The Einstein field equation itself is actually extremely simple:
G = 8piT
Sure, if we don’t mind that G and T take a full page to write out in terms of the derivatives of the metric tensor. By this logic every equation is extremely simple—it simply asserts that A=B for some A,B. :-)
Another urban legend, which I’ve heard told about various mathematicians, and which Misha Polyak self-effacingly tells about himself (and therefore might even be true), is the following:
As a young postdoc, Misha was giving a talk at a prestigious US university about his new diagrammatic formula for a certain finite type invariant, which had 158 terms. A famous (but unnamed) mathematician was sitting, sleeping, in the front row. “Oh dear, he doesn’t like my talk,” thought Misha.
But then, just as Misha’s talk was coming to a close, the famous professor wakes with a start. Like a man possessed, the famous professor leaps up out of his chair, and cries, “By golly! That looks exactly like the Grothendieck-Riemann-Roch Theorem!!!”
Misha didn’t know what to say. Perhaps, in his sleep, this great professor had simplified Misha’s 158 term diagrammatic formula for a topological invariant, and had discovered a deep mathematical connection with algebraic geometry? It was, after all, not impossible. Misha paced in front of the board silently, not knowing quite how to respond. Should he feign understanding, or admit his own ignorance? Finally, because the tension had become too great to bear, Misha asked in an undertone, “How so, sir?”
“Well,” explained the famous professor grandly. “There’s a left hand side to your formula on the left.”
“Yes,” agreed Misha meekly.
“And a right hand side to your formula on the right.”
“Indeed,” agreed Misha.
“And you claim that they are equal!” concluded the great professor. “Just like the Grothendieck-Riemann-Roch Theorem!”
Sure, if we don’t mind that G and T take a full page to write out in terms of the derivatives of the metric tensor.
Yeah, but there are only three objects you can write in terms of the metric tensor which transform the “right” way (G, T, and g itself). So the most general equation which satisfies those transformation laws is aG + bT + cg = 0.
Now, a is non-zero (otherwise you get an universe where there’s no matter/energy other than “dark energy”), so by redefining b and c we have G + bT + cg = 0; b is negative (because things attract each other rather than repelling each other) and we call it −8piG/c^4 (it’s just a matter of choice of units of measurement; we might as well set it to 1); and c is the cosmological constant.
I doubt that Scott will reply to this, 5 years later and on a different site, so let me try instead.
there are only three objects you can write in terms of the metric tensor which transform the “right” way (G, T, and g itself).
Hindsight bias? There are plenty of such objects. Google f(R) gravity, for example. There are also many different contractions of powers of products of R, T and G that fit. There is also torsion, and probably other things (supergravity and string theory tend to add a few).
You might want to argue that G=T is “the simplest”, but it is anything but, for the reasons Scott explained. Once you find something that works, you call it G and T, write G=T and call it “simple”. That’s what Einstein did, since his first attempt, R=T, did not work out.
The Einstein field equation itself is actually extremely simple:
G = 8piT
Sure, if we don’t mind that G and T take a full page to write out in terms of the derivatives of the metric tensor. By this logic every equation is extremely simple—it simply asserts that A=B for some A,B. :-)
http://mathoverflow.net/questions/53122/mathematical-urban-legends
Another urban legend, which I’ve heard told about various mathematicians, and which Misha Polyak self-effacingly tells about himself (and therefore might even be true), is the following:
As a young postdoc, Misha was giving a talk at a prestigious US university about his new diagrammatic formula for a certain finite type invariant, which had 158 terms. A famous (but unnamed) mathematician was sitting, sleeping, in the front row. “Oh dear, he doesn’t like my talk,” thought Misha. But then, just as Misha’s talk was coming to a close, the famous professor wakes with a start. Like a man possessed, the famous professor leaps up out of his chair, and cries, “By golly! That looks exactly like the Grothendieck-Riemann-Roch Theorem!!!” Misha didn’t know what to say. Perhaps, in his sleep, this great professor had simplified Misha’s 158 term diagrammatic formula for a topological invariant, and had discovered a deep mathematical connection with algebraic geometry? It was, after all, not impossible. Misha paced in front of the board silently, not knowing quite how to respond. Should he feign understanding, or admit his own ignorance? Finally, because the tension had become too great to bear, Misha asked in an undertone, “How so, sir?” “Well,” explained the famous professor grandly. “There’s a left hand side to your formula on the left.” “Yes,” agreed Misha meekly. “And a right hand side to your formula on the right.” “Indeed,” agreed Misha. “And you claim that they are equal!” concluded the great professor. “Just like the Grothendieck-Riemann-Roch Theorem!”
Yeah, but there are only three objects you can write in terms of the metric tensor which transform the “right” way (G, T, and g itself). So the most general equation which satisfies those transformation laws is aG + bT + cg = 0.
Now, a is non-zero (otherwise you get an universe where there’s no matter/energy other than “dark energy”), so by redefining b and c we have G + bT + cg = 0; b is negative (because things attract each other rather than repelling each other) and we call it −8piG/c^4 (it’s just a matter of choice of units of measurement; we might as well set it to 1); and c is the cosmological constant.
I doubt that Scott will reply to this, 5 years later and on a different site, so let me try instead.
Hindsight bias? There are plenty of such objects. Google f(R) gravity, for example. There are also many different contractions of powers of products of R, T and G that fit. There is also torsion, and probably other things (supergravity and string theory tend to add a few).
You might want to argue that G=T is “the simplest”, but it is anything but, for the reasons Scott explained. Once you find something that works, you call it G and T, write G=T and call it “simple”. That’s what Einstein did, since his first attempt, R=T, did not work out.
Interesting...