# Introducing Relativity

Ingrid Belanhas quoted3 years ago
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.
Ingrid Belanhas quoted3 years ago
CONVERSELY, ON A NEGATIVELY CURVED SPACE...
... THE SUM OF THE ANGLES OF A TRIANGLE IS LESS THAN 180 DEGREES
Ingrid Belanhas quoted3 years ago
This is a general characteristic of positively curved spaces – the sum of angles of triangles formed from their geodesics is GREATER than 180 degrees
Ingrid Belanhas quoted3 years ago
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
Ingrid Belanhas quoted3 years ago
So parallel lines can meet! In this case, the space is said to have POSITIVE curvature.
Ingrid Belanhas quoted3 years ago
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
Ingrid Belanhas quoted3 years ago
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
Ingrid Belanhas quoted3 years ago
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
Ingrid Belanhas quoted3 years ago
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.
Ingrid Belanhas quoted3 years ago
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.
Ingrid Belanhas quoted3 years ago
In particular, if there is no matter at a particular point (x,y,z,t) – a vacuum – then Tij(x,y,z,t) = 0.
Ingrid Belanhas quoted3 years ago
Now we can write down Einstein’s equations of General Relativity:
Gij = 8πGTij + Λgij
Ingrid Belanhas quoted3 years ago
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.
Ingrid Belanhas quoted3 years ago
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 …
Ingrid Belanhas quoted3 years ago
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.
Ingrid Belanhas quoted3 years ago
0-tensor is simply a single number: for example, the number “2”
Ingrid Belanhas quoted3 years ago
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
Ingrid Belanhas quoted3 years ago
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.
Ingrid Belanhas quoted3 years ago
Riemannian geometries, the distance between two points does not have to be positive – it can be zero or it can be negative!
Ingrid Belanhas quoted3 years ago
The Clay Foundation offers a million-dollar prize for proving it true (and nothing for proving it false
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