# I How can the stress tensor be non-zero where there is no matter?

#### SamRoss

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Summary
Curvature comes from the stress tensor so how can there be curvature when there is no mass?
You're on Earth. You throw a ball and watch its trajectory. It's curved. That's because the Earth is curving space-time at every point along the trajectory. But the Earth itself is not present along the trajectory - there is no matter along the trajectory (let's ignore the air and any radiation that might be present) - so how is it curving the space there? There's not supposed to be action at a distance. Does it have something to do with gravitational waves? If so (and perhaps even if not because I'm still curious), what part of the field equations point to the existence of gravitational waves?

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#### Dale

Mentor
Summary: Curvature comes from the stress tensor so how can there be curvature when there is no mass?
The stress energy tensor is proportional to the Einstein tensor. So the Einstein tensor must be zero in vacuum, but the Riemann curvature tensor need not be zero in vacuum.

#### Ibix

The curvature of spacetime at any time and place is fully described by the Riemann tensor. The Einstein field equations are differential equations that restrict certain combinations of derivatives of the Riemann tensor to be equal to the stress-energy tensor. So in vacuum, they say that some combination of derivatives of curvature are zero, not that curvature is zero. In fact you can divide the Riemann into two simpler tensors, the Ricci and Weyl tensors. In vacuum the former is zero but the latter need not be. Gravitational waves do not come into this.

Gravitational waves emerge from the field equations by treating the metric as flat space time plus a small perturbation. Neglecting second order terms in derivatives of the perturbation, the Einstein field equations become the wave equation. You can get gravitational waves in strong fields as well, but there you have to solve numerically.

#### Orodruin

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Just to also point out that the equivalent question in electromagnetism would be: ”The EM field comes from charges and currents so how can there be an EM field where there are no charges or currents?”

#### SamRoss

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The stress energy tensor is proportional to the Einstein tensor. So the Einstein tensor must be zero in vacuum, but the Riemann curvature tensor need not be zero in vacuum.
So in vacuum, they say that some combination of derivatives of curvature are zero, not that curvature is zero.
This makes sense. I'm still confused about the apparent action at a distance, though.

Gravitational waves do not come into this.
Let's say someone was just floating in space. Nothing is causing him to move. All of a sudden, the Earth blinks into existence nearby. According to Newton, the force from the Earth would traverse the distance between it and the man instantaneously, causing the man to move toward the Earth. According to Einstein, why is there a time delay between the moment the Earth blinked into existence and the moment the curvature at the man's location became non-zero? And where is that found in the field equations?

#### SamRoss

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Also, I can understand from the explanations how the curvature could be non-zero even when the stress tensor is zero. However, that curvature would still presumably be due to the Earth. So if the mass of the Earth is not entering into the field equations through the stress tensor, then where does it come in?

#### Orodruin

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All of a sudden, the Earth blinks into existence nearby.
This violates local energy-momentum conservation. You cannot do this in GR.

I'm still confused about the apparent action at a distance, though.
Did you understand the parallel to electromagnetism? It is exactly the same. The metric is governed by the Einstein field equations just as the electromagnetic field is governed by Maxwell's equation. The source term in Maxwell's equations are charges and currents, the source term in Einstein's equations is the stress-energy tensor.

So if the mass of the Earth is not entering into the field equations through the stress tensor, then where does it come in?
Why do you think the mass of the Earth would not enter into the stress-energy tensor?

#### Michael Price

The Einstein field equations only require that the non-trivial trace of the Riemann tensor (the Ricci tensor) vanish in the absence of matter. There is no requirement for the Riemann tensor itself to vanish.

#### SamRoss

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I think I have two sources of confusion. ..

1. In Einstein's theory, there is supposed to be a speed of gravity. Where does that come from?

This violates local energy-momentum conservation. You cannot do this in GR.
I know that the example I provided was absurd. I was trying to get at the mechanism for the speed of propogation of gravity.

2.
The source term in Maxwell's equations are charges and currents, the source term in Einstein's equations is the stress-energy tensor.
Why do you think the mass of the Earth would not enter into the stress-energy tensor?
I think my other source of confusion is that I don't see an analogous inverse square law in the field equations. I see a stress tensor which is supposed to be defined at every point. If there's no matter at a point, then that point is not creating any curvature. From what has been said, there may still be curvature at that point, but presumably the curvature would be coming from matter at a different point. So what is the analogous inverse square law? How can we see from the field equations that matter from one point will affect curvature at a different point?

