Janna Levin

But I'm happy just talking about the large dimensions, the three we see everywhere. Up, down, east, west, north, south, three spatial dimensions, and time is the fourth. Nobody can really visualize it, but we know mathematically how to unpack it on paper. I can mathematically suppress one of the spatial dimensions and then I can draw it pretty well.

Now, the problem is that we'd call it a Euclidean spacetime. A Euclidean spacetime is when all the dimensions are orthogonal and are treated equally. Time is not another Euclidean dimension. It's actually a Minkowskian spacetime. But it means that the space-time, we're misrepresenting it when we draw it, but we're misrepresenting it in a way that we deeply understand. I can give you an example.

Now, the problem is that we'd call it a Euclidean spacetime. A Euclidean spacetime is when all the dimensions are orthogonal and are treated equally. Time is not another Euclidean dimension. It's actually a Minkowskian spacetime. But it means that the space-time, we're misrepresenting it when we draw it, but we're misrepresenting it in a way that we deeply understand. I can give you an example.

Now, the problem is that we'd call it a Euclidean spacetime. A Euclidean spacetime is when all the dimensions are orthogonal and are treated equally. Time is not another Euclidean dimension. It's actually a Minkowskian spacetime. But it means that the space-time, we're misrepresenting it when we draw it, but we're misrepresenting it in a way that we deeply understand. I can give you an example.

The Earth, I can project onto a flat sheet of paper. I am now misrepresenting a map of the Earth. And I know that, but I understand the rules for how to add distances on this misrepresentation, because the Earth is not a flat sheet of paper. It's a sphere.

The Earth, I can project onto a flat sheet of paper. I am now misrepresenting a map of the Earth. And I know that, but I understand the rules for how to add distances on this misrepresentation, because the Earth is not a flat sheet of paper. It's a sphere.

The Earth, I can project onto a flat sheet of paper. I am now misrepresenting a map of the Earth. And I know that, but I understand the rules for how to add distances on this misrepresentation, because the Earth is not a flat sheet of paper. It's a sphere.

And as long as I understand the rules for how I get from the North Pole to the South Pole, that I'm moving along really a great arc, and I understand that the distance is not the distance I would measure on a flat sheet of paper... then I can do a really great job with a map and understanding the rules of addition, multiplication, and the geometries, not the geometry of a flat sheet of paper.

And as long as I understand the rules for how I get from the North Pole to the South Pole, that I'm moving along really a great arc, and I understand that the distance is not the distance I would measure on a flat sheet of paper... then I can do a really great job with a map and understanding the rules of addition, multiplication, and the geometries, not the geometry of a flat sheet of paper.

And as long as I understand the rules for how I get from the North Pole to the South Pole, that I'm moving along really a great arc, and I understand that the distance is not the distance I would measure on a flat sheet of paper... then I can do a really great job with a map and understanding the rules of addition, multiplication, and the geometries, not the geometry of a flat sheet of paper.

I can do the same thing with spacetime. I can draw it on a flat sheet of paper, but I know that it's not actually a flat Euclidean space. And so my rules for measuring distances are different than the rules I would use that, for instance, Cartesian rules of geometry. I would know to use the correct rules for Minkowski spacetime and

I can do the same thing with spacetime. I can draw it on a flat sheet of paper, but I know that it's not actually a flat Euclidean space. And so my rules for measuring distances are different than the rules I would use that, for instance, Cartesian rules of geometry. I would know to use the correct rules for Minkowski spacetime and

I can do the same thing with spacetime. I can draw it on a flat sheet of paper, but I know that it's not actually a flat Euclidean space. And so my rules for measuring distances are different than the rules I would use that, for instance, Cartesian rules of geometry. I would know to use the correct rules for Minkowski spacetime and

And that will allow me to calculate how long time has elapsed, which is now a kind of a length, a space-time length on my map, between two relative observers, and I will get the correct answer. But only if I use these different rules.

And that will allow me to calculate how long time has elapsed, which is now a kind of a length, a space-time length on my map, between two relative observers, and I will get the correct answer. But only if I use these different rules.

And that will allow me to calculate how long time has elapsed, which is now a kind of a length, a space-time length on my map, between two relative observers, and I will get the correct answer. But only if I use these different rules.

Right, exactly. So Einstein struggled for this completely general theory, not a specific solution like a black hole or an expanding space-time or galaxies make lenses. Those are all solutions. That's why what he did was so enormous. It's an entire paradigm that says, over here is matter and energy. I'm going to call that the right-hand side of the equation.

Right, exactly. So Einstein struggled for this completely general theory, not a specific solution like a black hole or an expanding space-time or galaxies make lenses. Those are all solutions. That's why what he did was so enormous. It's an entire paradigm that says, over here is matter and energy. I'm going to call that the right-hand side of the equation.

Right, exactly. So Einstein struggled for this completely general theory, not a specific solution like a black hole or an expanding space-time or galaxies make lenses. Those are all solutions. That's why what he did was so enormous. It's an entire paradigm that says, over here is matter and energy. I'm going to call that the right-hand side of the equation.

Everything on the right-hand side of Einstein's equations is how matter and energy are distributed in spacetime. On the left-hand side tells you how space and time deform in response to that matter and energy. And it can be impossible to solve some of those equations.