# What is intuitively the source of the gravitational field?

## Main Question or Discussion Point

In general relativity, the source of the gravitational field is the "stress-energy tensor"
I know that thing is not just energy.

Can some one explain what this quantity is?
It's just a 'thing' that just happens to work mathematically and means nothing else?

## Answers and Replies

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pervect
Staff Emeritus
It's a matrix of numbers. The various components are the energy density, momentum density, and pressure.

See http://en.wikipedia.org/wiki/Stress–energy_tensor

or

http://math.ucr.edu/home/baez/einstein/ specifically http://math.ucr.edu/home/baez/einstein/node3.html

The later only discusses the diagonal components by considering a special case where the momentum is zero and the 3x3 pressure tensor (also known as the stress tensor) is made diagonal by a "principal axis transformation".

http://en.wikipedia.org/wiki/Mohr's_circle

may be of some help in visualizing the "stress" part of the stress-energy tensor as well. The principle axis transformation just rotates the stress ellipsoid so that it's axis are aligned with the coordinate axes.

That part of the stress-energy tensor that isn't stress is the "energy momentum 4-vector"
http://en.wikipedia.org/wiki/Four-momentum - expressed, however, as a density.

phinds
Gold Member
2019 Award
What is intuitively the source of the gravitational field?
I don't think our intuition has any concept of the stress energy tensor. I think human intuition says the source of gravity is the ground. Intuition is often not very helpful in dealing with science.

I knew some of these links already, the Tensor just looks a collection of random stuff.

The stress-energy tensor represent the flow of momentum?
Not sure what that means.
the tensor represent the components in the various space time directions?
right?

tt is momentum that 'moves' in time only = rest mass?
here cinetic energy is excluded right?
tx = momentum
normal momentum of particles?
xx = Pressure
the same pressure we are familiar with? ....blood pressure has a gravitational field??? :P
It means some other abstract pressure related to energy?
xy = shear stress
like in mechanics???

basically shear stress, pressure and rest mass represent internal energy of the system?
The source of the gravitational field is the 'invariant energy' of a system, the amount of energy that can not be removed by changing reference frame of the observer?
How momentum fits in there? momentum alone doesn't have a gravitational field right?
a single photon has no gravitational field right? How it fits in the Tensor?
Two antiparallel photons considered separately have no gravitational field, but if considered together they do?

The Tensor is a conserved quantity right?

hmm so many questions .....

I don't think our intuition has any concept of the stress energy tensor. I think human intuition says the source of gravity is the ground. Intuition is often not very helpful in dealing with science.
its a matter of philosophical belief. I don't like the never ending mathematical abstraction of modern physics. I think its a social phenomenon caused by social problems in modern academia, nothing to do with reality it self.

Besides, the tensor came from somewhere right?

I don't think our intuition has any concept of the stress energy tensor. I think human intuition says the source of gravity is the ground. Intuition is often not very helpful in dealing with science.
Does this mean it is just better to "shut the heck up and calculate"? That is, to forget about trying to "intuit" these things and just learn how to apply them mechanically to solve problems and generate predictions for experiments?

WannabeNewton
It's very easy to get an intuition for the stress-energy tensor so that's not an issue; it's just an extension of basic concepts from fluid mechanics to fluids in space-time and more general matter fields like the electromagnetic field. The issue is that it's too big an order for a forum post. The OP needs to get a proper textbook on GR; I would recommend Schutz "A First Course in General Relativity"-see chapter 4.

It's just a 'thing' that just happens to work mathematically and means nothing else?
'sort of'.... that's how it may have started, to the first part; 'no' to the second part. Einstein tried a number of different formulations before arriving at the stress energy tensor in use today. He picked the right one because he was able, somehow, to foretell it would provide accurate predictions and conform to experimental results. Keep in mind it took him about a decade to figure this out, it was not an approach that started with first principles and led inexorably to the SET as the 'obvious' solution.

