U and V in the Kruskal Diagram

Thread starter Airsteve0
Start date Oct 18, 2012
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In summary, U and V represent the two variables used to plot the Kruskal Diagram, with U as the vertical axis representing distance from the black hole's center and V as the horizontal axis representing time experienced by an observer. These variables are related by the Schwarzschild coordinates, with the shape of the diagram representing the entire spacetime surrounding a black hole, including the event horizon and singularity. The diagram also illustrates the concept of time dilation, with time appearing to slow down near the event horizon due to intense gravitational pull. Finally, the Kruskal Diagram demonstrates the possibility of a black hole's singularity by showing all paths leading towards the center, representing the point of infinite density.

Oct 18, 2012

Airsteve0

I would just like to verify whether or not the variables u and v in the Kruskal diagram are functions of t. Thanks!

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Oct 18, 2012

Bill_K

Science Advisor

Insights Author

The transformation from Schwarzschild coordinates r, t to Kruskal coordinates u and v is:

u = -4M e^-u/4M
v = 4M e^v/4M

where u = t - r*, v = t + r*, and r* = r + 2M ln|(r/2M) -1|

Oct 18, 2012

Airsteve0

ok, that clears things up, thanks!

1. What do U and V represent in the Kruskal Diagram?

U and V represent the two variables used to plot the Kruskal Diagram. U is the vertical axis and represents the distance from a black hole's center, while V is the horizontal axis and represents the time experienced by an observer at that distance.

2. How are U and V related in the Kruskal Diagram?

U and V are related by the Schwarzschild coordinates, which are used to describe the curvature of spacetime around a non-rotating black hole. These coordinates are related by the equation V = ±c√(1 - 2M/U), where c is the speed of light and M is the mass of the black hole.

3. What does the shape of the Kruskal Diagram represent?

The shape of the Kruskal Diagram represents the entire spacetime surrounding a black hole. It shows the regions of space that are accessible to an observer at a given distance from the black hole, as well as the event horizon and the singularity.

4. How does the Kruskal Diagram illustrate the concept of time dilation?

The Kruskal Diagram illustrates time dilation by showing how time appears to slow down for an observer at a certain distance from the black hole. As the observer approaches the event horizon, time appears to slow down to a stop, indicated by the horizontal line on the diagram. This is due to the intense gravitational pull of the black hole, which warps the fabric of spacetime.

5. How does the Kruskal Diagram demonstrate the possibility of a black hole's singularity?

The Kruskal Diagram demonstrates the possibility of a black hole's singularity by showing that all paths lead towards the center of the diagram, representing the singularity. This implies that once an object crosses the event horizon, it is unable to escape the gravitational pull of the singularity and is ultimately crushed into an infinitely small point with infinite density.