#### Michael Price

I think my other source of confusion is that I don't see an analogous inverse square law in the field equations. I see a stress tensor which is supposed to be defined at every point. If there's no matter at a point, then that point is not creating any curvature. From what has been said, there may still be curvature at that point, but presumably the curvature would be coming from matter at a different point. So what is the analogous inverse square law? How can we see from the field equations that matter from one point will affect curvature at a different point?
To see how Newtonian gravity emerges from the Einstein field equations is not simple. You need to study GR at a much more technical level - there is no way to get around that.

#### Michael Price

I think I have two sources of confusion. ..

1. In Einstein's theory, there is supposed to be a speed of gravity. Where does that come from?
The speed of gravity comes from the Einstein field equations, in the same way as the speed of light comes from Maxwell's equations.

#### Ibix

All of a sudden, the Earth blinks into existence nearby.
The Einstein field equations enforce the local conservation of stress energy, and this scenario violates that. Thus there is no solution of the field equations for this situation - it's impossible in GR.
So if the mass of the Earth is not entering into the field equations through the stress tensor, then where does it come in?
The mass of the Earth enters the same way it does in Newtonian gravity if you express it using Poisson's equation - it's a boundary condition. In fact, simplifying the Einstein field equations gives you Poisson's equation in the low speed weak field limit.
1. In Einstein's theory, there is supposed to be a speed of gravity. Where does that come from?
It doesn't. There's a speed of gravitational waves, which falls out of the Einstein field equations as I described earlier. Chapter 6 of Sean Carroll's lecture notes covers it.

How can we see from the field equations that matter from one point will aff
Because the curvature isn't zero where the stress-energy is non-zero and the field equations are differential equations - their solutions must be smooth functions so cannot reach zero (except in limits or instantaneously).

#### Orodruin

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I think my other source of confusion is that I don't see an analogous inverse square law in the field equations.
Do you "see" an inverse square law in Maxwell's equations? You need to actually study the field equations to derive it.

I see a stress tensor which is supposed to be defined at every point. If there's no matter at a point, then that point is not creating any curvature.
Again, the electromagnetic equivalent of this statement is: "I see a charge density which is defined at every point. If there is no charge at a point, then that point is not creating any electromagnetic field."
This does not stop that point from having a non-zero electromagnetic field as the field can be influenced by charges located elsewhere.

How can we see from the field equations that matter from one point will affect curvature at a different point?
The field equations are a set of coupled partial differential equations, just like Maxwell's equations. You need to solve those equations to find the solution, but generally curvature will not be zero just because the stress-energy tensor is. The most stunning example would be the Schwarzschild solution, which is a vacuum solution (i.e., it has $T_{\mu\nu} = 0$ everywhere).

#### Dale

Mentor
I'm still confused about the apparent action at a distance, though.
There is no action at a distance involved. I suspect that you may be using this term but meaning something different than normal. Can you describe what you mean by action at a distance and what makes you think there is action at a distance here.

I know that the example I provided was absurd.
The problem isn’t that it is absurd. We can deal with absurdity just fine. The problem is that it is logically inconsistent with GR. It is like asking which real number is the square root of -1. There is nothing wrong with the square root of -1, but it just doesn’t fit anywhere with the real numbers. Similarly, if you specify GR then mass popping into existence doesn’t fit.

#### pervect

Staff Emeritus
This makes sense. I'm still confused about the apparent action at a distance, though.

Let's say someone was just floating in space. Nothing is causing him to move. All of a sudden, the Earth blinks into existence nearby. According to Newton, the force from the Earth would traverse the distance between it and the man instantaneously, causing the man to move toward the Earth. According to Einstein, why is there a time delay between the moment the Earth blinked into existence and the moment the curvature at the man's location became non-zero? And where is that found in the field equations?
You might want to study the electromagnetic equivalent problem first.

A charge is sitting out in empty space. All of a sudden, another charge pops into existence. According to Maxwell's equations, what happens?

Using arguments similar to yours, we could argue for the existence of a "paradox", and "action at a distance", but it'd be a distraction from the real issue. The main point is that Maxwell's equations do not allow charges to just appear or disappear like that.

Gravity is a bit more complicated, but the answer is basically the same.

Going back to the electromagnetic equations, the way we measure the speed of light is not to make charges appear and disappear out of nothingness, because that's impossible. Instead, we study the speed of electromagnetic radiation.

On a more technical note, you can find a proof that the continuity equations can be derived from Maxwell's equations <<here>>.

The continuity equations are the equations that say that charge can't just disappear.

$$\nabla \cdot \vec{j} + \frac{\partial \rho}{\partial t} = 0$$

Here j is the current density, and $\rho$ is the charge density.

