But a cylinder is clearly not flat! How can this be?

Well, because parallel geodesics remain equidistant, we know that the cylinder, like the flat sheet of paper, is INTRINSICALLY FLAT. However, it is intuitively clear that in some way the cylinder really is curved. And at the same time, it is intuitively obvious that a flat piece of paper really is flat.

Well, because parallel geodesics remain equidistant, we know that the cylinder, like the flat sheet of paper, is INTRINSICALLY FLAT. However, it is intuitively clear that in some way the cylinder really is curved. And at the same time, it is intuitively obvious that a flat piece of paper really is flat.

CONVERSELY, ON A NEGATIVELY CURVED SPACE...

... THE SUM OF THE ANGLES OF A TRIANGLE IS LESS THAN 180 DEGREES

... THE SUM OF THE ANGLES OF A TRIANGLE IS LESS THAN 180 DEGREES

This is a general characteristic of positively curved spaces – the sum of angles of triangles formed from their geodesics is GREATER than 180 degrees

It is also possible to construct spaces in which the parallel geodesics never intersect, but the distance between them increases the further along the geodesics you go

So parallel lines can meet! In this case, the space is said to have POSITIVE curvature.

ALL LINES OF LONGITUDE AND THE EQUATOR ARE GEODESICS-THEY ARE “GREAT CIRCLES”

ALL THE LINES OF LONGITUDE ARE PARALLEL AT THE EQUATOR, SINCE THEY ALL INTERSECT THE EQUATOR AT RIGHT ANGLES

HOWEVER, ALL OF THE LINES OF LONGITUDE INTERSECT AT THE NORTH & SOUTH POLES

ALL THE LINES OF LONGITUDE ARE PARALLEL AT THE EQUATOR, SINCE THEY ALL INTERSECT THE EQUATOR AT RIGHT ANGLES

HOWEVER, ALL OF THE LINES OF LONGITUDE INTERSECT AT THE NORTH & SOUTH POLES

WE THEN SAY TWO GEODESICS ARE PARALLEL IF THEY ARE PARALLEL AT SOME POINT...

...THAT IS THE ANGLES THEY MAKE ON INTERSECTION WITH A THIRD GEODESIC ARE THE SAME

...THAT IS THE ANGLES THEY MAKE ON INTERSECTION WITH A THIRD GEODESIC ARE THE SAME

In fact, it is true in general only if the space on which you draw the parallel lines is flat. Hence, Euclidean geometry is the study of flat-surface geometry

TWO PARALLEL LINES CAN NEVER MEET

THIS SEEMED INTUITIVELY OBVIOUS BUT HE WAS UNABLE TO PROVE IT

In the end, Euclid had to take it as an assumption – an axiom. This is because it is NOT generally true.

THIS SEEMED INTUITIVELY OBVIOUS BUT HE WAS UNABLE TO PROVE IT

In the end, Euclid had to take it as an assumption – an axiom. This is because it is NOT generally true.

From Einstein’s equations this means that Gij = g11 at the point (x,y,z,t). But crucially, even if = 0, this does NOT mean that the space is flat at the point (x,y,z).

This is very important since from our own daily experience, the earth goes round the sun, even though the space between the sun and the earth is almost a perfect vacuum.

This is very important since from our own daily experience, the earth goes round the sun, even though the space between the sun and the earth is almost a perfect vacuum.

In particular, if there is no matter at a particular point (x,y,z,t) – a vacuum – then Tij(x,y,z,t) = 0.

Now we can write down Einstein’s equations of General Relativity:

Gij = 8πGTij + Λgij

Gij = 8πGTij + Λgij

3-tensor is a three-dimensional block of numbers which we can denote with three indices. For example, Cijk where each of i, j, k can be any of 1, 2, 3 or 4.

2-tensor is a matrix or block of 4 × 4 = 16 numbers which we can denote by Bij. The two indices i and j tell us it is a block of numbers …

1-tensor is a string of four numbers (in four spacetime dimensions).

So, for example, A = (1 0 –1 3.14) is a 1-tensor or simply just a “vector”, which is also an arrow in spacetime.

Often we write Aj for the vector.

Here i = 1, 2, 3 or 4 so that A1 = 1, A2 = 0, etc. The electric and magnetic fields are described this way.

So, for example, A = (1 0 –1 3.14) is a 1-tensor or simply just a “vector”, which is also an arrow in spacetime.

Often we write Aj for the vector.

Here i = 1, 2, 3 or 4 so that A1 = 1, A2 = 0, etc. The electric and magnetic fields are described this way.

0-tensor is simply a single number: for example, the number “2”

BUT FROM SPECIAL RELATIVITY AND E2=m2c4+p2c2 IT FOLLOWS THAT MOMENTUM IS ENERGY AND ENERGY IS MASS

HENCE, IT SEEMS REASONABLE THAT ANY ENERGY IN THE UNIVERSE WILL CAUSE SPACETIME TO CURVE

HENCE, IT SEEMS REASONABLE THAT ANY ENERGY IN THE UNIVERSE WILL CAUSE SPACETIME TO CURVE

That missing ingredient is contained in the question: “How does spacetime know how to curve to give the right geodesics to send the moon sailing in an ellipse around the earth?”

Since it is the earth’s gravity that causes the moon to revolve around it, we know that mass must be doing the job of curving spacetime.

Since it is the earth’s gravity that causes the moon to revolve around it, we know that mass must be doing the job of curving spacetime.

Riemannian geometries, the distance between two points does not have to be positive – it can be zero or it can be negative!

The Clay Foundation offers a million-dollar prize for proving it true (and nothing for proving it false

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