# Is there a reason that the stress-energy-momentum tensor curves the spacetime?

The famous field equations of Albert Einstein describe, how spacetime is curved by the stress-energy-momentum tensor. The spacetime is curved by matter and energy. General relativity is the underlying law of Newtons law. One can infer Newtons law from GR. Is there an underlying law which results in general relativity?

Probably one would also have to answer "what actually is spacetime?" or "How does matter influence the spacetime?" either. Do we know the answer to those questions?

• Questions on why are technically philosophical, physics doesn't really answer the why, physics job is to make models and compare them to what we see and measure experimentally. Mar 15, 2022 at 16:47
• I think this question is valid and doesn't merit the down votes. There's no "why" question here.
– Cham
Mar 15, 2022 at 20:15
• The answer is actually very simple: We don't know the proper answer to this question (i.e how matter-energy is curving spacetime). For that, we probably need a quantum theory of gravity!
– Cham
Mar 15, 2022 at 20:17
• "What actually is spacetime?": it is a mathematical object called Lorentzian manifold. "How does matter influences spacetime?" Experiments say: consistently with Einstein equations, as far as we know. Mar 15, 2022 at 23:46
• This question has merit. Most of the observational evidence that we have for GR is in post Newtonian approximation limit in vacuum solutions, there are also evidences for equivalence principle. It is taken for granted, but we don't have evidence for why it is only the stress energy tensor of matter field which should couple to gravity, and whether it is indeed true for all systems. Mar 16, 2022 at 8:44

While there's no currently known answer to your question (since we don't have a quantum theory of the true vacuum and of gravity), I'm tempted to speculate a bit and very roughly (for what it's worth):

We usually think of matter as something placed "inside" space, and to define vacuum as the absence of matter/energy. Maybe it's better to interpret matter as "vacuum removed".

You start with an empty universe. There's a flat geometry vacuum. Vacuum isn't the same as nil, since there is geometry (lengths, time duration, volume, etc). In a sense, vacuum is a bit like some kind of weird inert fluid, that has the same properties for any observer (the fluid doesn't "flow": it doesn't have a local velocity for any observer).

Then add a small ball of matter. This ball "punches a hole" in the vacuum, since now there's something siting in space: there is less vacuum where the ball is, and the amount of vacuum removed is defined by the ball's energy content. Creating the ball is like "pushing" the vacuum to the outside, i.e. produces some deformation (curvature!) in the vacuum geometry. We could argue that gravity is a macroscopic effect of removing some vacuum by adding energy in it.

Notice that empty space may have a residual energy density everywhere (the cosmological constant), even in the absence of any content. I could elaborate on this idea but I think this is out of the scope of your question...

This "answer" may feels like pure non-sense speculation (for what it's worth!). Well then, too bad for the down votes.

• Interesting idea. Of course, the notion of the displacement of spacetime causing its warping would be more intuitive if the warping scaled with volume rather than matter/energy density. Any thoughts on how to make sense of this? Mar 16, 2022 at 3:30
• there's no currently known answer” - It is known. Spacetime and stress-energy are Fourier conjugates. Similarly, a wave and its spectrum are Fourier conjugates. We understand how the spectrum defines the wave. The same way stress-energy defines spacetime. And your answer is a very rough intuitive interpretation of this process. Mar 20, 2022 at 15:00

Is there a reason that the stress-energy-momentum tensor curves the spacetime?

The reason is the experimental discovery that presence of matter affects spacetime.

The stress-energy tensor in Einstein field equations describes how matter affects spacetime geometry and vice versa [1]. Spacetime and matter are two different things. Locally, spacetime can be curved without matter being there. Otherwise, the presence of matter changes spacetime curvature. The stress-energy tensor can be seen as some kind of interface between matter and spacetime.

[1] https://arxiv.org/abs/1803.09872v1, Dennis Lehmkuhl, "How Einstein saw the role of the energy-momentum tensor in GR", page 5.

• The reason is the experimental discovery that matter effects spacetime.” - Not really. The experiment part is only the specific value of the gravitational constant, but the fact that matter curves spacetime is a purely theoretical consequence of the Least Action Principle with the Hilbert action. Mar 20, 2022 at 15:05
• @safesphere, I appreciate you comment but then what about such experiments like scientificamerican.com/article/… and so many others? The Least Action Principle and the postulated Hilbert action is in my understanding just explanation of these experimental findings in terms of a mathematical model.
– JanG
Mar 20, 2022 at 15:22
• When you say that something is just an experimental result, it usually means that there is no theoretical explanation. It is not the case here. The phenomenon in question does have a theoretical explanation, so it is not just an experimental result. Every theory is created to explain certain observations. In this case, GR is based on the experimental equivalence principle, but not on the idea that stress-energy curves spacetime. It is a theoretical conclusion. BTW your link is about the time dilation in the flat Lorentzian spacetime where stress-energy is zero. Mar 21, 2022 at 3:10
• I should take another like link, like this one: sciencealert.com/… . You are of course right about the equivalence principle as base of GR. Einstein had a hard time trying to include the effect of matter on spacetime.
– JanG
Mar 21, 2022 at 9:38