Basically means that if the rate of change of charge density $\rho$ is nonzero, there must be a current flowing away in such a manner as to carry the charge away away, expressed mathematically by $\nabla \cdot \vec{J}$. That's not compatible with a charge suddenly disappearing for no reason. Intuitively, we can say that charge is conserved, so it can't just vanish, or appear from nowhere. The math just illustrates that the conservation of charge is already a logical consequence of Maxwell's equations, it is not a seprarate assumption that needs to be added to them.

Thus it is mathematically inconsistent to ask what happens when a charge dissappears according to Maxwell's equations. It's the same as asking "what happens if 0=1".

http://maxwells-equations.com/equations/continuity.php

#### SamRoss

Gold Member
To see how Newtonian gravity emerges from the Einstein field equations is not simple. You need to study GR at a much more technical level - there is no way to get around that.
I have attempted a rigorous understanding of GR. I admit that tensor calculus is not easy for me, but there are some intuitive misconceptions that I have as well. Those are what I am trying to work through with this discussion (before I try another go at the hard math).

Because the curvature isn't zero where the stress-energy is non-zero and the field equations are differential equations - their solutions must be smooth functions so cannot reach zero (except in limits or instantaneously).
The field equations are a set of coupled partial differential equations, just like Maxwell's equations.
Yes, I think I have been ignoring the differential aspect. So if the metric is curved at the Earth, it can’t possibly turn into the Minkowski metric at a neighboring point, correct?

There is no action at a distance involved. I suspect that you may be using this term but meaning something different than normal. Can you describe what you mean by action at a distance and what makes you think there is action at a distance here.
My understanding of action at a distance is that what happens in once place instantaneously affects what happens somewhere else. Here’s why it appears to me (erroneously, I’m sure) that this is happening here. I imagine being in a reference frame in which the Earth appears to be moving. With my superpowers, I can see the curvature caused by the Earth’s mass. At one instant in time, I focus on the Earth and the curvature of a point ten light years away. A second later, the Earth is somewhere else. Has the curvature at the point ten light years away changed? I’ve never heard anything leading me to believe it has not changed which makes it seem like the Earth sent a faster-than-light message to the point to tell it how to change.

I have an idea out of this mess but I’m not sure if it makes sense. Maybe I’m taking too literally the idea that the Earth is “causing” the curvature. Maybe I shouldn’t be viewing the field equations as a prescription for mass to send signals to different points in space and telling them how to bend. Rather, the two sides of the equations can be viewed separately. The stress tensor contains information on where the mass will be in the next instant of time (since it contains all the momenta) while the Einstein tensor contains information on how the curvature at each instant will be changing. The equal sign between the two should be taken more as a correlation than a causation. That’s my shot at it anyway. Does it make any sense at all? (Why do I get the feeling that someone’s going to tell me it doesn’t make sense or that it makes sense but it’s still not right or that "light year" should be one word?)

#### Dale

Mentor
My understanding of action at a distance is that what happens in once place instantaneously affects what happens somewhere else.
That is a correct understanding of action at a distance, but nothing like that happens in GR. The direct observation of gravitational waves essentially directly disproves the action at a distance.

I imagine being in a reference frame in which the Earth appears to be moving. With my superpowers, I can see the curvature caused by the Earth’s mass. At one instant in time, I focus on the Earth and the curvature of a point ten light years away. A second later, the Earth is somewhere else. Has the curvature at the point ten light years away changed?
That depends on the motion of the earth with respect to the point 10 years ago. If it was moving relative to the point 10 years ago then the curvature does change. If it was not moving 10 years ago then the curvature does not change. The current motion is not relevant.

#### SamRoss

Gold Member
There's not supposed to be action at a distance. Does it have something to do with gravitational waves?
The direct observation of gravitational waves essentially directly disproves the action at a distance.
So the answer to one of my original questions is "yes"? Gravitational waves are the mechanism whereby a mass would tell a point somewhere how to alter its curvature?

#### pervect

Staff Emeritus
So the answer to one of my original questions is "yes"? Gravitational waves are the mechanism whereby a mass would tell a point somewhere how to alter its curvature?
I would say no.

Let's consider a quote from Wheeler.

Wheeler said:
Space[-time] tells matter how to move
Matter tells space[-time] how to curve
I believe Wheelers original quote did use "space" , rather than space-time, though space-time is more accurate (but less accessible to the general reader), IMO.

I would say that the "mechanism" where matter tells space-time how to curve is not gravitational waves, but Einstein's field equations.

As an aside, note that Wheeler also said "matter" rather than "mass". Pressure, for instance, isn't mass, but it's one of the terms of the stress-energy tensor which tells space-time how to curve.

