize that both massless particles as well as massive particles can affect the geometry. However the equation also includes their momentum terms so their vectors or spinors also are involved.

Under the stress energy momentum tensor the energy density is the \(T_(00)\) component. The diagonal components, (orthogonal components are the Maximally symmetric components) 

However there are off diagonal components stress, strain,and vorticity these components have symmetry to gas and fluid flow ie through a pipe for everyday examples.

So take a vector field of particles each particle follows its own geodesic those geodesics can converge or diverge from one another as they do so they generate non linearity as they induce curvature terms. When curvature occurs you are naturally inducing acceleration (direction change is also part of acceleration its not just the change in the velocity magnitude) 

Edit cross posted with Markus were both providing the same answer 

(Stress energy momentum tensor relations of a multi particle field).

To make things more complicated each particle of the above field example is interacting with other particles so now our field now has continous changes in velocity.

We can now only average all the numerous curved paths (linearization of a nonlinear system) which is never exact.

The above demonstrates why a tensor field is required. As mentioned a tensor field includes magnitude, vector and spinor relations.

The same applies to a curve you can only average the length of the curve. The extrenums (Maxima and minima) a function  is always a graph (but not all graphs are functions for a graph to have a function it must pass a horizontal and vertical test (off topic).

 

https://tutorial.math.lamar.edu/classes/calcI/minmaxvalues.aspx#:~:text=The function will have an,domain or at relative extrema.

Edited October 19 by Mordred

[MJ kihara]

🙏 Thanks for the answers and explanations above... nothing else I can say more than that.

[Mordred]

Your welcome its often tricky to see beyond the mathematics so we're glad to help.

Lol I lost count on how many times getting lost in the math and lose sight of what the math is representing so can readily understand the difficulty 

Edited October 19 by Mordred