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By Root 585 0

z are the three coordinates of space, and ax, ay, and az can be numbers or they can be functions of x, y, and z. The metric determines lengths, distances, and angles between lines. For example, the length of a vector pointing from the origin to the point with coordinates (x, y, z) is √(axx2 + ayy2 + azz2). If ax = ay = az = 1, we have flat space, and distances and lengths would be measured in the familiar manner. For example, the length of a vector pointing from the origin to (x, y, z) would be √(x2 + y2 + z2). More complicated metrics can have cross terms, such as dxdy. In that case, the metric must be described by a tensor with two indices that tells the coefficients aij of each term in the metric of the form dxidxj. Later, when we discuss relativity, the metric will also have a term dt2 and could also have terms of the form dtdxi.

3. A hypersphere is defined by x12 + x22 +…+ xn2 = r2. Here xi refers to the ith coordinate (the location in the ith dimension) and r is the radius of the hypersphere. The cross-section of the hypersphere when it crosses a fixed location in the nth dimension, xn − d, is described by the equation x12 + x22 +…+ xn-12 = r2 − d2. This is the equation of a hypersphere of one lower dimension and radius √(r2 − d2). So, for example, when n = 3 and a sphere crosses Flatland, Flatlanders would see circles. (They would see disks if they saw the circles and their interiors, which would be mathematically described with an inequality.)

4. Calabi-Yau manifolds are not the only possible stringy hidden manifolds. We now know that others, such as ones called G2 holonomy manifolds, might also give acceptable models.

5. In string theory, we also sometimes use the word “brane” to mean space-filling branes that have the same number of dimensions as the higher-dimensional space. Here, however, we will concern ourselves only with branes that have fewer dimensions than the full higher-dimensional space, so I will restrict myself to the use of the term as described in the text.

6. A brane that extends in the dimensions x1,…, xj is described by the n − j equations xj + 1 = cj + 1, xj + 2 = cj + 2 =…, = xn = cn, where xi are coordinates, n is the number of dimensions of space, and ci are fixed constants describing the location of the brane. More complicated branes that curve in the given coordinate system are described by more complicated equations that describe the surface.

7. In equation form, Newton’s law says that the gravitational force is Gm1 m2/r2, where G is Newton’s constant of gravitation, m1 and m2 are the two masses that are attracted to each other, and r is the distance between them.

8. Newtonian gravity respects Euclidean geometry. In Euclidean geometry, x2 + y2 + z2, the length of a vector pointing to the point with coordinates (x, y, z), is independent of the coordinate frame. That is, you can rotate your coordinates but the distance to any point won’t change, even though the individual coordinates will. Special relativity puts time into the picture. It says that x2 + y2 + z2 − c2t2 is independent of your choice of inertial reference frame. Notice that this invariant quantity involves both space and time, but time is treated differently because of the minus sign in front of the term c2t2. Also notice that for this quantity to be independent of inertial frame, changes in reference frame must mix the values of the space and time coordinates. If one reference frame moves at speed v with respect to the other in the x direction, the transformation of coordinates from (t, x, y, z) to (t?, x?, y?, z?) would be x? γx − cβγt, t? = γt − βγx/c, y? = y, z? = z, where β = v/c, c is the velocity of light, and γ = 1/√(1 − β2).

9. Einstein’s equations tell us how to derive the metric gμν from a known distribution of matter and energy: Rμν −½ gμνR = 8πGTμν/c4. Rμν is the Ricci curvature tensor and is related to the metric gμν, Tμν is the stress-energy tensor describing the matter-energy distribution, G is Newton’s constant of gravitation, and c is the speed of light. For example, for matter of mass density at rest,