The New Field Equations

According to the theory of general relativity, mass causes spacetime to curve, and spacetime tells objects how to move. If you look at the equation below, it should be obvious why this is the case.

Well ... OK ... it is not obvious. Perhaps we can derive some field equations that are equivalent but more intuitive? Let's start with diagrams A and B below (r=radius; ct=spacetime portion of the radius; ct'=warped spacetime):

Diagram A shows an arbitrary sphere of space with no mass present. The field lines are straight (or flat) and connect the center with the outer edge. The field lines in diagram B curl like waves and pull all the space inward toward the center, shortening the spacetime wavelengths and creating a smaller sphere with more curvature. (To see more details on how this works, click here.)

The Lorentz equation above is pretty straight forward (G=Newton's constant). It shows how ct' is a function of mass (m). Add mass (m) and ct' shortens. We can use this equation as a model for our new field equations. Let's see what we can come up with:

The last equation above is kind of interesting. All the stuff on the left side must equal one. Let's multiply both sides by 8(pi)r, (the derivative of a sphere area) and do a few more steps (E=energy; T=energy density):

Add some Tensor indices:

We now have something equivalent to Einstein's field equations. Notice how each term contains an 8pi factor. We can do away with it.

What was not obvious before is now more obvious. If the stress-energy tensor (Tuv) changes, the spacetime variable (ct') also changes. Any particle in the vicinity will be affected by the changing ct', and move along a geodesic curve.

Notice there are a couple of 1/r^2's that can be factored. Time to dress this puppy up a little bit more:

We've come full circle. We now have a tensor version of the Lorentz factor we started with. To change things up a bit more, let's bring ct inside the parentheses.

Finally, we can name the ct and ct' tensors S and S', respectively:

Debunking D-branes and other Extra-dimension Myths

According to string theory, d-branes come in one or more dimensions. They provide an anchor for strings. Some string theories have only even-numbered d-branes; others have odd numbered d-branes. As we shall see later in this post, the odd d-branes, or systems with an odd number of space dimensions, have a better shot at being real. One real example is space in our universe. We perceive it as 3D, and it can be thought of as a giant 3-brane.

But what about 4-branes, 5-branes and beyond? Today we are going to put various multi-dimensional branes to a rigorous test. The test is designed to show whether extra dimensions truly exist. It is the cross-product test. The diagram below shows how the cross product works in three dimensions:

The results are in red. When we calculate the cross-product of two dimensions, we get a third dimension that is perpendicular to the others. To test for extra dimensions, we need to define what we mean when we say "dimension." Dimensions are lines in space that are perpendicular to each other.

Because they are perpendicular to each other, their cross-products must yield a kind of symmetry, i.e., there must be the same number of each dimension. The indexes can also be added or subtracted to get the cross-product. For example, in 3D space, D3D1 = D2 (3-1=2) and D1D2 = D3 (1+2=3) However, there must not be any index sharing. For example, D4D8 = D4 is invalid, since the D4 on the left side of the equation is clearly the same dimension as the D4 on the right. We could change the index(s) to cover our tracks, but then we lose the index symmetry--and such a loss reveals the problem in a different way.

We also need at least three dimensions to get started. Applying the test to 1-branes and 2-branes yields a bunch of zeros. But at least there is symmetry, so the dimensions in the 2-brane might be perpendicular. The fact there is no index sharing is also a good sign.

As you can see, I put the results in tables. Each row element and each column element yield a corresponding result in the tables (let your finger be your guide). Now let's look at a 3D system or 3-brane:

You'll notice cross-products that share the same index yield zero as they should. Three dimensions obey the index rule and have perfect symmetry. The bottom table above shows there are two results for each dimension. A 3-brane is perfect; it meets all the requirements, and we'll use it as a yard stick to judge systems and objects with extra dimensions.

Let's check out 4D:

The 4-brane isn't looking good. The results in red violate the index-sharing rule. There is also a lack of symmetry--there is not the same number of each dimension (see right-hand table). The cross-products are also fake. I'll explain what that means later when I'm done laying the groundwork. For now, let's move up to five dimensions:

Ah! Perfect symmetry! Four of each dimension. Could we be living in a 5D universe? A 5-brane sure looks feasible--but then there are those nasty index violations in red. Oh well.

The 5-brane does show, however, that odd numbers of dimensions have symmetry. What is true for five dimensions is also true for seven dimensions and the nine dimensions of E500 string theories:

Once again we have perfect symmetry and index violations in red. What about the 10 space dimensions of M-theory?

Holy d-brane, Batman! What a mess! Ten dimensions lack symmetry and have bleeding-red index violations. It's unlikely these dimensions are perpendicular to each other. But what about the 9D system? That one looks pretty good, not perfect like 3D, but close. Thus it is time to explain why its cross-products are fake as a presidential campaign promise.