It's very easy to get an intuition for the stress-energy tensor so that's not an issue
I don't think our intuition has any concept of the stress energy tensor.
Once an 'Einstein' put the formulation together, and his 'intuition' about physical phenomena was remarkable, physical interpretations of the components can be made by we mere mortals. But scientists of Einstein's era were not so confident; it took some years to make the interpretations we read about in a few minutes or hours today and for science to accept the formulation as accurate.

For example, I recall the SET is a 'torsion free formulation'....If there is a reason Einstein did it that way.... I've forgotten. Or maybe he got lucky??

It's a matrix of numbers. The various components are the energy density, momentum density, and pressure.
Simply put what is "pressure"? Seems like saying a ball squeezed in a vice has greater gravity then one that is not.

Maybe it's spacetime "pressure", in that an object in a gravitational potential, itself "exerts" a greater gravitational potential than if it was in flat spacetime.

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WannabeNewton
Once an 'Einstein' put the formulation together, and his 'intuition' about physical phenomena was remarkable, physical interpretations of the components can be made by we mere mortals...
You seem to be making $T^{\mu\nu}$ look like some chimeric conception. Take $T^{\mu\nu} = \rho u^{\mu}u^{\nu} + p(\eta^{\mu\nu} + u^{\mu}u^{\nu})$, the stress-energy tensor for a perfect fluid in Minkowski space-time with pressure $p$ and mass density $\rho$ in a comoving local Lorentz frame and 4-velocity $u^{\mu}$, and analyze the physical interpretation of the components from first principles using momentarily comoving inertial frames of fluid elements in the perfect fluid. If you know vector calculus and basic fluid mechanics then this is a straightforward yet highly instructive task from the standpoint of physics.

For example, I recall the SET is a 'torsion free formulation'....If there is a reason Einstein did it that way.... I've forgotten. Or maybe he got lucky??
I don't know of the historical details regarding why Einstein chose a torsion-free connection but in GR one works with a torsion-free connection by basic assumption (one that is subsequently backed up by experiment) and not as a consequence of more fundamental physical principles.

EDIT: see here for more: http://math.ucr.edu/home/baez/gr/torsion.html

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WannabeNewton
Simply put what is "pressure"?
Pressure is just force per unit area; it is a fundamental thermodynamic variable. There are many different relations between pressure and other fundamental extensive and intensive thermodynamic variables. See section 3.4 of Schroeder "An Introduction to Thermal Physics".

What you stated later in your post is not what pressure is. However keep in mind that pressure need not only be exerted by fluid elements of what we "normally" think of as fluids (e.g. liquids and gases). For example an electromagnetic wave incident on a surface can exert pressure on that surface.

Seems like saying a ball squeezed in a vice has greater gravity then one that is not.
correct!!

And a compressed spring has 'greater gravity' than when uncompressed...
due to additional potential energy....

edit:
Maybe it's spacetime "pressure", in that an object in a gravitational potential, itself "exerts" a greater gravitational potential than if it was in flat spacetime.
no.
Flat spacetime is zero gravity [no curvature]; curved spacetime results from gravitational sources already described in the SET.

1 person
in GR one works with a torsion-free connection by basic assumption (one that is subsequently backed up by experiment).....
Some also work with one:
I posted a query almost a year ago which did not draw all that much discussion:

Can torsion avoid the big bang singularity

.....The torsion of spacetime generates gravitational repulsion in the early Universe filled with quarks and leptons, preventing the cosmological singularity: the Universe expands from a state of minimum but finite radius........

We also propose that the torsion of spacetime, which is produced by the spin of quarks and leptons ﬁlling the Universe and prevents the formation of singularities (points of spacetime with inﬁnite curvature and matter density), provides a physical mechanism for a scenario in which each collapsing black hole gives birth to a new universe inside it. Gravitational repulsion induced by torsion, which becomes signiﬁcant at extremely high densities, prevents the cosmological singularity.....

