I would love to see another post on this focusing on the right side of the equation. Einstein's original formulation, as you noted, did not properly account for the conservation of mass and energy. Noether, working with Hilbert, would go on to formalize this in her famous theorem three years later, but for the time being Einstein had a big problem.
Conservation of energy is a funny thing. Leibniz was the first to point out that his vis viva quantity (m*v^2) was invariant, but it was only with the advent of the industrial revolution that scientists saw the connection between this quantity and heat. Joule famously showed the connection between grav.pot energy, kinetic energy and heat which capitulated years of theorizing.
Hilbert ALMOST got to the conservation of energy with his GR formulation but failed to see it yielding the conservation laws. Einstein instead used conservation of energy AS A CONSTRAINT, ie. an assumption, when (as would be discovered) it is a CONSEQUENCE of the general covariance. So what happened next?
In October 1916 Einstein revisited the issue and finally showed from his equations that for a matter-based Lagrangian, the mass-energy tensor T, that right side of his equation, must be divergenceless (ie. Div(T)=0). Okay, fine, but again he was assuming conservation of energy as a given. The real question was, would the left side, the curvature side, be divergenceless too?
In August 1917 Hermann Weyl finally showed that it was and gradually other physicists like Klein and Hilbert came on board. But something was still strange, as Arthur Eddington finally put the pieces together in 1920 after rederiving Div(G)=0 for fun. He remarked that as Div(G)=0 seems both consequential AND straight-forward, how had no one ever discovered this before??
In fact they had! By an obscure mathematician named Aurel Voss in 1880; then again by Ricci in 1889; and finally by Klein's student Bianchi in 1902. For reasons that are not entirely clear (and certainly not fair) the identity became known as the Bianchi Identity:
While this doesn't look like anything more than fodder for mathematicians this is perhaps one of the most profound formulations in mathematics given its implications for the "real" world. In this identity we have the birth of the divergenceless Einstein tensor, which then enforces divergencelessness on T and thus gives us the conservation laws!
As my GR prof once smirked, "don't tell the quantum guy down the hall, but his whole job is just the Bianchi Identities." 😂
-------------------------------------------------
PS:
There is something very profound and profoundly mal-understood here imho. The conservation laws apply to energy regardless of whether they deal with conserved forces or dissipative ones. When we think of entropy we tend to think of those hard-to-calculate dissipative forces that seem so chaotic and intractable. And yet based on a mathematical identity we know that even highly entropic, irreversible events (like a bomb explosion) yield to conservation laws. Maybe someone has something profound to say to squelch my awe, but to me this lends hope to any physicist pondering the tractability of any problem.
I have my hands full understanding how electromagnetism works (a topic I'll finally be able to get in to in Chapter 6 in a few months). My initial draft had only a short section on relativity and when I opened up the hood to take a closer look, it doubled, redoubled, and is on track to double again before I can complete even a reasonable summary of the scientific, historical, and philosophic issues here. I appreciate your thoughtful comment. The bottom line is that GR may align with observations, but it is far from proven, and there are any number of alternatives and extensions to it that might better describe what's going on. Detailing those is something I'll have to leave to someone else.
The thing that annoys me about the physics of the last century or so is the lack of introspection. For energy to be conserved, it must exist, and it must have both upper (Planck?) and lower (0) bounds. And it must be equal everywhere. Which means that spacetime is really a potential energy field, with (proper) time as the potential from which all other energies are drawn.
That's interesting. I ended my physics education 20 years ago a my Masters in Astronomy (observational cosmology). But GR always fascinated me. It's always taught poorly with very little reference to physical objects. I think we're still grasping in the dark and probably won't figure it out until our actual physical experiences start to match our intuition. For example, if someone had just sat down and transformed Maxwell's Laws with Galilean relativity they would have found they were NOT invariant. But no one really thought to test that until the popularization of steam locomotion when people started traveling faster than horseback (fwiw most trains only traveled at a horse's gallop, ie. 25mph, until the advent of diesel in the 1930s).
We probably won't figure it out until space travel becomes somewhat frequent and GR effects are more than just negligible.
Great stuff! Fascinating. But I'm confused. (I know, shocking). There's a big jump there near the end from gravity (curved space- that I can grasp) to curved space-time.
