The Causal Bootstrap Paradox in General Relativity

[Alien101]

TL;DR

Einstein's field equation G_μν = 8πG T_μν appears to establish a clear causal relationship: matter-energy (T_μν) determines spacetime curvature (G_μν).

The Fundamental Causality Problem:

In General Relativity, matter follows geodesics determined by the metric, which means spacetime geometry determines matter distribution. This creates a circular causality: geometry determines matter motion → matter motion determines stress-energy → stress-energy determines geometry.

How can Einstein's equation be truly "causal" when both sides are mutually determining each other? Is this circular dependency a fundamental flaw, or does it reveal something deeper about the nature of spacetime?

 

[PeroK]

This "paradox" applies to many equations: Newton's second law, Maxwell's equations etc. It describes a mutual interdependence at the core of many physical theories.

 

[Alien101]

This "paradox" applies to many equations: Newton's second law, Maxwell's equations etc. It describes a mutual interdependence at the core of many physical theories.

That's a fair point about mutual interdependence in physics, but I think the GR case has a unique twist that makes it more fundamentally problematic. The key difference in Newton's F = ma or Maxwell's equations, we have clear temporal causality:

Forces at time t determine acceleration at time t
Charges and currents at time t determine fields at time t+dt
The "mutual dependence" operates through time evolution

But Einstein's equation is different:
G_μν = 8πG T_μν is a constraint equation that must hold simultaneously at every spacetime point. The circularity isn't temporal—it's instantaneous logical circularity.

 

[PeroK]

Re Newton's second law, the mutual interdependence is between force and mass.

Re Maxwell, the field tells charged particles how to move and the charged particles determine the field.

 

[Alien101]

Re Newton's second law, the mutual interdependence is between force and mass.

Re Maxwell, the field tells charged particles how to move and the charged particles determine the field.

Consider this specific example:

I specify the metric g_μν on some initial surface
This determines geodesics, so matter motion is fixed
Matter motion determines T_μν everywhere
But Einstein's equation says T_μν determines G_μν (hence g_μν)
So the metric I "freely specified" in step 1 is actually constrained by the consequences of my own specification

The question is, does this mean spacetime geometry and matter content are mutually self-consistent in some profound way, or does it reveal that we're double-counting the same physical information in both the metric and stress-energy tensor?

This seems fundamentally different from F = ma, where force and acceleration are genuinely independent concepts that happen to be related. In GR, are geometry and matter-energy truly independent, or are we describing the same physical reality in two different mathematical languages?

What's your take on whether this represents genuine mutual causation versus redundant description of a single underlying reality?

 

[PeroK]

My take is that the argument is at best philosophical and in any case irrelevant to the physics. The search is for a quantum theory of gravity. Philosophical dissection of an emergent theory, such as GR, has little or no purpose as far as physics is concerned.

 

[renormalize]

But Einstein's equation is different:
G_μν = 8πG T_μν is a constraint equation that must hold simultaneously at every spacetime point.

Sorry, I don't see the difference. Using your logic, why can't I just as well say about Maxwell electromagnetism that $\nabla_{\nu}F^{\mu\nu}=4\pi J^{\mu}$ "is a constraint equation that must hold simultaneously at every spacetime point"? Or alternatively, why can't I regard both $G_{\mu\nu}=8\pi T_{\mu\nu}$ and $\nabla_{\nu}F^{\mu\nu}=4\pi J^{\mu}$ as time-evolution equations for the gravitational and electromagnetic fields, respectively?

 

[martinbn]

I think you have a misunderstanding of what an equation is. The equation doesn't say that $T$ determines or causes $G$. It says that $G$ and $T$ are equal (when multiplied by the appropriate constant). That's what an equation is, the left side is equal to the right side. How this relates to physics, for instance how does initial date determine the future is a separate question. In general relativity the evolution is more complicated because you don' t have a background space time. You can look up the initial value problem in general relativity.

https://arxiv.org/abs/1304.1960
https://ems.press/books/esi/66

 

[Dale]

TL;DR Summary: Einstein's field equation G_μν = 8πG T_μν appears to establish a clear causal relationship: matter-energy (T_μν) determines spacetime curvature (G_μν).

This isn’t a paradox, it is simply a mistake. The EFE does not establish a causal relationship. A thought that it does is a mistake.

 

[Dale]

Forces at time t determine acceleration at time t

Not really. Many times, for instance, in statics, the acceleration is 0 at time t and that acceleration is what determines the forces at time t. Even Newton’s laws are not causal relationships.

To see an actual causal relationship, see Jefimenko’s equations.

https://en.m.wikipedia.org/wiki/Jefimenko's_equations

In Jefimenko’s equations charges and currents in the past (retarded time) cause fields at some time t. That is the form of a causal relationship. Causes do not occur at the same time as effects. Causes occur before effects.

 

[PeterDonis]

I specify the metric g_μν on some initial surface
This determines geodesics, so matter motion is fixed

Wrong. You've only specified the metric on an initial spacelike hypersurface. That's not enough to specify geodesics of the spacetime.

You seem to be reasoning from false premises. Of course that's going to give you false conclusions.