If a traveler with a speed of 99.9999c (extremely close to the speed of light) passes by a stationary object with extreme gravity, both the traveler and the stationary object will experience significant time dilation, but in different ways.
For the traveler: Due to their high velocity relative to the stationary object, the traveler will experience time dilation according to the principles of special relativity. Time dilation is given by the Lorentz factor γ, which depends on the velocity relative to an observer. As the traveler's velocity approaches the speed of light, γ becomes extremely large, resulting in significant time dilation.
From the perspective of the stationary object: From the perspective of the stationary object, the traveler's clock will appear to be running much slower than their own clock. This is because the gravitational field of the stationary object affects the flow of time in its vicinity, causing time dilation. The stronger the gravitational field, the more pronounced the time dilation.
When these two effects—high velocity time dilation and gravitational time dilation—occur simultaneously, the time dilation experienced by the traveler will depend on the specific details of the situation, such as the exact velocity of the traveler and the strength of the gravitational field of the stationary object.
In general, as the traveler moves closer to the stationary object, both effects will amplify each other, resulting in even greater time dilation. However, without specific values for the velocity and the strength of the gravitational field, it is challenging to provide precise quantitative information on the exact magnitude of the time dilation experienced by the traveler or the stationary object in this scenario.