"Relativity deals in four-vectors: mathematical constructs comprising not only three spacial quantities but also time"
Why did 'time' need to be included? Why can't gravity 'bend light' without 'time'?
Space and time are inseparable. In order for movement to exist, they *must* be two sides of the same coin - a change in position and a change in time.
Spacetime is very like electromagnetism - you cannot have space without time, nor can you have electricity without magnetism, and for remarkably similar reasons. Primarily, the finite and independent speed of causation, AKA the speed at which the fields deform.
It was Einstein's math professor, Minkowski, (the one who said Einstein was a "lazy dog" who skipped class and showed no talent) who noted special relativity could be expressed as a geometry of space and time together: spacetime.
Gravity makes clocks run slower - an effect seen in GPS whose satellites have atomic clocks set faster for the lower gravity they experience in orbit relative to the ground (and a correction slower for their orbital velocity, but the gravity contribution dominates).
Light refracts, bending into glass because it travels slower in glass. Similarly from the time effect alone, light traveling deeper in a gravity well will appear to slow down from the perspective of an outside observer and will therefore tend to bend.
suppose aether is real, would it also be at a higher density near a gravity well? This would cause light to appear to slow down to an outside observer but from its own persepective still be traveling through c aether untis per second. (Is that even a thing?) Trying to get a grasp thanks.
Possibly. We should always keep Newton's advice in mind and avoid framing hypotheses particularly just to fit one data point. A hypothesis demonstrates value the more it demonstrates consilience and ties together a wide variety of different observations.
Yes, gravity can bend light; according to Einstein's theory of general relativity, massive objects warp spacetime, causing light to follow a curved path when passing near them, a phenomenon known as gravitational lensing; essentially, gravity bends light by curving the space through which it travels.
I would love to see another post on this focusing on the right side of the equation. Einstein's original formulation, as you noted, did not properly account for the conservation of mass and energy. Noether, working with Hilbert, would go on to formalize this in her famous theorem three years later, but for the time being Einstein had a big problem.
Conservation of energy is a funny thing. Leibniz was the first to point out that his vis viva quantity (m*v^2) was invariant, but it was only with the advent of the industrial revolution that scientists saw the connection between this quantity and heat. Joule famously showed the connection between grav.pot energy, kinetic energy and heat which capitulated years of theorizing.
Hilbert ALMOST got to the conservation of energy with his GR formulation but failed to see it yielding the conservation laws. Einstein instead used conservation of energy AS A CONSTRAINT, ie. an assumption, when (as would be discovered) it is a CONSEQUENCE of the general covariance. So what happened next?
In October 1916 Einstein revisited the issue and finally showed from his equations that for a matter-based Lagrangian, the mass-energy tensor T, that right side of his equation, must be divergenceless (ie. Div(T)=0). Okay, fine, but again he was assuming conservation of energy as a given. The real question was, would the left side, the curvature side, be divergenceless too?
In August 1917 Hermann Weyl finally showed that it was and gradually other physicists like Klein and Hilbert came on board. But something was still strange, as Arthur Eddington finally put the pieces together in 1920 after rederiving Div(G)=0 for fun. He remarked that as Div(G)=0 seems both consequential AND straight-forward, how had no one ever discovered this before??
In fact they had! By an obscure mathematician named Aurel Voss in 1880; then again by Ricci in 1889; and finally by Klein's student Bianchi in 1902. For reasons that are not entirely clear (and certainly not fair) the identity became known as the Bianchi Identity:
Div_e(R_a,b,c,d) + Div_d(R_a,b,e,c) + Div_c(R_a,b,d,e) = 0
While this doesn't look like anything more than fodder for mathematicians this is perhaps one of the most profound formulations in mathematics given its implications for the "real" world. In this identity we have the birth of the divergenceless Einstein tensor, which then enforces divergencelessness on T and thus gives us the conservation laws!
