Help with the curvature of space-time meaning and equations.

[steuben]

I'm working to try to firm up some of the technobabble background of some of my KSP fan-fic work. Or at least figure out if it doesn't introduce any inconsistencies within the setting.

Quote

Equipped with a linear FTL drive with a EC-factor of 3 and a CS-T maximum of .10

Which translated into almost English means: it can go 10^3x (I've yet to figure out what x is) times the speed of light as long as the slope of the curvature of space-time is less than 0.10

The last half is still pretty technobabbly. But it generates a couple of questions. How far out from something like the Sun would the curve of space-time be less than 0.10? How do you find out?

I think I know the second answer. I will need to figure out the four gradients of the field equations with respect to each dimension of space time. Assemble those into a matrix and then find the norm, probably the Frobenius norm of that matrix.

Which now brings me to my list of questions:

1. Does my chain of vector calculus, and matrix algebra make sense, or at the least a series of valid operations? That portion of my education is long ago and far away. I know I'm in for a big world of hurt to perform the calculations.

2. Does there exist a example/explanation on how to calculate the gravity field around a body using general relativity?

3. Do I need to approach this using the general relativity? Can I get away with using the much simpler Newtonian Equations? Half of my reflex is that, yes I can as the differences between Newtonian Gravity and GR Gravity only really begin to appear when your playing close to the sun.

4. Have I gone and chucked several different  and unrelated concepts and phrases and chucked them into the idea-puree maker?

 

[Terwin]

On 9/14/2018 at 9:14 AM, steuben said:

The last half is still pretty technobabbly. But it generates a couple of questions. How far out from something like the Sun would the curve of space-time be less than 0.10? How do you find out?

0.10 what?

0.10g?

0.10m/s/s?

Able to reverse the direction of any object travelling at less then 0.10c in no more than 1m?

0.10 Mcguffins per Foo?

 

[Gargamel]

4 hours ago, Terwin said:

0.10 what?

0.10g?

0.10m/s/s?

Able to reverse the direction of any object travelling at less then 0.10c in no more than 1m?

0.10 Mcguffins per Foo?

 

I believe curvature is a value usually expressed without units.    Kinda like the slope of a graph. 

[K^2]

In general, curvature of space-time is a 4-dimensional rank-4 tensor called Riemann Tensor. Which is an exceptionally ugly object. (Takes 4^4 = 256 individual real numbers to describe in a particular coordinate system!)

Fortunately, there are some scalar quantities that encode useful information about curvature. The simplest of these is Ricci Scalar, which is just a full inner product of the Ricci Tensor with itself. Problem is, vacuum is Ricci flat, R = 0. In KSP, value above zero would indicate that you are experiencing aero/lithobraking.

Fortunately, there are good reasons to try and gauge how curved space-time is in places where it is Ricci flat. In particular Kretschmann Scalar tells you how "bent" the space-time is. In general, computing it still requires you to evaluate the full Riemann Tensor, and then do some inner products with itself. For a general space-time configuration, a very complex task. Fortunately, KSP simplifies things a lot. Gravity within a particular sphere of influence is perfectly mapped to Schwarzschild Metric. Getting from there to Kretschmann Scalar is a lot of work, but given how frequently the two concepts creep up in GR, no surprise, it's been done already. And the general solution is K = 12 r_s² / r⁶. Here, r_s = 2 μ / c² is the Schwarzschild Radius, and r is distance from center of the body that produces gravity in this sphere of influence. In other words, altitude + body radius.

So in principle, if you want to limit how fast your FTL system can operate, or whether it can operate at all, based on "curvature of space-time", using Kretschmann Scalar is a decent option. Just keep that sixth power in denominator in mind. The value of K will drop really fast as you move away from the relevant body.

Does this make any physical sense? Some, definitely. Alcubierre is designed to operate under K = 0 (Minkowski, to be precise). And as that value increases, weird things do start happening. Can some variant of warp work with K > 0, and fail past a certain limit? It seems plausible. There are angular momentum conservation problems with FTL in vicinity of a source of gravity, which suggests that the drive will be radiating and losing energy that way. So there will probably be an energy drain as you move near a planet/star, and it will probably increase with K. Although, probably in some very odd fashion.

I wish I could give you more concrete answers on that bit, but this is so very not my field of study, and all the papers I've seen on warp avoid topic of warp near planetary bodies like a plague. No wonder, really, as the math goes from very difficult to absolutely nightmarish. The above is the best I can come up with based on simplified models, using Alcubierre as a base, and trying to account for conservation laws enforced by relevant symmetries.

Hope this helps a little.

[steuben]

Don't worry about it being an area of study. You're closer to it then I am, who is thinking in terms of rubber sheets and bowling balls, and read a Brief History of Time thirty years ago. I have to try an figure out something with enough string, glue, and gaffer tape for the concepts to hold together till the reader gets to the fridge.  

Okay...

If I'm reading it right, and doing the math right. using the Sun at Earth's orbit I get something like 9.3 10^-60, and using Kerbol at Kerbin's orbit 1.3 10^-57.

Either I'm going to have to deal with Ningis, or figure out a renormalization.

[Gargamel]

8 hours ago, steuben said:

read a Brief History of Time thirty years ago.

I too read that 30+ years ago.   I've read his latest, but I wonder how BHT holds up....

8 hours ago, steuben said:

 fridge.  

Nice link.   Thanks.