Black Hole Time Dilation Calculator

Black Hole Time Dilation (Schwarzschild):

\[ t = \frac{t_0}{\sqrt{1 - \frac{3GM}{c^2 r}}} \]

Proper Time (t0):

Gravitational Constant (G):

m³ kg⁻¹ s⁻²

Mass (M):

Speed of Light (c):

m/s

Radius (r):

Dilated Time (t):

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1. What is Black Hole Time Dilation?

Black Hole Time Dilation, based on Schwarzschild metric in general relativity, describes how time passes at different rates in regions of varying gravitational potential. Near a black hole, time appears to slow down from the perspective of a distant observer.

2. How Does the Calculator Work?

The calculator uses the Schwarzschild time dilation formula:

\[ t = \frac{t_0}{\sqrt{1 - \frac{3GM}{c^2 r}}} \]

Where:

\( t \) — Dilated time as measured by distant observer (s)
\( t_0 \) — Proper time measured by local observer (s)
\( G \) — Gravitational constant (6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²)
\( M \) — Mass of the black hole (kg)
\( c \) — Speed of light in vacuum (299,792,458 m/s)
\( r \) — Radial coordinate from center (m)

Explanation: The formula shows how time dilation increases as one approaches the event horizon (where denominator approaches zero).

3. Importance of Time Dilation Calculation

Details: Understanding time dilation is crucial for astrophysics, GPS satellite synchronization, and testing general relativity predictions. It has practical implications for space travel and communication near massive objects.

4. Using the Calculator

Tips: Enter proper time in seconds, gravitational constant, black hole mass in kg, speed of light in m/s, and radius in meters. All values must be positive. The calculator automatically provides commonly used values for G and c.

5. Frequently Asked Questions (FAQ)

Q1: What happens at the event horizon?
A: At the event horizon (r = 2GM/c²), time dilation becomes infinite, meaning time appears to stop for a distant observer.

Q2: Why is there a factor of 3 in the formula?
A: The factor 3 comes from the specific orbital radius (3GM/c²) known as the photon sphere, where light can orbit the black hole.

Q3: How accurate is this formula?
A: This is an exact solution for non-rotating (Schwarzschild) black holes. For rotating black holes, the Kerr metric would be needed.

Q4: Can this be used for other massive objects?
A: Yes, the formula applies to any spherical mass, but the effects are only significant near extremely dense objects like black holes or neutron stars.

Q5: What are the units for the result?
A: The result is in seconds, same as the proper time input. The dilation factor shows how much slower time passes compared to infinity.