'gravitational repulsion' can be interpreted as 'negative pressure'....a cosmological constant in the FLRW/Einstein cosmological model....which offers an additonal prspective on 'what is pressure'.... but I've not seen it described as originating from torsion in most descriptions.

For those who may be interested, in the above linked discussion there are some other related discussions including this post:

Stevendaryl:
In General Relativity, the assumption is made that the stress-energy tensor Tαβis symmetric. However, if there are particles with intrinsic spin, then this assumption is false, as described
here: http://en.wikipedia.org/wiki/Spin_tensor

It is interesting, for sure, how much we can infer from mathematics.

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pervect
Staff Emeritus
Simply put what is "pressure"? Seems like saying a ball squeezed in a vice has greater gravity then one that is not.
Yes - this is true. Though it turns out that part of the jaw of the vice will be under tension, so that part of the vice will have "less gravity".

Maybe it's spacetime "pressure"...
No - it's perfectly ordinary "pressure". I think Baez talks about pressure casusing gravity in "The Meaning of Einstein's Equation". http://math.ucr.edu/home/baez/einstein/

pervect
Staff Emeritus
correct!!

And a compressed spring has 'greater gravity' than when uncompressed...
due to additional potential energy....
A subtle but important point here - in the case of the compressed spring, you have additional energy stored due to the work that the pressure did on the spring. This stored energy causes more gravity. You have in addition the effects of the pressure in the spring, which also causes more gravity, along with the extra stored energy.

So in GR pressure does not have to do work to cause gravity.

pervect
Staff Emeritus
I knew some of these links already, the Tensor just looks a collection of random stuff.
A tensor is far from a "random collection of stuff". I am not aware of any simple non-mathematical way of describing why a tensor is so special, without appealing to the details of its transformation properties, however.

In broad strokes, I can say that you can express any physical law in the mathematics of tensors. I can say that tensors transform, mathematically, in the simplest possible manner when you change coordinates. I can say that when you express a physical law in terms of tensors (which is always possible), you gain the freedom to choose arbitrary coordinates.

Example: if you describe a position in space with generalized coordinates a,b, and c, the distance between the points (a,b,c) = (1,0,0), and (a,b,c) = (0,0,0) doesn't have to be one. Due to the non-trivial and non-obvious effects of space-time curvature, this freedom to use arbitrary coordinates becomes vital to the development of GR.

I can also add that tensors have a long history of use before GR, particularly in fluid mechanics. Of course, the classical tensors transform via the Gallilean transformation - the relativistic tensors transform via the Lorentz transform.

I suppose I can add a bit more. In special relativity, space and time "mix together". One person's space becomes another person's time. That's why space and time become a unified concept, space-time. It means also that if you have only the space part of an observation, you can't transform between observers. You need all the information, both the space and time parts of a tensor, to be able to the transform between observers.

Energy and momentum mix together in the same way - what manifests as momentum to one person is energy to another. The mathematical construct which describes this is the energy-momentum 4-vector (which is also a tensor, but a rank 1 tensor).

In a similar manner, all of the components of the stress-energy tensor "mix together". Energy mixing with momentum is already familiar to the student of SR - but it turns out that pressure mixes with energy density in a similar manner, so one needs to add energy density, momentum density, and pressure to get a consistent and complete view of an object that can be transformed from one observer to another.

Is it correct to say, that this thing represent the energy of a system that never go away when you change reference frame? The tensor represents, 'invariant energy'? I really want an answer to this one.

For a single photon, when you change coordinates, its energy vary, in general kinetic energy vary etc..... So pure kinetic energy is not valid here. If you have a gas of photons, no matter what is your reference frame, you can't have a reference frame where the gas has arbitrarily low energy....

Its not apparent why pure kinetic energy is invalid here. An explanation? A single photon has only zeros in the tensor?

@ pervect
i didn't meant what are tensors, i meant this particular tensor

pervect
Staff Emeritus
Is it correct to say, that this thing represent the energy of a system that never go away when you change reference frame? The tensor represents, 'invariant energy'? I really want an answer to this one.
The tensor as a whole is regarded as representing the momentum and energy of a system (they can't be separated - what is perceived as energy by one observer is perceived as momentum by another).

