Lifelong person Re-IDentification (L-ReID) exploits sequentially collected data to continuously train and update a ReID model, focusing on the overall performance of all data. Its main challenge is to avoid the catastrophic forgetting problem of old knowledge while training on new data. Existing L-ReID methods typically re-extract new features for all historical gallery images for inference after each update, known as "re-indexing". However, historical gallery data typically suffers from direct saving due to the data privacy issue and the high re-indexing costs for large-scale gallery images. As a result, it inevitably leads to incompatible retrieval between query features extracted by the updated model and gallery features extracted by those before the update, greatly impairing the re-identification performance. To tackle the above issue, this paper focuses on a new task called Re-index Free Lifelong person Re-IDentification (RFL-ReID), which requires performing lifelong person re-identification without re-indexing historical gallery images. Therefore, RFL-ReID is more challenging than L-ReID, requiring continuous learning and balancing new and old knowledge in diverse streaming data, and making the features output by the new and old models compatible with each other. To this end, we propose a Bidirectional Continuous Compatible Representation (Bi-C2R) framework to continuously update the gallery features extracted by the old model to perform efficient L-ReID in a compatible manner. We verify our proposed Bi-C2R method through theoretical analysis and extensive experiments on multiple benchmarks, which demonstrate that the proposed method can achieve leading performance on both the introduced RFL-ReID task and the traditional L-ReID task.