Question

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=mc²

Einstein first interpreted it not as an equivalence, but as cause and result: an energy amount of E causes a mass of E/c². 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 mc². 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/c⁴)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?

Answer 1

The equation E = mc² from special relativity is an accepted bidirectional equivalence between energy and mass. In general relativity, the equation G = (8πG/c⁴)T is seen as spacetime curvature being caused by energy-momentum, but the idea of curvature causing energy-momentum, representing a deeper bilateral equivalence, remains less explicitly recognized but inherent in the symmetrical structure of the theory.

Step-by-step explanation:

Albert Einstein's theory of special relativity gives us the well-known equation E = mc², indicating that energy and mass are equivalent. Initially, Einstein saw this as energy causing a corresponding mass. Over time, the scientific community has accepted the bidirectional equivalence, where not only does energy produce mass, but mass can also be converted into energy. With regards to general relativity, the formula G = (8πG/c⁴)T is often seen as a one-way relationship where energy-momentum causes spacetime curvature. However, the interpretation of this as a complete equivalence, suggesting that curvature can induce energy-momentum, or that energy-momentum is essentially how we perceive curvature, has been less explicitly discussed but is implied in the theory's symmetrical treatment of the spacetime and energy-momentum tensors.

Einstein's exploration of general relativity was driven by the Principle of Equivalence, which played a crucial role in his thought experiments, such as considering the experience in an accelerating elevator versus in a gravitational field. His development of general relativity involved extending the concept of relativity to non-inertial frames of reference, leading to the revolutionary insight that gravitation and acceleration are indistinguishable at the local level.

The question of whether spacetime curvature could equally be said to cause energy-momentum is intriguing and suggests a deeper level of symmetry in general relativity that possibly hints at avenues towards quantum gravity. This symmetry is reflective of the underlying physics that governs the universe and represents complex concepts challenging to grasp fully.