If there are really nine dimensions of space then any cross-product between any two dimensions should yield the remaining seven--since the remaining seven are allegedly perpendicular to the two. The above tables only show single results for each cross-product. Such a strategy is useful in that it exposes which cross-products aren't perpendicular (see index violations marked in red). That being said, D1D2 should yield D9 and the rest. D3D6 should yield D8 and the rest. If these extra dimensions were real, there would be more uncertainty due to multiple results. We would not get a nice, single result from any cross-product. If we could choose the ideal d-brane or universe, 3D would be the best choice. The marvelous thing about 3D is you have perfect symmetry and the same number of dot-products as cross-products. With higher dimensions this is never the case.

The diagram above shows how D1D2 equals seven possible results in a 9D system. Such a system has 81 dot-products and 567 cross-products! Not exactly a balanced system. Then again, what if all those extra dimensions are tiny and curled up? That would sweep the multiple results under the rug. However, if dimensions are curled, their angles are no longer perpendicular. The diagram below shows the x-axis being curled to the y-axis. Note how the angle is no longer ninety degrees.

If dimensions don't have to be right angles to each other, then we can have an infinite number of them within right-angled 2D or 3D space.

The above diagram shows how a 2D triangle can be expanded into a 4D triangle that occupies a flat 2D space. But why stop at four dimensions when we can have eight or more?

As you can see, it is easy to fool ourselves into believing there are extra dimensions. What we call extra dimensions could really just be a bunch of posers occupying a flat 2D or 3D space. If there are really four or more dimensions that are right angles to each other, cross-product results would be uncertain. A lack of symmetry implies extra dimensions do not exist.

For more on this topic, check out theses links: "Debunking Bosonic String Theory's 26 Dimensions" and "Are String Theory's Extra Dimensions Real?" and How to Derive M-Theory's Eleven Dimensions and Reduce Them to Four Dimensions.

Einstein's Field Equations Simplified

Start with light speed (c) squared equal to velocity (v) squared:

Divide both sides of the equation by c^2:

We know that g is acceleration due to gravity, and it is equal to Gm/r^2. Remove an r from the denominator and we get a velocity squared:

Now we can make a substitution:

If we take the derivative of the sphere area, we get a line with a magnitude of 8(pi)r:

Multiply both sides by 8(pi)r:

We need to convert the mass (m) to energy (E), so we find mass in terms of energy:

Replace m with E/c^2 and divide both sides of the equation by r^3. Doing so gives us a number over an area or r^2. E is replaced by T (energy density). When T is increased, the r^2 on the left side must decrease. If we think of space as an imaginary sphere with radius (r), the smaller r gets, the greater the curvature. An arbitrary distance around the circumference covers a greater angle.

You might have noticed that when r gets smaller, the energy density grows larger, which in turn causes r to shrink even more. Here's where the field equation predicts a star collapsing into a black hole.

If we don't want to dwell on black holes, we need to multiply the left side by a coefficient (g), so when T increases, g increases. The variable g will also be a measure of curvature.

Convert g into the metric tensor and T into the Energy-stress tensor, and we have the equivalent of the field equations in simplified form:

But why settle for simple when you could have complicated? LOL! What about the Einstein tensor which contains the Ricci tensor and all those lovely Christoffel symbols? I show how to derive those here and here.

But what is the logic behind the left side of the field equations? Let's start with the gravitational energy minus the escape velocity energy. When those two values are equal, spacetime is considered flat and gravity is zero.

As you can see, we don't have to think of gravity in terms of spacetime. We could think of gravity as the difference between two energies. But Einstein insisted upon spacetime, we need to express energy in terms of space, or a number over r^2, so we multiply the left side by 8(pi)G/(C^4)(r^3):

Let's replace the energies with R's, then make tensors out of them and add a cosmological constant (since empty space still has energy). In the final step, those are replaced by Gij--the Einstein tensor.

Update: Below is the Lagrangian (L) derived from the field equations--further clarifying the logic behind the left side of the field equations.

Are the Laws of Physics Unchanged Under Time Reversal?

The CPT Theorem says if you replace matter with anti-matter, reverse charge, parity, and time, you will have a universe that is a mirror image of our own. That seems reasonable on the face of it. But then it goes on to say that the laws of physics will be the same when time is reversed????

Let’s pretend you are bored surfing the internet and you decide to go outside and watch paratroopers jump out of an airplane. You are standing on solid ground. Gravity is holding you firm to the earth. It has an attractive force. You look up. You see a guy jump out of the plane. He falls, his chute opens and you watch him glide to the ground. Gravity works the same way for him as it does for you: it’s an attractive force pulling him to earth.

Now, somebody flips the switch, and time reverses. You are still standing on firm, solid ground. Gravity is still an attractive force for you. But the paratrooper guy? Well he’s now falling up! Gravity is a repulsive force for him. Gravity is no longer consistent; it has changed!

You suddenly wake up and realize you were just having a bad dream. You are in your bedroom, time is moving forward and all is well with the laws of physics. You decide to play with your magnet. There are some iron filings stuck to it and there is another iron filing flying toward it. Electromagnetism seems to work as you expect. Then someone flips the switch. Time reverses. The iron filings that were stuck to the magnet stay stuck. So far, so good--but the iron filing that was flying toward your magnet is now flying away! Electromagnetism is messed up! You can’t cure it by waking up this time. The only cure is forward time.

Conclusion: It is no accident the arrow of time goes forward. The laws of physics depend on it.