EFE: Stress-Energy & Spacetime Curvature

[MattRob]

So, rather than causality and time travel paradoxes and the like that are usually discussed about relativity, I'm curious about something else.

On one side of the Einstein Field Equations is the Stress-Energy Tensor, along with some constant coefficients (G, c^-4, etc), which essentially describes the distribution of mass-energy.

On the other side, is the tensors and such that describe the curvature of spacetime.

Typically, people picture spacetime curvature with the rubber sheet analogy - a mass sitting on the rubber sheet causes the rubber sheet to curve.

But an equal sign doesn't have an arrow pointing in the direction of causality. So wouldn't it make just as much sense to say that spacetime curvature causes stress-energy, as it would be to say that stress-energy causes spacetime curvature?

 

[Dale]

But an equal sign doesn't have an arrow pointing in the direction of causality. So wouldn't it make just as much sense to say that spacetime curvature causes stress-energy, as it would be to say that stress-energy causes spacetime curvature?

I agree. We often say "cause" when we mean "implies"

 

[PeterDonis]

wouldn't it make just as much sense to say that spacetime curvature causes stress-energy, as it would be to say that stress-energy causes spacetime curvature?

I would say no, because the Einstein Field Equation is a differential equation--more precisely, it's a set of ten differential equations, since a symmetric 2nd rank tensor in 4 dimensions has ten independent components. (Actually, not all ten are completely independent, for reasons which are too complicated to fit in the margin of this post, but we can leave the exact number aside for this discussion.) The LHS, the Einstein tensor, contains all the derivatives--roughly, it's the first and second derivatives of the metric tensor. The RHS, the stress-energy tensor, contains no derivatives of the metric. (It might contain derivatives of non-gravitational fields.)

Physically, the usual interpretation of such differential equations, in terms of causality, is that the LHS, the differential part, is the "effect", and the RHS, the source, is the "cause". One way of viewing it is that we can't manipulate the geometry of spacetime, the LHS of the EFE, directly. We can only manipulate matter and energy, the RHS. In other words, we change the source, so we view the source as the cause and the field--the spacetime geometry--as the effect. (Consider the analogy with electromagnetism: Maxwell's Equations also just have an equals sign, with no explicit direction of causality; but we say charges and currents cause electromagnetic fields, not the other way around.)

(I should note that there is a possible retort to this. I won't give it here because I'm curious what others' reactions will be.)

 

[MattRob]

[snip]

Okay, just to be clear, I'll refer to it written like this;

So the LHS has the metric and derivatives of it (in the connections in the Ricci tensor), but the RHS has no metric.

Granted, objects follow geodesics which are given by the connections which are functions of the metric and derivatives of the metric, so geodesics arise in part from derivatives of the metric, but I could just as well put every term on one side of the equal sign and set it equal to zero, and it would still be true, so really it just describes a relationship in-between what we call stress-energy and what we call spacetime curvature. Still don't see the causality. Also, why would differentiation imply going from cause to effect when taking the derivative? Why not when taking an integral?*

Though it really is useful to think in terms of stress-energy causing curvature, it's kind of interesting to note that it need not necessarily be the case.

Though I guess you could argue that you can have curvature without stress-energy (strictly speaking, the Schwarzschild, or more intuitively gravity waves or other vacuum solutions), but you cannot have stress-energy without curvature. Isn't that a somewhat effective way to argue causality?

*Kind of a funny, perhaps unrelated thought, but what marks the passage of time is entropy, moving from a state of high order to one of low order. Aside from the coincidental nomenclature of "order" meaning the power of a function which is lowered when taking a derivative, this implies some kind of loss of information. A high-order (order as in the inverse of entropy) system has all the information to specify a lower-order system, but a lower-order system lacks the information to specify a higher-order system. In the same way, an equation has everything needed to specify its derivative, but its derivative does not have everything needed to specify its integral due to the integration constant. Hence, there is some interesting parallel in-between the passage of time and taking a derivative.

 

[Dale]

Consider the analogy with electromagnetism: Maxwell's Equations also just have an equals sign, with no explicit direction of causality; but we say charges and currents cause electromagnetic fields, not the other way around.

Jefimenko's equations show that Maxwell's equations can be written in a truly causal form, where a charge configuration at one time gives a field at a later time. To me, this is the justification for saying that charges cause fields, not the order of the derivative.

I am not sure if there is an equivalent formulation of the EFE.

 

[Mister T]

But an equal sign doesn't have an arrow pointing in the direction of causality.

So that tells you that there can be more to a relationship between two expressions than just equality. Those expressions represent something physical. In other words, the equation is just part of the relationship, it is not the entire relationship. An equation is a mathematical entity. There's more to physics than just the math, although it's an essential and very important part.
.

 

[Paul Colby]

In EM causality is imposed as a boundary condition on the solution. A radiating dipole radiates because the solution chosen has that property. One may equally well choose incoming waves from infinity. Much of the physics is in the boundary conditions.