The components of the energy do change when you change coordinates, though. It is only in an abstract sense that we can regard either an energy-momentum four vector or a stress energy tensor as having an "existence" independent of any coordinates.

For a single photon, when you change coordinates, its energy vary, in general kinetic energy vary etc.....
The numerical values of the various components of the stress energy tensor also change when you change coordinates. But they change in a highly standardized way. When you regard the choice of coordinates as a convention, then the physical representation of a particular physical system is represented by a tensor, the components of which transform in a standardized way when you choose (at your whim) a specific coordinate system. We "abstract away" the coordiate system, and regard the tensor as representing the underlying physical reality in some sense that is independent of a specific choice of coordinates.

@ pervect
i didn't meant what are tensors, i meant this particular tensor
None of what's below is going to make sense unless you are familiar with 4-vectors. Since I already wrote it, I'm going to leave it and hope for the best. Unfortunately I don't have any idea of your background.

There are several ways of talking about the stress energy tensor. One way of looking at it is this - the stress energy tensor , when multiplied by the 4-velocity of an observer, gives the purely spatial energy momentum density (which is a 4-vector) measured by an observer moving at that four velocity. This is the approach used by MTW in "Gravitation", basically

What do I mean by purely spatial? Intuitively, an moving observer has some natural notion of "space". Formally, this "space" is just a submanifold of the 4-manifold of space-time whose basis vectors are all perpendicular to the 4-velocity. So the 4-vector defines a way of locally slicing space-time into time and space.. You then take the "natural" 3-volume element on the "spatial" submanifold (we'll skip over some fine-print items here, such as assuming we are on an orientable manifold, not a non-orientable manifold like a mobius strip).

This gives you the volume element I'm talking about, which you use in the way you'd normally use a spatial volume element - note that a space-time volume element is something different, this volume element is purely spatial.

The difference in relativity is that the volume element isn't universal as it is in Newtonian mechanics, but it depends on the observer.

One simple way of realizing why this observer dependence exists is to note how boxes (which form volume elements) get "squashed" by Lorentz contraction.

If you want a more formal definition of the stress-energy tensor, along with a proof that it is a tensor, I'd suggest http://web.mit.edu/edbert/GR/gr2b.pdf, which talks about how you can get the stress-energy tensor of a swarm of particles as the tensor product of the number-flux vector and the energy-momentum 4-vector. Because both the number-flux vector and the energy-momentum 4-vector are tensors, the stress-energy tensor is also a tensor.

pervect
Staff Emeritus
I might have said this before, but one of the simplest semi-rigorous explanations of GR and the "source" of gravity I'm aware of is Baez's http://math.ucr.edu/home/baez/einstein/ "The Meaning of Einstein's Equation".

The tensor as a whole is regarded as representing the momentum and energy of a system (they can't be separated - what is perceived as energy by one observer is perceived as momentum by another).

The components of the energy do change when you change coordinates, though. It is only in an abstract sense that we can regard either an energy-momentum four vector or a stress energy tensor as having an "existence" independent of any coordinates.
The stress-energy tensor doesn't represent just energy. I got that. In some reference frame you can have momentum, in an other you are at rest.

I'll explain it differently.
The tensor represent momentum, that doesn't go away by changing reference frame??? Here, i see energy/mass as a form of "internal momentum".

It doesn't mater what is your reference frame, all observers must agree on the curvature of space time. Right???
Do they also all agree about the source of the gravitational field?

The numerical values of the various components of the stress energy tensor also change when you change coordinates. But they change in a highly standardized way. When you regard the choice of coordinates as a convention, then the physical representation of a particular physical system is represented by a tensor, the components of which transform in a standardized way when you choose (at your whim) a specific coordinate system. We "abstract away" the coordiate system, and regard the tensor as representing the underlying physical reality in some sense that is independent of a specific choice of coordinates.
I meant reference frame. I think you mean the same with coordinate system....

