A massive quantity of energy.

Before talking about coordinate systems, tensors, and other cool stuff I need to know better, let's have an engineering look at the equations, specially at energy numbers.

I said before that G thing stands for describing curvature through several mathematical tools we'll see later. Let's focus, for the moment, in the T thing.

The T summarizes something called stress-energy tensor and can be descomposed as:

 

Where 'T' with the tiny greek letters is the tensor per se and the fraction is a constant. An important constant:

• G stands for the Newtonian cosmological constant. This number is in the order of 10^-11 in SI units.
• The other is actually the speed of light to the 4^th power. The order of this number is 10²¹ in SI units.

A quick glance at the units would tell us in order to counteract a 10^-32 factor, energy applied should be huge. But this could be a wrong assumption: I don't know yet what order of magnitude a typical spacetime curvature is.

Before screaming in horror, let's have a closer look to the units I must use.

Units in spacetime: not the usual way of measuring things.

This concept deserves its own post, but my only information source is "Gravitation" for now, so better I wait until having a more global point of view. By now, a quick summary:

• In spacetime, it seems a good idea using geometrized units: this means in practice, everything is measured as lenghts. "Gravitation" use cm -1/100^th part of a meter- as basic unit.
• Mass is the lenght of the 4-momentum vector. Converting this lenght to usual mass (kg), requires of dividing the lenght by the factor G/c²
• Time is technically a lenght too. In special relativity light speed is considered constant -in vaccum- so I can always define time as the lenght a light ray travels. The more time the travel represents, the more lenght light represents.

Now, you have to believe -you can check Gravitation's Box 1.5- that  near the surface of Earth:

This is the term of curvature tensor which gives relative acceleration between 2 test particles -given in cm- separated 1 cm in z direction per cm of light travel, squared. 

The goal of this post is actually obtaining an order of magnitude of energy (Joules). In order to do so:

• I assume sumations of different terms in curvature tensor, give us the same order of magnitude, so I just take into account the number in the previous formula as order reference. 
• Using the geometrized unit system, I can say:
  
• I convert to ergs the previous number knowing G/c⁴=0.826·10^-49 cm/erg
• I convert (again) to Joules the previous number knowing 1 erg=10^-7 J

This estimation yields an order about... 10³⁰ MJ.  
That means in order to obtain a spacetime curvature equal to that generated by Earth, between two points 1 cm away in z direction, I must be able to generate that level of energy -topological and practical issues excluded.

Comparing this estimation with some numbers of energy "existent" in Nature...

• Lightning strike: order about 100 MJ
• Annual USA electricity consumption (2009): order about 10¹³ MJ
• Annual global energy use (2008): order about 10¹⁴ MJ

... Make the amount of energy needed to curve spacetime huge. Huge. HUGE!!!

Some unconcluded conclusions...

So, time to scream in horror. A carefully calculation maybe is able to retrieve a couple of orders in that exponent. An error in my calculations -which is more than likely- a couple more. But the fact stands:
if you wanted to warp spacetime to produce, at least, an acceleration similar to Earth acceleration in a free fall body, you would have to gather the energy the world consumes in a whole year...
... 2 times.

Don't feel discouraged!! You already knew it was impossible!

There's an important issue in this calculation, anyway. Prior to working on this blog, I used to do this estimation with the famous equation...

 ... Which leads to considerate 5 orders of magnitude more -even asumming momentum zero- in energy needed: that's... 10³⁵ MJ .
You can check for yourself by inserting Earth mass in the m in the equation. It's actually a much more fast calculation than previous one. Anyway, they differ in 5 orders of magnitude. How is that possible?

This could be explained by the fact "Gravitation" number I have used in the first estimation considers the curvature between 2 free particles separated 1 cm -close to Earth's surface-, and it's possible curvature between 2 free particles away more distance be higher.

But that idea doesn't convince me neither. I must learn more about that G thing in order to refine this estimation...