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Some time ago I read “Analogieën”, a Dutch translation of 'Surfaces and Essences' by Douglas Hofstadter and Emmanuel Sander (Analogy is the core of all thinking). In one of the last chapters of this book they describe the development of Einstein's thinking about his theory of relativity. The most famous result of the theory of special relativity is the formula

$$ E=mc^2 $$

Einstein first interpreted it not as an equivalence, but as cause and result: an energy amount of $E$ causes a mass of $E/c^2$. Not until a few years later, he interpreted it also in the other direction: a mass amount of $m$ is equivalent to an amount energy of $mc^2$. Nowadays this equivalence is generally accepted. The difference between mass and energy has almost completely disappeared. In elementary particle physics, even the same units are used for mass and energy.

The book then continues to describe the development in Einstein’s thinking in formulating the theory of general relativity. They repeatedly mention a parallel development. The theory of special relativity resulted from the broadening of Galileo's relativity postulate from the field of mechanical physics to the whole field of physics, including optics. It is impossible to construct a physics experiment to determine the absolute velocity within a frame that moves with a constant velocity. Therefore, the velocity of light, $c$, must be an invariant. Similarly, the theory of general relativity results from a similar broadening. Not only for mechanical physics, but for all physics there is no way to construct an experiment to determine the difference between a frame in rest in a gravitational field and a frame without a gravitational field that is accelerating. Taking this as a starting point, it took Einstein about 8 years to formulate his theory of general relativity, which resulted in the formula

$$ G=(8πG/c^4) T $$

Here $G$ is the curvature tensor of space-time, $T$ is the energy-momentum density tensor and $G$ and $c$ are constants.

It struck me that the book does not tell about a similar development in the interpretation of this equation of general relativity as for that of special relativity. When I searched on the Internet, I saw that it is usually interpreted in a cause and result way: energy and momentum cause a curvature in space-time. Nowhere I see that it is interpreted as an equivalence, so that it can also go in the other direction: a certain curvature in space-time will cause energy-momentum. Or even a full equivalence: energy-momentum is the way in which we perceive such a space-time curvature. I wonder whether I have searched enough. Has this equivalence already been discussed, or am I the first one to think about it after more than a century? That is hard to believe.

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    $\begingroup$ It is not an equivalence. I think you will find Denis Lehmkuhl paper about this issue quite interesting. Please look into physics.stackexchange.com/a/699116/281096 . $\endgroup$
    – JanG
    Oct 21, 2023 at 16:09
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    $\begingroup$ The only people who think that there is an “equivalence” between mass and energy are the people who don’t know the actual formula: $E^2-(\mathbf pc)^2=(mc^2)^2$ or $E^2-\mathbf p^2=m^2$ in units where $c=1$. $\endgroup$
    – Ghoster
    Oct 21, 2023 at 23:31
  • $\begingroup$ This Pythagorean-style relationship, except with the negative sign characteristic of Minkowskian geometry, says that mass is the Lorentz-invariant length of the momentum 4-vector $(E,\mathbf p)$, which is a much more beautiful relationship than a pointless “equivalence”. If two things are equivalent, you don’t need both of them. Mass and energy are entirely different physical/geometric things, and you need both. A single component of a vector ($E$) is not equivalent to its length ($m$)! $\endgroup$
    – Ghoster
    Oct 21, 2023 at 23:41
  • $\begingroup$ To add, there is no equivalence because curvature is one thing, whereas energy-momentum has many distinct species (mass, electromagnetic energy, strong energy, weak energy, etc.) $\endgroup$ Oct 22, 2023 at 13:26
  • $\begingroup$ History/philosophy of physics without enough actual physics can be a misleading teacher of physics. The relation $E=mc^2$ is a definition of rest energy of a body with mass $m$ in special relativity. This is not a deeper equivalence between the concepts of energy and mass in the general sense. There are cases where there is energy, but there is no mass, e.g. a plane EM wave. $\endgroup$ Oct 22, 2023 at 23:58

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First, neither of these equations that you have discussed here is in the correct form for a cause-effect relation. In a cause-effect relationship, by definition, the cause precedes the effect. So causal relationships are given by equations like this one $$B(t)=\int_{-\infty}^{t} A(t_r) dt_r$$ or even $$ B(t)=A(t_r) $$ where $A$ is the cause and $B$ is the effect and $t > t_r$. These equations are not symmetric; the cause and the effect are unambiguously identified by the fact that the cause is earlier than the effect.

In electromagnetism, for example, Maxwell’s equations are in a non-causal form, while Jefimenko’s equations and the retarded potentials are in a causal form. Although commonly stated, it is incorrect to describe Maxwell’s as saying that a changing E field causes a curling B field etc. It is correct to describe Jefimenko’s equations as saying that charges and currents cause E and B fields.

When I searched on the Internet, I saw that it is usually interpreted in a cause and result way: energy and momentum cause a curvature in space-time.

It may be common, but it is an incorrect interpretation. Causes precede effects. In that equation the energy and momentum do not precede the curvature. So Einstein field equation (EFE) does not support that claim.

The ADM formalism is closer to a causal equation for GR than the EFE.

Nowhere I see that it is interpreted as an equivalence, so that it can also go in the other direction: a certain curvature in space-time will cause energy-momentum

This is, in fact, common in the literature. Not in the incorrect sense that you have stated here where curvature causes stress-energy, but in the correct sense that curvature implies stress-energy.

For one specific example, in the derivation of the Alcubierre spacetime, the metric was stated first and the implied stress-energy tensor was then derived from that. Many other spacetimes were derived using that metric-based approach, which is founded on the fact that the equivalence relationship in the EFE does go both ways. So this equivalence is, in fact, both well known and commonly used.

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