As my GR prof once smirked, "don't tell the quantum guy down the hall, but his whole job is just the Bianchi Identities." 😂
-------------------------------------------------
PS:
There is something very profound and profoundly mal-understood here imho. The conservation laws apply to energy regardless of whether they deal with conserved forces or dissipative ones. When we think of entropy we tend to think of those hard-to-calculate dissipative forces that seem so chaotic and intractable. And yet based on a mathematical identity we know that even highly entropic, irreversible events (like a bomb explosion) yield to conservation laws. Maybe someone has something profound to say to squelch my awe, but to me this lends hope to any physicist pondering the tractability of any problem.
A.J.R. Klopp (Aaron)
I have my hands full understanding how electromagnetism works (a topic I'll finally be able to get in to in Chapter 6 in a few months). My initial draft had only a short section on relativity and when I opened up the hood to take a closer look, it doubled, redoubled, and is on track to double again before I can complete even a reasonable summary of the scientific, historical, and philosophic issues here. I appreciate your thoughtful comment. The bottom line is that GR may align with observations, but it is far from proven, and there are any number of alternatives and extensions to it that might better describe what's going on. Detailing those is something I'll have to leave to someone else.
Maybe I'll take a stab at a short post on this then. Will be sure to give you a shout out if I do.
Well stated, sir.
The thing that annoys me about the physics of the last century or so is the lack of introspection. For energy to be conserved, it must exist, and it must have both upper (Planck?) and lower (0) bounds. And it must be equal everywhere. Which means that spacetime is really a potential energy field, with (proper) time as the potential from which all other energies are drawn.
That's interesting. I ended my physics education 20 years ago a my Masters in Astronomy (observational cosmology). But GR always fascinated me. It's always taught poorly with very little reference to physical objects. I think we're still grasping in the dark and probably won't figure it out until our actual physical experiences start to match our intuition. For example, if someone had just sat down and transformed Maxwell's Laws with Galilean relativity they would have found they were NOT invariant. But no one really thought to test that until the popularization of steam locomotion when people started traveling faster than horseback (fwiw most trains only traveled at a horse's gallop, ie. 25mph, until the advent of diesel in the 1930s).
We probably won't figure it out until space travel becomes somewhat frequent and GR effects are more than just negligible.
In a few sections, I'll be discussing Ron Hatch and his claim that how GPS works is inconsistent with GR.
I'll keep an eye out
Great stuff! Fascinating. But I'm confused. (I know, shocking). There's a big jump there near the end from gravity (curved space- that I can grasp) to curved space-time.
"Relativity deals in four-vectors: mathematical constructs comprising not only three spacial quantities but also time"
Why did 'time' need to be included? Why can't gravity 'bend light' without 'time'?
Working my way back thru the most recent posts.
Space and time are inseparable. In order for movement to exist, they *must* be two sides of the same coin - a change in position and a change in time.
Spacetime is very like electromagnetism - you cannot have space without time, nor can you have electricity without magnetism, and for remarkably similar reasons. Primarily, the finite and independent speed of causation, AKA the speed at which the fields deform.
It was Einstein's math professor, Minkowski, (the one who said Einstein was a "lazy dog" who skipped class and showed no talent) who noted special relativity could be expressed as a geometry of space and time together: spacetime.
Gravity makes clocks run slower - an effect seen in GPS whose satellites have atomic clocks set faster for the lower gravity they experience in orbit relative to the ground (and a correction slower for their orbital velocity, but the gravity contribution dominates).
Light refracts, bending into glass because it travels slower in glass. Similarly from the time effect alone, light traveling deeper in a gravity well will appear to slow down from the perspective of an outside observer and will therefore tend to bend.
suppose aether is real, would it also be at a higher density near a gravity well? This would cause light to appear to slow down to an outside observer but from its own persepective still be traveling through c aether untis per second. (Is that even a thing?) Trying to get a grasp thanks.
Possibly. We should always keep Newton's advice in mind and avoid framing hypotheses particularly just to fit one data point. A hypothesis demonstrates value the more it demonstrates consilience and ties together a wide variety of different observations.
well I agree with that. But is the fabric of reality compressible or uniform? How would one test such a thing?
Good question!
Yes, gravity can bend light; according to Einstein's theory of general relativity, massive objects warp spacetime, causing light to follow a curved path when passing near them, a phenomenon known as gravitational lensing; essentially, gravity bends light by curving the space through which it travels.