With the proper choice of coordinates, the so-called "harmonic gauge condition", Einstein's Field equations are a wave equation. But it takes some effort to force Einstein's field equations into the form of a wave equation. This is very similar to how in electromagnetism, the Lorenz gauge reduces Maxwell's equations to a wave equation. However, in the Coulomb gauge, this is not the case. Furthermore, when discussing the coulomb force, the 1/r^2 force, in electromagnetism, one typically uses the Coulomb gauge rather than the Lorentz gauge.

At a deep level, the observation that gravity and electromagnetism can be reduced to the wave equation is a very powerful observation. Assuming that it happens all the time without effort can lead to confusion.

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#### SamRoss

Gold Member
At a deep level, the observation that gravity and electromagnetism can be reduced to the wave equation is a very powerful observation. Assuming that it happens all the time without effort can lead to confusion.
It seems I need to study the derivation of the wave equation from the field equations before I can gain any more insight. Thanks for your help.

#### PeterDonis

Mentor
Gravitational waves are the mechanism whereby a mass would tell a point somewhere how to alter its curvature?
Gravitational waves can be though of as a mechanism whereby information about changes in curvature driven by certain motions of matter propagates through spacetime. Similarly, electromagnetic waves can be thought of as a mechanism whereby information about changes in the electromagnetic field driven by certain motions of charges propagates through spacetime.

The "can be" above is because of various caveats in using such an interpretation, some of which @pervect described in his post.

#### vanhees71

Gold Member
I have attempted a rigorous understanding of GR. I admit that tensor calculus is not easy for me, but there are some intuitive misconceptions that I have as well. Those are what I am trying to work through with this discussion (before I try another go at the hard math).
Your most serious misconception is that you might be able to understand physics without learning the only adequate language to describe it we know. It's math! Admittedly it's difficult, but on the other hand it's great fun to learn it, "not because it's easy but because it's hard" (I couldn't resist to quote this on a 20th July ;-)).

#### rrogers

This makes sense. I'm still confused about the apparent action at a distance, though.

Let's say someone was just floating in space. Nothing is causing him to move. All of a sudden, the Earth blinks into existence nearby. According to Newton, the force from the Earth would traverse the distance between it and the man instantaneously, causing the man to move toward the Earth. According to Einstein, why is there a time delay between the moment the Earth blinked into existence and the moment the curvature at the man's location became non-zero? And where is that found in the field equations?
Take a look at how Finite element analysis software works; the differential equations govern how adjacent cells interact. Stitching the cells together, and obeying the differential relations, allows you to see how a source can propagate out into the surroundings; and go on and on. All without "action at a distance". The GR equations perform the same functions; local (infinitesimal if you will) conditions propagate along with the rules. Now looking how an FEA develops you can see no "action at a distance" is necessary if you have adjacent cell/points/etc... and the accompanying rules.

#### alantheastronomer

The speed of gravity comes from the Einstein field equations, in the same way as the speed of light comes from Maxwell's equations.
…and amazingly enough the two are one and the same!
SamRoss said:
I'm still confused about the apparent action at a distance, though.
SamRoss said:
If there's no matter at a point, then then that point is not creating any curvature. From what's been said, there may still be curvature at that point, but presumably the curvature would be coming from matter at a different point.
Yes you're right, the curvature is due to the mass that is at some distance away - that's why there doesn't need to be any source of stress-energy at that point; but that's not action at a distance - you're confusing action-at-a-distance with simply the influence of the gravitational potential as it extends throughout spacetime. You don't need to examine the full blown Einstein equations to understand - they were meant to cover all possible types of complex scenarios of masses and motions. All you need to look at is the simple case of a point mass at rest; The stress-energy tensor then simply reduces to the mass m; it's gravitational potential is given by P=GM/r which you can see extends out a distance r from the point mass yielding a curvature at a point where there are no other sources. Take the derivative with respect to r and you get your force equation F=GMm/r^2 .

#### PeterDonis

Mentor
t's gravitational potential is given by P=GM/r which you can see extends out a distance r from the point mass yielding a curvature at a point where there are no other sources. Take the derivative with respect to r and you get your force equation F=GMm/r^2
This is a Newtonian analysis, not a GR analysis. In GR gravity is not a force. Also, there is no "point mass" in GR; a Schwarzschild black hole is vacuum everywhere, and a GR model of an ordinary gravitating body like a planet is not a point mass, it's a finite region of nonzero stress-energy surrounded by vacuum.

Also, none of the expressions you wrote down are expressions for the spacetime curvature. Spacetime curvature is tidal gravity, which goes like $M / r^3$.

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