You have a point mass. In one reference frame it is at rest, in an other it has momentum. In the first case, the tensor just has the rest mass, in the other it also has the momentum?
Both have the same space time curvature around them, but the tensor doesn't look the same.
????

Should they be the same?
It follows some invariance?
If not, how can you tell its the same reality here?

None of what's below is going to make sense unless you are familiar with 4-vectors. Since I already wrote it, I'm going to leave it and hope for the best. Unfortunately I don't have any idea of your background.
you just discovered quantum death, how to truly kill a quantum immortal :P

hmmm

The stress energy tensor, represent all the non gravitational energy momentum.
To have the whole of energy momentum, you need to add the energy momentum stored in the curvature of space time. The whole energy is ill defined in GR, not the non gravitational part of it when considered alone.
The left side of the equation does the weird stuff.

Is this right or i'm still confused?

WannabeNewton
Reference frames and coordinate systems are two different things. But yes your statements in post #20 are more or less correct. Local gravitational energy density is not well-defined in arbitrary space-times because it is not gauge invariant. There are ways to get around it by defining pseudo-tensors but generally one works with Komar integrals to deal with dynamical properties of space-time like energy, momentum, and angular momentum of space-times.

Anyways I really suggest getting a textbook on GR because that's the best way to learn these things.

K^2
The best way to understand Stress Energy Tensor is to first understand current density. Say you have a bunch of particles with some number density $\small \rho$. Now, lets say you go to a different coordinate system, one moving at some speed $\small \vec{u}$ with respect to the original. What's the new density? Well, that all depends on how fast the particles were moving in the original frame of reference. More specifically, on the current. So lets say that the number of particles passing through YZ plane was $\small j_x$. Similarly, we define $\small j_y$ and $\small j_z$. This gives us the current density vector $\small \vec{j}$. From this, we can construct the four-current $\small j^{\mu} = (\rho, \vec{j})$. Finally, number density in a different coordinate system is simply given by taking a dot product with relative four-velocity. $\small \rho' = u_{\mu}j^{\mu}$.

Hopefully, all of this looks familiar. What's important is that number density can be thought of as number of particles flowing through a certain area of a surface with constant-time. And in order to understand how that transforms with change of coordinate system, you have to also know how many particles are passing through surfaces of constant x, y, and z. Then you can figure out the number density in any coordinate system.

In order to describe how a scalar density transforms, you need a four-current vector. But what if the quantity whose density you are interested in is not a scalar, but a vector itself. Specifically, we are interested in a four-momentum density vector and how that transforms. And it turns out that we have exactly the same situation. Momentum and energy can be carried by currents, and so in order to describe a transformation, we need a sort of vector of vectors. If we put that notion into more precise mathematical language, we end up with the Stress Energy Tensor.

So if you want an intuitive picture of what Stress Energy Tensor is, think of it as a four-vector current analogy for the four-momentum. It's encodes the density and flow of energy and momentum.

pervect
Staff Emeritus
hmmm

The stress energy tensor, represent all the non gravitational energy momentum.
To have the whole of energy momentum, you need to add the energy momentum stored in the curvature of space time. The whole energy is ill defined in GR, not the non gravitational part of it when considered alone.
The left side of the equation does the weird stuff.

Is this right or i'm still confused?
That's basically right!

Ok, i'm not confused anymore :D

Is this correct now???
The stress energy tensor is just all the conservations laws of mechanics.
its the energy and momentum in the time components.
Shear stress is just tork
Pressure is just the forces
The divergence of this, its simply all of mechanics....

further more. I'm asking about the source, not just the stress energy tensor.
Is the "strees energy of the gravitational field" included automatically in the field part of the equation?
Or GR is just violating the equivalence of inertial mass and gravitational mass???
Gravity attracts gravity, just because its spacetime curvature??? Or also because of its energy content???

Pervect, i would like to see your diagrams too :*